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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linear programming</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For the retronym referring to television broadcasting, see <a href="Broadcast_programming" title="Broadcast programming">Broadcast programming</a>.</div>

<p><b>Linear programming</b> (<b>LP</b>), also called <b>linear optimization</b>, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a <a href="Mathematical_model" title="Mathematical model">mathematical model</a> whose requirements and objective are represented by <a href="Linear_function#As_a_polynomial_function" title="Linear function">linear relationships</a>. Linear programming is a special case of mathematical programming (also known as <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>).

</p><p>More formally, linear programming is a technique for the <a href="Mathematical_optimization" title="Mathematical optimization">optimization</a> of a <a href="Linear" class="mw-redirect" title="Linear">linear</a> <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a>, subject to <a href="Linear_equality" class="mw-redirect" title="Linear equality">linear equality</a> and <a href="Linear_inequality" title="Linear inequality">linear inequality</a> <a href="Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a>. Its <a href="Feasible_region" title="Feasible region">feasible region</a> is a <a href="Convex_polytope" title="Convex polytope">convex polytope</a>, which is a set defined as the <a href="Intersection_(mathematics)" class="mw-redirect" title="Intersection (mathematics)">intersection</a> of finitely many <a href="Half-space_(geometry)" title="Half-space (geometry)">half spaces</a>, each of which is defined by a linear inequality. Its objective function is a <a href="Real_number" title="Real number">real</a>-valued <a href="Affine_function" class="mw-redirect" title="Affine function">affine (linear) function</a> defined on this polytope. A linear programming <a href="Algorithm" title="Algorithm">algorithm</a> finds a point in the <a href="Polytope" title="Polytope">polytope</a> where this function has the largest (or smallest) value if such a point exists.

</p><p>Linear programs are problems that can be expressed in <a href="Canonical_form" title="Canonical form">standard form</a> as:

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<mtext>Find a vector</mtext>

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<mtext>that maximizes</mtext>

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<mtext>subject to</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\text{Find a vector}}&&\mathbf {x} \\&{\text{that maximizes}}&&\mathbf {c} ^{\mathsf {T}}\mathbf {x} \\&{\text{subject to}}&&A\mathbf {x} \leq \mathbf {b} \\&{\text{and}}&&\mathbf {x} \geq \mathbf {0} .\end{aligned}}}</annotation>

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</math></span><img src="./2cc3fcf5b820bf2341c5f8d69bb086bc9a7ffc75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:28.516ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}&{\text{Find a vector}}&&\mathbf {x} \\&{\text{that maximizes}}&&\mathbf {c} ^{\mathsf {T}}\mathbf {x} \\&{\text{subject to}}&&A\mathbf {x} \leq \mathbf {b} \\&{\text{and}}&&\mathbf {x} \geq \mathbf {0} .\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Here the components of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> are the variables to be determined, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} }</annotation>

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</math></span><img src="./8798d172f59e21f2ce193a3118d4063d19353ded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {c} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \mapsto \mathbf {c} ^{\mathsf {T}}\mathbf {x} }</annotation>

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</math></span><img src="./3cf441d5722c9264b7cef14c7dddbaeb9763a592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} \mapsto \mathbf {c} ^{\mathsf {T}}\mathbf {x} }" loading="lazy"></span> in this case) is called the <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a>. The constraints <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} \leq \mathbf {b} }">

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</p><p>Linear programming can be applied to various fields of study. It is widely used in mathematics and, to a lesser extent, in business, <a href="Economics" title="Economics">economics</a>, and some engineering problems. There is a close connection between linear programs, eigenequations, <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a>'s general equilibrium model, and structural equilibrium models (see <a href="Dual_linear_program" title="Dual linear program">dual linear program</a> for details).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proven useful in modeling diverse types of problems in <a href="Automated_planning_and_scheduling" title="Automated planning and scheduling">planning</a>, <a href="Routing" title="Routing">routing</a>, <a href="Scheduling_(production_processes)" title="Scheduling (production processes)">scheduling</a>, <a href="Assignment_problem" title="Assignment problem">assignment</a>, and design.

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The problem of solving a system of linear inequalities dates back at least as far as <a href="Joseph_Fourier" title="Joseph Fourier">Fourier</a>, who in 1827 published a method for solving them,<sup id="cite_ref-SierksmaZwols2015_4-0" class="reference"><a href="#cite_note-SierksmaZwols2015-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and after whom the method of <a href="Fourier%E2%80%93Motzkin_elimination" title="Fourier–Motzkin elimination">Fourier–Motzkin elimination</a> is named.

</p><p>In the late 1930s, Soviet mathematician <a href="Leonid_Kantorovich" title="Leonid Kantorovich">Leonid Kantorovich</a> and American economist <a href="Wassily_Leontief" title="Wassily Leontief">Wassily Leontief</a> independently delved into the practical applications of linear programming. Kantorovich focused on manufacturing schedules, while Leontief explored economic applications. Their groundbreaking work was largely overlooked for decades.

</p><p>The turning point came during World War II when linear programming emerged as a vital tool. It found extensive use in addressing complex wartime challenges, including transportation logistics, scheduling, and resource allocation. Linear programming proved invaluable in optimizing these processes while considering critical constraints such as costs and resource availability.

</p><p>Despite its initial obscurity, the wartime successes propelled linear programming into the spotlight. Post-WWII, the method gained widespread recognition and became a cornerstone in various fields, from operations research to economics. The overlooked contributions of Kantorovich and Leontief in the late 1930s eventually became foundational to the broader acceptance and utilization of linear programming in optimizing decision-making processes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>Kantorovich's work was initially neglected in the <a href="USSR" class="mw-redirect" title="USSR">USSR</a>.<sup id="cite_ref-dantzig1982_6-0" class="reference"><a href="#cite_note-dantzig1982-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> About the same time as Kantorovich, the Dutch-American economist <a href="Tjalling_Koopmans" title="Tjalling Koopmans">T. C. Koopmans</a> formulated classical economic problems as linear programs. Kantorovich and Koopmans later shared the 1975 <a href="Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a>.<sup id="cite_ref-SierksmaZwols2015_4-1" class="reference"><a href="#cite_note-SierksmaZwols2015-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In 1941, <a href="Frank_Lauren_Hitchcock" title="Frank Lauren Hitchcock">Frank Lauren Hitchcock</a> also formulated transportation problems as linear programs and gave a solution very similar to the later <a href="Simplex_method" class="mw-redirect" title="Simplex method">simplex method</a>.<sup id="cite_ref-Schrijver1998_7-0" class="reference"><a href="#cite_note-Schrijver1998-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Hitchcock had died in 1957, and the Nobel Memorial Prize is not awarded posthumously.

</p><p>From 1946 to 1947 <a href="George_Dantzig" title="George Dantzig">George B. Dantzig</a> independently developed general linear programming formulation to use for planning problems in the US Air Force.<sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In 1947, Dantzig also invented the <a href="Simplex_algorithm" title="Simplex algorithm">simplex method</a> that, for the first time efficiently, tackled the linear programming problem in most cases.<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> When Dantzig arranged a meeting with <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> to discuss his simplex method, von Neumann immediately conjectured the theory of <a href="#Duality">duality</a> by realizing that the problem he had been working in <a href="Game_theory" title="Game theory">game theory</a> was equivalent.<sup id="cite_ref-:0_8-2" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January 5, 1948.<sup id="cite_ref-dantzig1982_6-1" class="reference"><a href="#cite_note-dantzig1982-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Dantzig's work was made available to public in 1951. In the post-war years, many industries applied it in their daily planning.

</p><p>Dantzig's original example was to find the best assignment of 70 people to 70 jobs. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the <a href="Abundance_of_the_chemical_elements" title="Abundance of the chemical elements">number of particles</a> in the <a href="Observable_universe" title="Observable universe">observable universe</a>. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>. The theory behind linear programming drastically reduces the number of possible solutions that must be checked.

</p><p>The linear programming problem was first shown to be solvable in polynomial time by <a href="Leonid_Khachiyan" title="Leonid Khachiyan">Leonid Khachiyan</a> in 1979,<sup id="cite_ref-khachiyan79_9-0" class="reference"><a href="#cite_note-khachiyan79-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> but a larger theoretical and practical breakthrough in the field came in 1984 when <a href="Narendra_Karmarkar" title="Narendra Karmarkar">Narendra Karmarkar</a> introduced a new <a href="Interior-point_method" title="Interior-point method">interior-point method</a> for solving linear-programming problems.<sup id="cite_ref-karmarkar84_10-0" class="reference"><a href="#cite_note-karmarkar84-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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<p>Linear programming is a widely used field of optimization for several reasons. Many practical problems in <a href="Operations_research" title="Operations research">operations research</a> can be expressed as linear programming problems.<sup id="cite_ref-dantzig1982_6-2" class="reference"><a href="#cite_note-dantzig1982-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Certain special cases of linear programming, such as <i><a href="Network_flow_problem" title="Network flow problem">network flow</a></i> problems and <a href="Multi-commodity_flow_problem" title="Multi-commodity flow problem"><i>multicommodity flow</i> problems</a>, are considered important enough to have much research on specialized algorithms. A number of algorithms for other types of optimization problems work by solving linear programming problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as <i>duality,</i> <i>decomposition,</i> and the importance of <i>convexity</i> and its generalizations. Likewise, linear programming was heavily used in the early formation of <a href="Microeconomics" title="Microeconomics">microeconomics</a>, and it is currently utilized in company management, such as planning, production, transportation, and technology. Although the modern management issues are ever-changing, most companies would like to <a href="Profit_maximization" title="Profit maximization">maximize profits</a> and minimize costs with limited resources. Google also uses linear programming to stabilize YouTube videos.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Standard_form">Standard form</h2></div>

<p><i>Standard form</i> is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts:

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<ul><li>A <b>linear (or affine) function to be maximized</b></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle f(x_{1},x_{2})=c_{1}x_{1}+c_{2}x_{2}}</annotation>

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<ul><li><b>Problem constraints</b> of the following form</li></ul>

<dl><dd>e.g.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&\leq b_{1}\\a_{21}x_{1}+a_{22}x_{2}&\leq b_{2}\\a_{31}x_{1}+a_{32}x_{2}&\leq b_{3}\\\end{matrix}}}</annotation>

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</math></span><img src="./28772e37b1aeb348f1a34b87d3c8d4a97d841d8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.399ex; height:9.176ex;" alt="{\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}&\leq b_{1}\\a_{21}x_{1}+a_{22}x_{2}&\leq b_{2}\\a_{31}x_{1}+a_{32}x_{2}&\leq b_{3}\\\end{matrix}}}" loading="lazy"></span></dd></dl></dd></dl>

<ul><li><b>Non-negative variables</b></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x_{1}\geq 0\\x_{2}\geq 0\end{matrix}}}</annotation>

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</math></span><img src="./4de5f0757bee798b2cab81a6f605f047db5fd94d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.396ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}x_{1}\geq 0\\x_{2}\geq 0\end{matrix}}}" loading="lazy"></span></dd></dl></dd></dl>

<p>The problem is usually expressed in <i><a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> form</i>, and then becomes:

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<annotation encoding="application/x-tex">{\displaystyle \max\{\,\mathbf {c} ^{\mathsf {T}}\mathbf {x} \mid \mathbf {x} \in \mathbb {R} ^{n}\land A\mathbf {x} \leq \mathbf {b} \land \mathbf {x} \geq 0\,\}}</annotation>

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</math></span><img src="./3b12fffe269bf36fdec149de2f34cba54154777f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.036ex; height:3.176ex;" alt="{\displaystyle \max\{\,\mathbf {c} ^{\mathsf {T}}\mathbf {x} \mid \mathbf {x} \in \mathbb {R} ^{n}\land A\mathbf {x} \leq \mathbf {b} \land \mathbf {x} \geq 0\,\}}" loading="lazy"></span></dd></dl>

<p>Other forms, such as minimization problems, problems with constraints on alternative forms, and problems involving negative <a href="Variable_(programming)" class="mw-redirect" title="Variable (programming)">variables</a> can always be rewritten into an equivalent problem in standard form.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

<p>Suppose that a farmer has a piece of farm land, say <i>L</i> <a href="Hectare" title="Hectare">hectares</a>, to be planted with either wheat or barley or some combination of the two. The farmer has <i>F</i> kilograms of fertilizer and <i>P</i> kilograms of pesticide. Every hectare of wheat requires <i>F</i><sub>1</sub> kilograms of fertilizer and <i>P</i><sub>1</sub> kilograms of pesticide, while every hectare of barley requires <i>F</i><sub>2</sub> kilograms of fertilizer and <i>P</i><sub>2</sub> kilograms of pesticide. Let S<sub>1</sub> be the selling price of wheat and S<sub>2</sub> be the selling price of barley, per hectare. If we denote the area of land planted with wheat and barley by <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub> respectively, then profit can be maximized by choosing optimal values for <i>x</i><sub>1</sub> and <i>x</i><sub>2</sub>. This problem can be expressed with the following linear programming problem in the standard form:

</p>

<table>

<td valign="top">Maximize:

</td>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S_{1}\cdot x_{1}+S_{2}\cdot x_{2}}</annotation>

</semantics>

</math></span><img src="./7dec056bd019a7cb780aba29201b6e2e68f2b89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.925ex; height:2.509ex;" alt="{\displaystyle S_{1}\cdot x_{1}+S_{2}\cdot x_{2}}" loading="lazy"></span>

</td>

<td>(maximize the revenue (the total wheat sales plus the total barley sales) – revenue is the "objective function")

</td></tr>

<tr>

<td><span class="nowrap">Subject to:</span>

</td>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}\leq L}</annotation>

</semantics>

</math></span><img src="./6d67b19bb362ed9055a186d4480fa279ad37e6a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.29ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}\leq L}" loading="lazy"></span>

</td>

<td>(limit on total area)

</td></tr>

<tr>

<td>

</td>

<msub>

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</msub>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{1}\cdot x_{1}+F_{2}\cdot x_{2}\leq F}</annotation>

</semantics>

</math></span><img src="./4ac3bb61bc71ac8c41a5163964ca10ffc9eef9ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.903ex; height:2.509ex;" alt="{\displaystyle F_{1}\cdot x_{1}+F_{2}\cdot x_{2}\leq F}" loading="lazy"></span>

</td>

<td>(limit on fertilizer)

</td></tr>

<tr>

<td>

</td>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle P_{1}\cdot x_{1}+P_{2}\cdot x_{2}\leq P}</annotation>

</semantics>

</math></span><img src="./da5cc1437a98288e22b1af4f5502ceeeef4b54ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.903ex; height:2.509ex;" alt="{\displaystyle P_{1}\cdot x_{1}+P_{2}\cdot x_{2}\leq P}" loading="lazy"></span>

</td>

<td>(limit on pesticide)

</td></tr>

<tr>

<td>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1}\geq 0,x_{2}\geq 0}</annotation>

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</math></span><img src="./9697ec736eb734f866bf36231f5126358d8d4c87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.324ex; height:2.509ex;" alt="{\displaystyle x_{1}\geq 0,x_{2}\geq 0}" loading="lazy"></span>

</td>

<td>(cannot plant a negative area).

</td></tr></tbody></table>

<p>In matrix form this becomes:

</p>

<dl><dd>maximize <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}S_{1}&S_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}">

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}S_{1}&S_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./9c476520119e4bf358d8ac36d1ebcd06f81fee69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.916ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}S_{1}&S_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}" loading="lazy"></span></dd>

<dd>subject to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&1\\F_{1}&F_{2}\\P_{1}&P_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\leq {\begin{bmatrix}L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\geq {\begin{bmatrix}0\\0\end{bmatrix}}.}">

<mrow>

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&1\\F_{1}&F_{2}\\P_{1}&P_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\leq {\begin{bmatrix}L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\geq {\begin{bmatrix}0\\0\end{bmatrix}}.}</annotation>

</semantics>

</math></span><img src="./6c9b6befef7460da6b47d4c1048e394879c08ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:40.684ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&1\\F_{1}&F_{2}\\P_{1}&P_{2}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\leq {\begin{bmatrix}L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\geq {\begin{bmatrix}0\\0\end{bmatrix}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Augmented_form_(slack_form)">Augmented form (slack form)</h2></div>

<p>Linear programming problems can be converted into an <i>augmented form</i> in order to apply the common form of the <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>. This form introduces non-negative <i><a href="Slack_variable" title="Slack variable">slack variables</a></i> to replace inequalities with equalities in the constraints. The problems can then be written in the following <a href="Block_matrix" title="Block matrix">block matrix</a> form:

</p>

<dl><dd>Maximize <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>:</dd>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&-\mathbf {c} ^{\mathsf {T}}&0\\0&\mathbf {A} &\mathbf {I} \end{bmatrix}}{\begin{bmatrix}z\\\mathbf {x} \\\mathbf {s} \end{bmatrix}}={\begin{bmatrix}0\\\mathbf {b} \end{bmatrix}}}</annotation>

</semantics>

</math></span><img src="./4f7faea608d12cd6df93bb7a817328d4a5ec8eed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.577ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&-\mathbf {c} ^{\mathsf {T}}&0\\0&\mathbf {A} &\mathbf {I} \end{bmatrix}}{\begin{bmatrix}z\\\mathbf {x} \\\mathbf {s} \end{bmatrix}}={\begin{bmatrix}0\\\mathbf {b} \end{bmatrix}}}" loading="lazy"></span></dd>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \geq 0,\mathbf {s} \geq 0}</annotation>

</semantics>

</math></span><img src="./27bae5d01bc43b5013d8c2e87e3584b222a4cc09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.022ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} \geq 0,\mathbf {s} \geq 0}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {s} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {s} }</annotation>

</semantics>

</math></span><img src="./644ae690160e658898a141e568a7fb0ee6040004.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.056ex; height:1.676ex;" alt="{\displaystyle \mathbf {s} }" loading="lazy"></span> are the newly introduced slack variables, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> are the decision variables, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span> is the variable to be maximized.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3></div>

<p>The example above is converted into the following augmented form:

</p>

<td colspan="2">Maximize: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}\cdot x_{1}+S_{2}\cdot x_{2}}">

<msub>

</mrow>

</msub>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle S_{1}\cdot x_{1}+S_{2}\cdot x_{2}}</annotation>

</semantics>

</math></span><img src="./7dec056bd019a7cb780aba29201b6e2e68f2b89c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.925ex; height:2.509ex;" alt="{\displaystyle S_{1}\cdot x_{1}+S_{2}\cdot x_{2}}" loading="lazy"></span>

</td>

<td>(objective function)

</td></tr>

<tr>

<td>subject to:

</td>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}+x_{3}=L}</annotation>

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</math></span><img src="./a9e39fd387ea3ca20b6379cc678794c067146836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.514ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}+x_{3}=L}" loading="lazy"></span>

</td>

<td>(augmented constraint)

</td></tr>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle F_{1}\cdot x_{1}+F_{2}\cdot x_{2}+x_{4}=F}</annotation>

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</math></span><img src="./29ba9b42deaad7b209eea515ae863b278e0d7237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.128ex; height:2.509ex;" alt="{\displaystyle F_{1}\cdot x_{1}+F_{2}\cdot x_{2}+x_{4}=F}" loading="lazy"></span>

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<td>(augmented constraint)

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<annotation encoding="application/x-tex">{\displaystyle P_{1}\cdot x_{1}+P_{2}\cdot x_{2}+x_{5}=P}</annotation>

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</math></span><img src="./6acebda76368cc11ad2dae7c2fbaae2c3d993bf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.128ex; height:2.509ex;" alt="{\displaystyle P_{1}\cdot x_{1}+P_{2}\cdot x_{2}+x_{5}=P}" loading="lazy"></span>

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<td>(augmented constraint)

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<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0.}</annotation>

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</math></span><img src="./3ab08e7d2cd117f25b841a243cba2764e91b231e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.963ex; height:2.509ex;" alt="{\displaystyle x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0.}" loading="lazy"></span>

</td></tr></tbody></table></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3},x_{4},x_{5}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{3},x_{4},x_{5}}</annotation>

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</math></span><img src="./e7038fea54005cc397744a38f3787349cbe6cf49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.22ex; height:2.009ex;" alt="{\displaystyle x_{3},x_{4},x_{5}}" loading="lazy"></span> are (non-negative) slack variables, representing in this example the unused area, the amount of unused fertilizer, and the amount of unused pesticide.

</p><p>In matrix form this becomes:

</p>

<dl><dd>Maximize <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math></span><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"></span>:</dd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&-S_{1}&-S_{2}&0&0&0\\0&1&1&1&0&0\\0&F_{1}&F_{2}&0&1&0\\0&P_{1}&P_{2}&0&0&1\\\end{bmatrix}}{\begin{bmatrix}z\\x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}}={\begin{bmatrix}0\\L\\F\\P\end{bmatrix}},\,{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}}\geq 0.}</annotation>

</semantics>

</math></span></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Duality">Duality</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dual_linear_program" title="Dual linear program">Dual linear program</a></div>

<p>Every linear programming problem, referred to as a <i>primal</i> problem, can be converted into a <a href="Dual_problem" class="mw-redirect" title="Dual problem">dual problem</a>, which provides an upper bound to the optimal value of the primal problem. In matrix form, we can express the <i>primal</i> problem as:

</p>

<dl><dd>Maximize <b>c</b><sup>T</sup><b>x</b> subject to <i>A</i><b>x</b> ≤ <b>b</b>, <b>x</b> ≥ 0;

<dl><dd>with the corresponding <b>symmetric</b> dual problem,</dd></dl></dd>

<dd>Minimize <b>b</b><sup>T</sup><b>y</b> subject to <i>A</i><sup>T</sup><b>y</b> ≥ <b>c</b>, <b>y</b> ≥ 0.</dd></dl>

<p>An alternative primal formulation is:

</p>

<dl><dd>Maximize <b>c</b><sup>T</sup><b>x</b> subject to <i>A</i><b>x</b> ≤ <b>b</b>;

<dl><dd>with the corresponding <b>asymmetric</b> dual problem,</dd></dl></dd>

<dd>Minimize <b>b</b><sup>T</sup><b>y</b> subject to <i>A</i><sup>T</sup><b>y</b> = <b>c</b>, <b>y</b> ≥ 0.</dd></dl>

<p>There are two ideas fundamental to duality theory. One is the fact that (for the symmetric dual) the dual of a dual linear program is the original primal linear program. Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. The <a href="Weak_duality" title="Weak duality">weak duality</a> theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution. The <a href="Strong_duality" title="Strong duality">strong duality</a> theorem states that if the primal has an optimal solution, <b>x</b><sup>*</sup>, then the dual also has an optimal solution, <b>y</b><sup>*</sup>, and <b>c</b><sup>T</sup><b>x</b><sup>*</sup>=<b>b</b><sup>T</sup><b>y</b><sup>*</sup>.

</p><p>A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible. See <a href="Dual_linear_program" title="Dual linear program">dual linear program</a> for details and several more examples.

</p>

<div class="mw-heading mw-heading2"><h2 id="Variations">Variations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Covering/packing_dualities">Covering/packing dualities</h3></div>

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</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size:130%;"><a class="mw-selflink-fragment" href="#Covering/packing_dualities">Covering/packing-problem pairs</a></th></tr><tr><td class="sidebar-content">

<table style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;display:block;margin-top:0.4em;"><tbody><tr style="vertical-align:top"><td style="font-weight:bold;background:#ddf;text-align:center;border:1px #fafafa solid;"> <a href="Covering_problems" title="Covering problems">Covering problems</a></td><td style="font-weight:bold;background:#ddf;text-align:center;border:1px #fafafa solid;"> <a href="Packing_problems" title="Packing problems">Packing problems</a></td></tr><tr style="vertical-align:top"><td style="padding-top:0.15em;"> <a href="Set_cover_problem" title="Set cover problem">Minimum set cover</a></td><td style="padding-top:0.15em;"> <a href="Set_packing" title="Set packing">Maximum set packing</a></td></tr><tr style="vertical-align:top"><td> <a href="Edge_cover" title="Edge cover">Minimum edge cover</a></td><td> <a href="Matching_(graph_theory)" title="Matching (graph theory)">Maximum matching</a></td></tr><tr style="vertical-align:top"><td> <a href="Vertex_cover" title="Vertex cover">Minimum vertex cover</a></td><td> <a href="Independent_set_(graph_theory)" title="Independent set (graph theory)">Maximum independent set</a></td></tr><tr style="vertical-align:top"><td> <a href="Bin_covering_problem" title="Bin covering problem">Bin covering</a></td><td> <a href="Bin_packing_problem" title="Bin packing problem">Bin packing</a></td></tr><tr style="vertical-align:top"><td> <a href="Polygon_covering" title="Polygon covering">Polygon covering</a></td><td> <a href="Rectangle_packing" title="Rectangle packing">Rectangle packing</a></td></tr></tbody></table></td>

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<p>A <a href="Covering_problem" class="mw-redirect" title="Covering problem">covering LP</a> is a linear program of the form:

</p>

<dl><dd>Minimize: <big><b>b</b><sup>T</sup><b>y</b></big>,</dd>

<dd>subject to: <big><i>A</i><sup>T</sup><b>y</b> ≥ <b>c</b>, <b>y</b> ≥ 0</big>,</dd></dl>

<p>such that the matrix <i>A</i> and the vectors <b>b</b> and <b>c</b> are non-negative.

</p><p>The dual of a covering LP is a <a href="Packing_problem" class="mw-redirect" title="Packing problem">packing LP</a>, a linear program of the form:

</p>

<dl><dd>Maximize: <big><b>c</b><sup>T</sup><b>x</b></big>,</dd>

<dd>subject to: <big><i>A</i><b>x</b> ≤ <b>b</b>, <b>x</b> ≥ 0</big>,</dd></dl>

<p>such that the matrix <i>A</i> and the vectors <b>b</b> and <b>c</b> are non-negative.

</p>

<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>

<p>Covering and packing LPs commonly arise as a <a href="Linear_programming_relaxation" title="Linear programming relaxation">linear programming relaxation</a> of a combinatorial problem and are important in the study of <a href="Approximation_algorithms" class="mw-redirect" title="Approximation algorithms">approximation algorithms</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> For example, the LP relaxations of the <a href="Set_packing" title="Set packing">set packing problem</a>, the <a href="Independent_set_problem" class="mw-redirect" title="Independent set problem">independent set problem</a>, and the <a href="Matching_(graph_theory)" title="Matching (graph theory)">matching problem</a> are packing LPs. The LP relaxations of the <a href="Set_cover_problem" title="Set cover problem">set cover problem</a>, the <a href="Vertex_cover_problem" class="mw-redirect" title="Vertex cover problem">vertex cover problem</a>, and the <a href="Dominating_set_problem" class="mw-redirect" title="Dominating set problem">dominating set problem</a> are also covering LPs.

</p><p>Finding a <a href="Fractional_coloring" title="Fractional coloring">fractional coloring</a> of a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> is another example of a covering LP. In this case, there is one constraint for each vertex of the graph and one variable for each <a href="Independent_set_(graph_theory)" title="Independent set (graph theory)">independent set</a> of the graph.

</p>

<div class="mw-heading mw-heading2"><h2 id="Complementary_slackness">Complementary slackness</h2></div>

<p>It is possible to obtain an optimal solution to the dual when only an optimal solution to the primal is known using the complementary slackness theorem. The theorem states:

</p><p>Suppose that <b>x</b> = (<b>x</b><sub>1</sub>, <b>x</b><sub>2</sub>, ... , <b>x</b><sub><i>n</i></sub>) is primal feasible and that <b>y</b> = (<b>y</b><sub>1</sub>, <b>y</b><sub>2</sub>, ... , <b>y</b><sub><i>m</i></sub>) is dual feasible. Let (<b>w</b><sub>1</sub>, <b>w</b><sub>2</sub>, ..., <b>w</b><sub><i>m</i></sub>) denote the corresponding primal slack variables, and let (<b>z</b><sub>1</sub>, <b>z</b><sub>2</sub>, ... , <b>z</b><sub><i>n</i></sub>) denote the corresponding dual slack variables. Then <b>x</b> and <b>y</b> are optimal for their respective problems if and only if

</p>

<p>So if the <i>i</i>-th slack variable of the primal is not zero, then the <i>i</i>-th variable of the dual is equal to zero. Likewise, if the <i>j</i>-th slack variable of the dual is not zero, then the <i>j</i>-th variable of the primal is equal to zero.

</p><p>This necessary condition for optimality conveys a fairly simple economic principle. In standard form (when maximizing), if there is slack in a constrained primal resource (i.e., there are "leftovers"), then additional quantities of that resource must have no value. Likewise, if there is slack in the dual (shadow) price non-negativity constraint requirement, i.e., the price is not zero, then there must be scarce supplies (no "leftovers").

</p>

<div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Existence_of_optimal_solutions">Existence of optimal solutions</h3></div>

<p>Geometrically, the linear constraints define the <a href="Feasible_region" title="Feasible region">feasible region</a>, which is a <a href="Convex_polytope" title="Convex polytope">convex polytope</a>. A <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear function</a> is a <a href="Convex_function" title="Convex function">convex function</a>, which implies that every <a href="Local_minimum" class="mw-redirect" title="Local minimum">local minimum</a> is a <a href="Global_minimum" class="mw-redirect" title="Global minimum">global minimum</a>; similarly, a linear function is a <a href="Concave_function" title="Concave function">concave function</a>, which implies that every <a href="Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a> is a <a href="Global_maximum" class="mw-redirect" title="Global maximum">global maximum</a>.

</p><p>An optimal solution need not exist, for two reasons. First, if the constraints are inconsistent, then no feasible solution exists: For instance, the constraints <b>x</b> ≥ 2 and <b>x</b> ≤ 1 cannot be satisfied jointly; in this case, we say that the LP is <i>infeasible</i>. Second, when the <a href="Polytope" title="Polytope">polytope</a> is unbounded in the direction of the gradient of the objective function (where the gradient of the objective function is the vector of the coefficients of the objective function), then no optimal value is attained because it is always possible to do better than any finite value of the objective function.

</p>

<div class="mw-heading mw-heading3"><h3 id="Optimal_vertices_(and_rays)_of_polyhedra">Optimal vertices (and rays) of polyhedra</h3></div>

<p>Otherwise, if a feasible solution exists and if the constraint set is bounded, then the optimum value is always attained on the boundary of the constraint set, by the <i><a href="Maximum_principle" title="Maximum principle">maximum principle</a></i> for <i><a href="Convex_function" title="Convex function">convex functions</a></i> (alternatively, by the <i>minimum</i> principle for <i><a href="Concave_function" title="Concave function">concave functions</a></i>) since linear functions are both convex and concave. However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere). For this feasibility problem with the zero-function for its objective-function, if there are two distinct solutions, then every convex combination of the solutions is a solution.

</p><p>The vertices of the polytope are also called <i>basic feasible solutions</i>. The reason for this choice of name is as follows. Let <i>d</i> denote the number of variables. Then the fundamental theorem of linear inequalities implies (for feasible problems) that for every vertex <b>x</b><sup>*</sup> of the LP feasible region, there exists a set of <i>d</i> (or fewer) inequality constraints from the LP such that, when we treat those <i>d</i> constraints as equalities, the unique solution is <b>x</b><sup>*</sup>. Thereby we can study these vertices by means of looking at certain subsets of the set of all constraints (a discrete set), rather than the continuum of LP solutions. This principle underlies the <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a> for solving linear programs.

</p>

<div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="List_of_numerical_analysis_topics#Linear_programming" title="List of numerical analysis topics">List of numerical analysis topics § Linear programming</a></div>

<div class="mw-heading mw-heading3"><h3 id="Basis_exchange_algorithms">Basis exchange algorithms</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Simplex_algorithm_of_Dantzig">Simplex algorithm of Dantzig</h4></div>

<p>The <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>, developed by <a href="George_Dantzig" title="George Dantzig">George Dantzig</a> in 1947, solves LP problems by constructing a feasible solution at a vertex of the <a href="Polytope" title="Polytope">polytope</a> and then walking along a path on the edges of the polytope to vertices with non-decreasing values of the objective function until an optimum is reached for sure. In many practical problems, "<a href="Simplex_algorithm#Degeneracy:_stalling_and_cycling" title="Simplex algorithm">stalling</a>" occurs: many pivots are made with no increase in the objective function.<sup id="cite_ref-DT03_13-0" class="reference"><a href="#cite_note-DT03-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Padberg_14-0" class="reference"><a href="#cite_note-Padberg-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In rare practical problems, the usual versions of the simplex algorithm may actually "cycle".<sup id="cite_ref-Padberg_14-1" class="reference"><a href="#cite_note-Padberg-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> To avoid cycles, researchers developed new pivoting rules.<sup id="cite_ref-FukudaTerlaky_15-0" class="reference"><a href="#cite_note-FukudaTerlaky-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p><p>In practice, the simplex <a href="Algorithm" title="Algorithm">algorithm</a> is quite efficient and can be guaranteed to find the global optimum if certain precautions against <i>cycling</i> are taken. The simplex algorithm has been proved to solve "random" problems efficiently, i.e. in a cubic number of steps,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> which is similar to its behavior on practical problems.<sup id="cite_ref-DT03_13-1" class="reference"><a href="#cite_note-DT03-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Todd_17-0" class="reference"><a href="#cite_note-Todd-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p><p>However, the simplex algorithm has poor worst-case behavior: Klee and Minty constructed a family of linear programming problems for which the simplex method takes a number of steps exponential in the problem size.<sup id="cite_ref-DT03_13-2" class="reference"><a href="#cite_note-DT03-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Murty_18-0" class="reference"><a href="#cite_note-Murty-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PS_19-0" class="reference"><a href="#cite_note-PS-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In fact, for some time it was not known whether the linear programming problem was solvable in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>, i.e. of <a href="P_(complexity)" title="P (complexity)">complexity class P</a>.

</p>

<div class="mw-heading mw-heading4"><h4 id="Criss-cross_algorithm">Criss-cross algorithm</h4></div>

<p>Like the simplex algorithm of Dantzig, the <a href="Criss-cross_algorithm" title="Criss-cross algorithm">criss-cross algorithm</a> is a basis-exchange algorithm that pivots between bases. However, the criss-cross algorithm need not maintain feasibility, but can pivot rather from a feasible basis to an infeasible basis. The criss-cross algorithm does not have <a href="Time_complexity" title="Time complexity">polynomial time-complexity</a> for linear programming. Both algorithms visit all 2<sup><i>D</i></sup> corners of a (perturbed) <a href="Unit_cube" title="Unit cube">cube</a> in dimension <i>D</i>, the <a href="Klee%E2%80%93Minty_cube" title="Klee–Minty cube">Klee–Minty cube</a>, in the <a href="Worst-case_complexity" title="Worst-case complexity">worst case</a>.<sup id="cite_ref-FukudaTerlaky_15-1" class="reference"><a href="#cite_note-FukudaTerlaky-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Roos_20-0" class="reference"><a href="#cite_note-Roos-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Interior_point">Interior point</h3></div>

<p>In contrast to the simplex algorithm, which finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible region.

</p>

<div class="mw-heading mw-heading4"><h4 id="Ellipsoid_algorithm,_following_Khachiyan">Ellipsoid algorithm, following Khachiyan</h4></div>

<p>This is the first <a href="Worst-case_complexity" title="Worst-case complexity">worst-case</a> <a href="Polynomial-time" class="mw-redirect" title="Polynomial-time">polynomial-time</a> algorithm ever found for linear programming. To solve a problem which has <i>n</i> variables and can be encoded in <i>L</i> input bits, this algorithm runs in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{6}L)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle O(n^{6}L)}</annotation>

</semantics>

</math></span><img src="./8b415f36ac1c25ad1609e767279717826fc236fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.614ex; height:3.176ex;" alt="{\displaystyle O(n^{6}L)}" loading="lazy"></span> time.<sup id="cite_ref-khachiyan79_9-1" class="reference"><a href="#cite_note-khachiyan79-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="Leonid_Khachiyan" title="Leonid Khachiyan">Leonid Khachiyan</a> solved this long-standing complexity issue in 1979 with the introduction of the <a href="Ellipsoid_method" title="Ellipsoid method">ellipsoid method</a>. The convergence analysis has (real-number) predecessors, notably the <a href="Iterative_method" title="Iterative method">iterative methods</a> developed by <a href="Naum_Z._Shor" title="Naum Z. Shor">Naum Z. Shor</a> and the <a href="Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a> by Arkadi Nemirovski and D. Yudin.

</p>

<div class="mw-heading mw-heading4"><h4 id="Projective_algorithm_of_Karmarkar">Projective algorithm of Karmarkar</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Karmarkar's algorithm</a></div>

<p>Khachiyan's algorithm was of landmark importance for establishing the polynomial-time solvability of linear programs. The algorithm was not a computational break-through, as the simplex method is more efficient for all but specially constructed families of linear programs.

</p><p>However, Khachiyan's algorithm inspired new lines of research in linear programming. In 1984, <a href="Narendra_Karmarkar" title="Narendra Karmarkar">N. Karmarkar</a> proposed a <a href="Projective_method" class="mw-redirect" title="Projective method">projective method</a> for linear programming. Karmarkar's algorithm<sup id="cite_ref-karmarkar84_10-1" class="reference"><a href="#cite_note-karmarkar84-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> improved on Khachiyan's<sup id="cite_ref-khachiyan79_9-2" class="reference"><a href="#cite_note-khachiyan79-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> worst-case polynomial bound (giving <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{3.5}L)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle O(n^{3.5}L)}</annotation>

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</math></span><img src="./75bfa3b481f545e25ae3d8d60ef6ca14d51050ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.894ex; height:3.176ex;" alt="{\displaystyle O(n^{3.5}L)}" loading="lazy"></span>). Karmarkar claimed that his algorithm was much faster in practical LP than the simplex method, a claim that created great interest in interior-point methods.<sup id="cite_ref-Strang_21-0" class="reference"><a href="#cite_note-Strang-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Since Karmarkar's discovery, many interior-point methods have been proposed and analyzed.

</p>

<div class="mw-heading mw-heading4"><h4 id="Vaidya's_87_algorithm">Vaidya's 87 algorithm</h4></div>

<p>In 1987, Vaidya proposed an algorithm that runs in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{3})}">

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle O(n^{3})}</annotation>

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</math></span><img src="./6b04f5c5cfea38f43406d9442387ad28555e2609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{3})}" loading="lazy"></span> time.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading4"><h4 id="Vaidya's_89_algorithm">Vaidya's 89 algorithm</h4></div>

<p>In 1989, Vaidya developed an algorithm that runs in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2.5})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n^{2.5})}</annotation>

</semantics>

</math></span><img src="./ace57227cac24373d5d91e9fbb5ef6ebdbaadce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.311ex; height:3.176ex;" alt="{\displaystyle O(n^{2.5})}" loading="lazy"></span> time.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Formally speaking, the algorithm takes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O((n+d)^{1.5}nL)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O((n+d)^{1.5}nL)}</annotation>

</semantics>

</math></span><img src="./6874202657ad85e546419778c0fc3b42e0b4634b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.154ex; height:3.176ex;" alt="{\displaystyle O((n+d)^{1.5}nL)}" loading="lazy"></span> arithmetic operations in the worst case, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math></span><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> is the number of constraints, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is the number of variables, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> is the number of bits.

</p>

<div class="mw-heading mw-heading4"><h4 id="Input_sparsity_time_algorithms">Input sparsity time algorithms</h4></div>

<p>In 2015, Lee and Sidford showed that linear programming can be solved in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}((nnz(A)+d^{2}){\sqrt {d}}L)}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}((nnz(A)+d^{2}){\sqrt {d}}L)}</annotation>

</semantics>

</math></span><img src="./b28f5af38de50f081b1763eb33c72e5feabb35f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.669ex; height:3.176ex;" alt="{\displaystyle {\tilde {O}}((nnz(A)+d^{2}){\sqrt {d}}L)}" loading="lazy"></span> time,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}}</annotation>

</semantics>

</math></span><img src="./838fcae65623915485e4245a38d787c4a99f9be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.676ex;" alt="{\displaystyle {\tilde {O}}}" loading="lazy"></span> denotes the <a href="Soft_O_notation" class="mw-redirect" title="Soft O notation">soft O notation</a>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nnz(A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nnz(A)}</annotation>

</semantics>

</math></span><img src="./487e860cf976f9010543f8d6655e2e1760ed1db6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.43ex; height:2.843ex;" alt="{\displaystyle nnz(A)}" loading="lazy"></span> represents the number of non-zero elements, and it remains taking <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2.5}L)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n^{2.5}L)}</annotation>

</semantics>

</math></span><img src="./5f3c8031399ff33ff9d8e02f4bf7349d1cbdca2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.894ex; height:3.176ex;" alt="{\displaystyle O(n^{2.5}L)}" loading="lazy"></span> in the worst case.

</p>

<div class="mw-heading mw-heading4"><h4 id="Current_matrix_multiplication_time_algorithm">Current matrix multiplication time algorithm</h4></div>

<p>In 2019, Cohen, Lee and Song improved the running time to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}((n^{\omega }+n^{2.5-\alpha /2}+n^{2+1/6})L)}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}((n^{\omega }+n^{2.5-\alpha /2}+n^{2+1/6})L)}</annotation>

</semantics>

</math></span><img src="./76ecc5fab68418127b1e4ea0e9398b0ec4d6afca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.201ex; height:3.343ex;" alt="{\displaystyle {\tilde {O}}((n^{\omega }+n^{2.5-\alpha /2}+n^{2+1/6})L)}" loading="lazy"></span> time, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> is the exponent of <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is the dual exponent of <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is (roughly) defined to be the largest number such that one can multiply an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>

</semantics>

</math></span><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"></span> matrix by a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n^{\alpha }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle n\times n^{\alpha }}</annotation>

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</math></span><img src="./8e4a40b0f0d07019fafc22a7dd163fc4732a8dc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.914ex; height:2.343ex;" alt="{\displaystyle n\times n^{\alpha }}" loading="lazy"></span> matrix in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(n^{2})}</annotation>

</semantics>

</math></span><img src="./6cd9594a16cb898b8f2a2dff9227a385ec183392.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{2})}" loading="lazy"></span> time. In a followup work by Lee, Song and Zhang, they reproduce the same result via a different method.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> These two algorithms remain <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}(n^{2+1/6}L)}">

<mover>

</mover>

</mrow>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(n^{2+1/6}L)}</annotation>

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</math></span><img src="./8450538557d72c8cfad1dcab89c3dcee0763bc81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.359ex; height:3.343ex;" alt="{\displaystyle {\tilde {O}}(n^{2+1/6}L)}" loading="lazy"></span> when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =2}</annotation>

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</math></span><img src="./b636606fdbc7771698df48d99d6cd90cc60135f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.707ex; height:2.176ex;" alt="{\displaystyle \omega =2}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =1}</annotation>

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</math></span><img src="./03d67a45a44be8b8f15e99b7def2b0cf0aba1717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =1}" loading="lazy"></span>. The result due to Jiang, Song, Weinstein and Zhang improved <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}(n^{2+1/6}L)}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(n^{2+1/6}L)}</annotation>

</semantics>

</math></span><img src="./8450538557d72c8cfad1dcab89c3dcee0763bc81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.359ex; height:3.343ex;" alt="{\displaystyle {\tilde {O}}(n^{2+1/6}L)}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {O}}(n^{2+1/18}L)}">

<mover>

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</mrow>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {O}}(n^{2+1/18}L)}</annotation>

</semantics>

</math></span><img src="./a19071edfea2a00c4a765ba968b4421f9ebeab83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.181ex; height:3.343ex;" alt="{\displaystyle {\tilde {O}}(n^{2+1/18}L)}" loading="lazy"></span>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Comparison_of_interior-point_methods_and_simplex_algorithms">Comparison of interior-point methods and simplex algorithms</h3></div>

<p>The current opinion is that the efficiencies of good implementations of simplex-based methods and interior point methods are similar for routine applications of linear programming. However, for specific types of LP problems, it may be that one type of solver is better than another (sometimes much better), and that the structure of the solutions generated by interior point methods versus simplex-based methods are significantly different with the support set of active variables being typically smaller for the latter one.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Open_problems_and_recent_work">Open problems and recent work</h2></div>

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</style>

<div><span class="unsolved-label" id="unsolved-label-computer_science">Unsolved problem in computer science</span></div>

<div class="unsolved-body">Does linear programming admit a strongly polynomial-time algorithm?</div>

<div class="unsolved-more"><a href="List_of_unsolved_problems_in_computer_science" title="List of unsolved problems in computer science">More unsolved problems in computer science</a></div>

</div>

<p>There are several open problems in the theory of linear programming, the solution of which would represent fundamental breakthroughs in mathematics and potentially major advances in our ability to solve large-scale linear programs.

</p>

<ul><li>Does LP admit a <a href="Strongly-polynomial_time" title="Strongly-polynomial time">strongly polynomial</a>-time algorithm?</li>

<li>Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution?</li>

<li>Does LP admit a polynomial-time algorithm in the real number (unit cost) model of computation?</li></ul>

<p>This closely related set of problems has been cited by <a href="Stephen_Smale" title="Stephen Smale">Stephen Smale</a> as among the <a href="Smale's_problems" title="Smale's problems">18 greatest unsolved problems</a> of the 21st century. In Smale's words, the third version of the problem "is the main unsolved problem of linear programming theory." While algorithms exist to solve linear programming in <a href="Strongly-polynomial_time" title="Strongly-polynomial time">weakly polynomial time</a>, such as the <a href="Ellipsoid_method" title="Ellipsoid method">ellipsoid methods</a> and <a href="Interior_point_method" class="mw-redirect" title="Interior point method">interior-point techniques</a>, no algorithms have yet been found that allow strongly polynomial-time performance in the number of constraints and the number of variables. The development of such algorithms would be of great theoretical interest, and perhaps allow practical gains in solving large LPs as well.

</p><p>Although the <a href="Hirsch_conjecture" title="Hirsch conjecture">Hirsch conjecture</a> was recently disproved for higher dimensions, it still leaves the following questions open.

</p>

<ul><li>Are there pivot rules which lead to polynomial-time simplex variants?</li>

<li>Do all polytopal graphs have polynomially bounded diameter?</li></ul>

<p>These questions relate to the performance analysis and development of simplex-like methods. The immense efficiency of the simplex algorithm in practice despite its exponential-time theoretical performance hints that there may be variations of simplex that run in polynomial or even strongly polynomial time. It would be of great practical and theoretical significance to know whether any such variants exist, particularly as an approach to deciding if LP can be solved in strongly polynomial time.

</p><p>The simplex algorithm and its variants fall in the family of edge-following algorithms, so named because they solve linear programming problems by moving from vertex to vertex along edges of a polytope. This means that their theoretical performance is limited by the maximum number of edges between any two vertices on the LP polytope. As a result, we are interested in knowing the maximum <a href="Graph_diameter" class="mw-redirect" title="Graph diameter">graph-theoretical diameter</a> of polytopal <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. It has been proved that all polytopes have subexponential diameter. The recent disproof of the Hirsch conjecture is the first step to prove whether any polytope has superpolynomial diameter. If any such polytopes exist, then no edge-following variant can run in polynomial time. Questions about polytope diameter are of independent mathematical interest.

</p><p>Simplex pivot methods preserve primal (or dual) feasibility. On the other hand, criss-cross pivot methods do not preserve (primal or dual) feasibility – they may visit primal feasible, dual feasible or primal-and-dual infeasible bases in any order. Pivot methods of this type have been studied since the 1970s.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Essentially, these methods attempt to find the shortest pivot path on the arrangement polytope under the linear programming problem. In contrast to polytopal graphs, graphs of arrangement polytopes are known to have small diameter, allowing the possibility of strongly polynomial-time criss-cross pivot algorithm without resolving questions about the diameter of general polytopes.<sup id="cite_ref-FukudaTerlaky_15-2" class="reference"><a href="#cite_note-FukudaTerlaky-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Integer_unknowns">Integer unknowns</h2></div>

<p>If all of the unknown variables are required to be integers, then the problem is called an <a href="Integer_programming" title="Integer programming">integer programming</a> (IP) or <b>integer linear programming</b> (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in many practical situations (those with bounded variables) <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>. <b>0–1 integer programming</b> or <b>binary integer programming</b> (BIP) is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This problem is also classified as NP-hard, and in fact the decision version was one of <a href="Karp's_21_NP-complete_problems" title="Karp's 21 NP-complete problems">Karp's 21 NP-complete problems</a>.

</p><p>If only some of the unknown variables are required to be integers, then the problem is called a <b>mixed integer (linear) programming</b> (MIP or MILP) problem. These are generally also NP-hard because they are even more general than ILP programs.

</p><p>There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is <a href="Totally_unimodular" class="mw-redirect" title="Totally unimodular">totally unimodular</a> and the right-hand sides of the constraints are integers or – more general – where the system has the <a href="Total_dual_integrality" title="Total dual integrality">total dual integrality</a> (TDI) property.

</p><p>Advanced algorithms for solving integer linear programs include:

</p>

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">cutting-plane method</a></li>

<li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a></li>

<li><a href="Branch_and_cut" title="Branch and cut">Branch and cut</a></li>

<li><a href="Branch_and_price" title="Branch and price">Branch and price</a></li>

<li>if the problem has some extra structure, it may be possible to apply <a href="Delayed_column_generation" class="mw-redirect" title="Delayed column generation">delayed column generation</a>.</li></ul>

<p>Such integer-programming algorithms are discussed by <a href="Manfred_W._Padberg" title="Manfred W. Padberg">Padberg</a> and in Beasley.

</p>

<div class="mw-heading mw-heading2"><h2 id="Integral_linear_programs">Integral linear programs</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Integral_polytope" title="Integral polytope">Integral polytope</a></div>

<p>A linear program in real variables is said to be <i><b>integral</b></i> if it has at least one optimal solution which is integral, i.e., made of only integer values. Likewise, a polyhedron <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{x\mid Ax\geq 0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P=\{x\mid Ax\geq 0\}}</annotation>

</semantics>

</math></span><img src="./a93eb6c51c5ee7e8f16d606914981c746c89426c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.769ex; height:2.843ex;" alt="{\displaystyle P=\{x\mid Ax\geq 0\}}" loading="lazy"></span> is said to be <i><b>integral</b></i> if for all bounded feasible objective functions <i>c</i>, the linear program <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\max cx\mid x\in P\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\max cx\mid x\in P\}}</annotation>

</semantics>

</math></span><img src="./08d3cb25b7596814b9233c836756ac16496e4e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.227ex; height:2.843ex;" alt="{\displaystyle \{\max cx\mid x\in P\}}" loading="lazy"></span> has an optimum <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{*}}</annotation>

</semantics>

</math></span><img src="./e5be23ee5d433f8b576e63bcb47518128ee0b6bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.343ex;" alt="{\displaystyle x^{*}}" loading="lazy"></span> with integer coordinates. As observed by Edmonds and Giles in 1977, one can equivalently say that the polyhedron <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is integral if for every bounded feasible integral objective function <i>c</i>, the optimal <i>value</i> of the linear program <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\max cx\mid x\in P\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\max cx\mid x\in P\}}</annotation>

</semantics>

</math></span><img src="./08d3cb25b7596814b9233c836756ac16496e4e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.227ex; height:2.843ex;" alt="{\displaystyle \{\max cx\mid x\in P\}}" loading="lazy"></span> is an integer.

</p><p>Integral linear programs are of central importance in the polyhedral aspect of <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> since they provide an alternate characterization of a problem. Specifically, for any problem, the convex hull of the solutions is an integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find the optimal feasible solution under any linear objective. Conversely, if we can prove that a <a href="Linear_programming_relaxation" title="Linear programming relaxation">linear programming relaxation</a> is integral, then it is the desired description of the convex hull of feasible (integral) solutions.

</p><p>Terminology is not consistent throughout the literature, so one should be careful to distinguish the following two concepts,

</p>

<ul><li>in an <i>integer linear program,</i> described in the previous section, variables are forcibly constrained to be integers, and this problem is NP-hard in general,</li>

<li>in an <i>integral linear program,</i> described in this section, variables are not constrained to be integers but rather one has proven somehow that the continuous problem always has an integral optimal value (assuming <i>c</i> is integral), and this optimal value may be found efficiently since all polynomial-size linear programs can be solved in polynomial time.</li></ul>

<p>One common way of proving that a polyhedron is integral is to show that it is <a href="Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">totally unimodular</a>. There are other general methods including the integer decomposition property and <a href="Total_dual_integrality" title="Total dual integrality">total dual integrality</a>. Other specific well-known integral LPs include the matching polytope, lattice polyhedra, <a href="Submodular_flow" title="Submodular flow">submodular flow</a> polyhedra, and the intersection of two generalized polymatroids/<i>g</i>-polymatroids – e.g. see Schrijver 2003.

</p>

<div class="mw-heading mw-heading2"><h2 id="Solvers_and_scripting_(programming)_languages">Solvers and scripting (programming) languages</h2></div>

<p><b><a href="Permissive_free_software_licence" class="mw-redirect" title="Permissive free software licence">Permissive</a> licenses:</b>

</p>

<th>Name

</th>

<th>License

</th>

<th>Brief info

</th></tr>

<tr>

<td><a href="Gekko_(optimization_software)" title="Gekko (optimization software)">Gekko</a></td>

<td><a href="MIT_License" title="MIT License">MIT License</a></td>

<td>Open-source library for solving large-scale LP, <a href="Quadratic_programming" title="Quadratic programming">QP</a>, <a href="Quadratically_constrained_quadratic_program" title="Quadratically constrained quadratic program">QCQP</a>, <a href="Nonlinear_programming" title="Nonlinear programming">NLP</a>, and <a href="Mixed_integer_programming" class="mw-redirect" title="Mixed integer programming">MIP</a> optimization

</td></tr>

<tr>

<td><a href="Apache_License" title="Apache License">Apache v2</a></td>

<td>Google's open-source linear programming solver

</td></tr>

<tr>

<td><a href="MPL_License" class="mw-redirect" title="MPL License">MPL License</a></td>

<td>Open-source modeling language with solvers for large-scale LP, <a href="Quadratic_programming" title="Quadratic programming">QP</a>, <a href="Quadratically_constrained_quadratic_program" title="Quadratically constrained quadratic program">QCQP</a>, <a href="Semidefinite_programming" title="Semidefinite programming">SDP</a>, <a href="Second-order_cone_programming" title="Second-order cone programming">SOCP</a>, <a href="Nonlinear_programming" title="Nonlinear programming">NLP</a>, and <a href="Mixed_integer_programming" class="mw-redirect" title="Mixed integer programming">MIP</a> optimization

</td></tr>

<tr>

<td><a href="Pyomo" title="Pyomo">Pyomo</a></td>

<td>An open-source modeling language for large-scale linear, mixed integer and nonlinear optimization

</td></tr>

<tr>

<td><a href="Apache_License" title="Apache License">Apache v2</a></td>

<td>A general-purpose constraint integer programming solver with an emphasis on MIP. Compatible with Zimpl modelling language.

</td></tr>

<tr>

<td><a href="SuanShu_numerical_library" title="SuanShu numerical library">SuanShu</a></td>

<td><a href="Apache_License" title="Apache License">Apache v2</a></td>

<td>An open-source suite of optimization algorithms to solve LP, <a href="Quadratic_programming" title="Quadratic programming">QP</a>, <a href="SOCP" class="mw-redirect" title="SOCP">SOCP</a>, <a href="Semidefinite_programming" title="Semidefinite programming">SDP</a>, <a href="Sequential_quadratic_programming" title="Sequential quadratic programming">SQP</a> in Java

</td></tr></tbody></table>

<p><b><a href="Copyleft" title="Copyleft">Copyleft (reciprocal)</a> licenses:</b>

</p>

<th>Name

</th>

<th>License

</th>

<th>Brief info

</th></tr>

<tr>

<td><a href="ALGLIB" title="ALGLIB">ALGLIB</a></td>

<td>An LP solver from ALGLIB project (C++, C#, Python)

</td></tr>

<tr>

<td><a href="Cassowary_constraint_solver" class="mw-redirect" title="Cassowary constraint solver">Cassowary constraint solver</a></td>

<td>An incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities

</td></tr>

<tr>

<td>An LP solver from COIN-OR

</td></tr>

<tr>

<td>GNU Linear Programming Kit, an LP/MILP solver with a native C <a href="API" title="API">API</a> and numerous (15) third-party wrappers for other languages. Specialist support for <a href="Flow_network" title="Flow network">flow networks</a>. Bundles the <a href="AMPL" title="AMPL">AMPL</a>-like <a href="GNU_MathProg" title="GNU MathProg">GNU MathProg</a> modelling language and translator.

</td></tr>

<tr>

<td><a href="Lp_solve" title="Lp solve">lp solve</a></td>

<td>An LP and <a href="Mixed-integer_programming" class="mw-redirect" title="Mixed-integer programming">MIP</a> solver featuring support for the <a href="MPS_(format)" title="MPS (format)">MPS format</a> and its own "lp" format, as well as custom formats through its "eXternal Language Interface" (XLI).<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Translating between model formats is also possible.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>A library for incrementally solving systems of linear equations with various goal functions

</td></tr>

<tr>

<td><a href="R-Project" class="mw-redirect" title="R-Project">R-Project</a></td>

<td>A programming language and software environment for statistical computing and graphics

</td></tr></tbody></table>

<p><a href="MINTO" title="MINTO">MINTO</a> (Mixed Integer Optimizer, an <a href="Integer_programming" title="Integer programming">integer programming</a> solver which uses branch and bound algorithm) has publicly available source code<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> but is not open source.

</p><p><b><a href="Proprietary_software" title="Proprietary software">Proprietary</a> licenses:</b>

</p>

<th>Name

</th>

<th>Brief info

</th></tr>

<tr>

<td><a href="AIMMS" title="AIMMS">AIMMS</a></td>

<td>A modeling language that allows to model linear, mixed integer, and nonlinear optimization models. It also offers a tool for constraint programming. Algorithm, in the forms of heuristics or exact methods, such as Branch-and-Cut or Column Generation, can also be implemented. The tool calls an appropriate solver such as CPLEX or similar, to solve the optimization problem at hand. Academic licenses are free of charge.

</td></tr>

<tr>

<td><a href="ALGLIB" title="ALGLIB">ALGLIB</a></td>

<td>A commercial edition of the copyleft licensed library. C++, C#, Python.

</td></tr>

<tr>

<td>A popular modeling language for large-scale linear, mixed integer and nonlinear optimisation with a free student limited version available (500 variables and 500 constraints).

</td></tr>

<tr>

<td><a href="Analytica_(software)" title="Analytica (software)">Analytica</a></td>

<td>A general modeling language and interactive development environment. Its influence diagrams enable users to formulate problems as graphs with nodes for decision variables, objectives, and constraints. Analytica Optimizer Edition includes linear, mixed integer, and nonlinear solvers and selects the solver to match the problem. It also accepts other engines as plug-ins, including <a href="XPRESS" title="XPRESS">XPRESS</a>, Gurobi, <a href="Artelys_Knitro" title="Artelys Knitro">Artelys Knitro</a>, and <a href="MOSEK" title="MOSEK">MOSEK</a>.

</td></tr>

<tr>

<td><a href="APMonitor" title="APMonitor">APMonitor</a></td>

<td>API to MATLAB and Python. Solve example Linear Programming (LP) problems through MATLAB, Python, or a web-interface.

</td></tr>

<tr>

<td><a href="CPLEX" title="CPLEX">CPLEX</a></td>

<td>Popular solver with an API for several programming languages, and also has a modelling language and works with AIMMS, AMPL, <a href="General_Algebraic_Modeling_System" class="mw-redirect" title="General Algebraic Modeling System">GAMS</a>, MPL, OpenOpt, OPL Development Studio, and <a href="TOMLAB" title="TOMLAB">TOMLAB</a>. Free for academic use.

</td></tr>

<tr>

<td><a href="Microsoft_Excel" title="Microsoft Excel">Excel</a> Solver Function</td>

<td>A nonlinear solver adjusted to spreadsheets in which function evaluations are based on the recalculating cells. Basic version available as a standard add-on for Excel.

</td></tr>

<tr>

<td><a href="FortMP" title="FortMP">FortMP</a></td>

<td>

</td></tr>

<tr>

<td>

</td></tr>

<tr>

<td><a href="Gurobi_Optimizer" title="Gurobi Optimizer">Gurobi Optimizer</a></td>

<td>

</td></tr>

<tr>

<td><a href="IMSL_Numerical_Libraries" title="IMSL Numerical Libraries">IMSL Numerical Libraries</a></td>

<td>Collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programming algorithms.

</td></tr>

<tr>

<td><a href="LINDO" title="LINDO">LINDO</a></td>

<td>Solver with an API for large scale optimization of linear, integer, quadratic, conic and general nonlinear programs with stochastic programming extensions. It offers a global optimization procedure for finding guaranteed globally optimal solution to general nonlinear programs with continuous and discrete variables. It also has a statistical sampling API to integrate Monte-Carlo simulations into an optimization framework. It has an algebraic modeling language (<a href="Lingo_(programming_language)" title="Lingo (programming language)">LINGO</a>) and allows modeling within a spreadsheet (What'sBest).

</td></tr>

<tr>

<td><a href="Maple_(software)" title="Maple (software)">Maple</a></td>

<td>A general-purpose programming-language for symbolic and numerical computing.

</td></tr>

<tr>

<td><a href="MATLAB" title="MATLAB">MATLAB</a></td>

<td>A general-purpose and matrix-oriented programming-language for numerical computing. Linear programming in MATLAB requires the <a href="Optimization_Toolbox" title="Optimization Toolbox">Optimization Toolbox</a> in addition to the base MATLAB product; available routines include INTLINPROG and LINPROG

</td></tr>

<tr>

<td><a href="Mathcad" title="Mathcad">Mathcad</a></td>

<td>A WYSIWYG math editor. It has functions for solving both linear and nonlinear optimization problems.

</td></tr>

<tr>

<td><a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a></td>

<td>A general-purpose programming-language for mathematics, including symbolic and numerical capabilities.

</td></tr>

<tr>

<td><a href="MOSEK" title="MOSEK">MOSEK</a></td>

<td>A solver for large scale optimization with API for several languages (C++, java, .net, Matlab and python).

</td></tr>

<tr>

<td><a href="NAG_Numerical_Library" title="NAG Numerical Library">NAG Numerical Library</a></td>

<td>A collection of mathematical and statistical routines developed by the <a href="Numerical_Algorithms_Group" class="mw-redirect" title="Numerical Algorithms Group">Numerical Algorithms Group</a> for multiple programming languages (C, C++, Fortran, Visual Basic, Java and C#) and packages (MATLAB, Excel, R, LabVIEW). The Optimization chapter of the NAG Library includes routines for linear programming problems with both sparse and non-sparse linear constraint matrices, together with routines for the optimization of quadratic, nonlinear, sums of squares of linear or nonlinear functions with nonlinear, bounded or no constraints. The NAG Library has routines for both local and global optimization, and for continuous or integer problems.

</td></tr>

<tr>

<td><a href="OptimJ" title="OptimJ">OptimJ</a></td>

<td>A Java-based modeling language for optimization with a free version available.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>

</td></tr>

<tr>

<td>A suite of solvers for Linear, Integer, Nonlinear, Derivative-Free, Network, Combinatorial and Constraint Optimization; the <a href="Algebraic_modeling_language" title="Algebraic modeling language">Algebraic modeling language</a> OPTMODEL; and a variety of vertical solutions aimed at specific problems/markets, all of which are fully integrated with the <a href="SAS_System" class="mw-redirect" title="SAS System">SAS System</a>.

</td></tr>

<tr>

<td><a href="FICO_Xpress" title="FICO Xpress">XPRESS</a></td>

<td>Solver for large-scale linear programs, quadratic programs, general nonlinear and mixed-integer programs. Has API for several programming languages, also has a modelling language Mosel and works with AMPL, <a href="General_Algebraic_Modeling_System" class="mw-redirect" title="General Algebraic Modeling System">GAMS</a>. Free for academic use.

</td></tr>

<tr>

<td><a href="VisSim" title="VisSim">VisSim</a></td>

<td>A visual <a href="Block_diagram" title="Block diagram">block diagram</a> language for simulation of <a href="Dynamical_system" title="Dynamical system">dynamical systems</a>.

</td></tr></tbody></table>

/* start https://en.wikipedia.org/ */

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<ul><li><a href="Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Expected_shortfall#Optimization_of_expected_shortfall" title="Expected shortfall">Expected shortfall § Optimization of expected shortfall</a></li>

<li><a href="Input%E2%80%93output_model" title="Input–output model">Input–output model</a></li>

<li><a href="Job_shop_scheduling" class="mw-redirect" title="Job shop scheduling">Job shop scheduling</a></li>

<li><a href="Least_absolute_deviations" title="Least absolute deviations">Least absolute deviations</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Linear_algebra" title="Linear algebra">Linear algebra</a></li>

<li><a href="Linear_production_game" title="Linear production game">Linear production game</a></li>

<li><a href="Linear-fractional_programming_(LFP)" class="mw-redirect" title="Linear-fractional programming (LFP)">Linear-fractional programming (LFP)</a></li>

<li><a href="LP-type_problem" title="LP-type problem">LP-type problem</a></li>

<li><a href="Mathematical_programming" class="mw-redirect" title="Mathematical programming">Mathematical programming</a></li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li>

<li><a href="Odds_algorithm" title="Odds algorithm">Odds algorithm</a> used to solve optimal stopping problems</li>

<li><a href="Oriented_matroid" title="Oriented matroid">Oriented matroid</a></li>

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a>, a superset of linear programming</li>

<li><a href="Semidefinite_programming" title="Semidefinite programming">Semidefinite programming</a></li>

<li><a href="Shadow_price" title="Shadow price">Shadow price</a></li>

<li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a>, used to solve LP problems</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFKemenyMorgensternThompson1956" class="citation journal cs1">Kemeny, J. G.; Morgenstern, O.; Thompson, G. L. (1956). "A Generalization of the von Neumann Model of an Expanding Economy". <i>Econometrica</i>. <b>24</b> (2): <span class="nowrap">115–</span>135. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1905746">10.2307/1905746</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1905746">1905746</a>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFLi2019" class="citation book cs1 cs1-prop-foreign-lang-source">Li, Wu (2019). <i>General Equilibrium and Structural Dynamics: Perspectives of New Structural Economics</i> (in Chinese). Beijing: Economic Science Press. pp. <span class="nowrap">122–</span>125. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-7-5218-0422-5</bdi>.</cite></span>

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<li id="cite_note-SierksmaZwols2015-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-SierksmaZwols2015_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SierksmaZwols2015_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGerard_SierksmaYori_Zwols2015" class="citation book cs1">Gerard Sierksma; Yori Zwols (2015). <i>Linear and Integer Optimization: Theory and Practice</i> (3rd ed.). CRC Press. p. 1. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1498710169</bdi>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/linear-programming-mathematics">"Linear programming | Definition & Facts | Britannica"</a>. <i>www.britannica.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-11-20</span></span>.</cite></span>

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<li id="cite_note-dantzig1982-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-dantzig1982_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-dantzig1982_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-dantzig1982_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGeorge_B._Dantzig1982" class="citation journal cs1">George B. Dantzig (April 1982). <a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/pdfs/ADA112060.pdf">"Reminiscences about the origins of linear programming"</a> <span class="cs1-format">(PDF)</span>. <i>Operations Research Letters</i>. <b>1</b> (2): <span class="nowrap">43–</span>48. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-6377%2882%2990043-8">10.1016/0167-6377(82)90043-8</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150520183722/http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA112060">Archived</a> from the original on May 20, 2015.</cite></span>

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<li id="cite_note-Schrijver1998-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schrijver1998_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAlexander_Schrijver1998" class="citation book cs1">Alexander Schrijver (1998). <i>Theory of Linear and Integer Programming</i>. John Wiley & Sons. pp. <span class="nowrap">221–</span>222. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-98232-6</bdi>.</cite></span>

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<li id="cite_note-:0-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFDantzigThapa1997" class="citation book cs1">Dantzig, George B.; Thapa, Mukund Narain (1997). <i>Linear programming</i>. New York: Springer. p. xxvii. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0387948333</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/35318475">35318475</a>.</cite></span>

</li>

<li id="cite_note-khachiyan79-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-khachiyan79_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-khachiyan79_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-khachiyan79_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFLeonid_Khachiyan1979" class="citation journal cs1">Leonid Khachiyan (1979). "A Polynomial Algorithm for Linear Programming". <i>Doklady Akademii Nauk SSSR</i>. <b>224</b> (5): <span class="nowrap">1093–</span>1096.</cite></span>

</li>

<li id="cite_note-karmarkar84-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-karmarkar84_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-karmarkar84_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNarendra_Karmarkar1984" class="citation journal cs1">Narendra Karmarkar (1984). "A New Polynomial-Time Algorithm for Linear Programming". <i>Combinatorica</i>. <b>4</b> (4): <span class="nowrap">373–</span>395. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02579150">10.1007/BF02579150</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7257867">7257867</a>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFM._GrundmannV._KwatraI._Essa2011" class="citation book cs1">M. Grundmann; V. Kwatra; I. Essa (2011). "Auto-directed video stabilization with robust L1 optimal camera paths". <a rel="nofollow" class="external text" href="https://static.googleusercontent.com/media/research.google.com/en//pubs/archive/37041.pdf"><i>CVPR 2011</i></a> <span class="cs1-format">(PDF)</span>. pp. <span class="nowrap">225–</span>232. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FCVPR.2011.5995525">10.1109/CVPR.2011.5995525</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4577-0394-2</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17707171">17707171</a>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFVazirani2001">Vazirani (2001</a>, p. 112)</span>

</li>

<li id="cite_note-DT03-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-DT03_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DT03_13-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-DT03_13-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDantzigThapa2003">Dantzig & Thapa (2003)</a></span>

</li>

<li id="cite_note-Padberg-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Padberg_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Padberg_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFPadberg1999">Padberg (1999)</a></span>

</li>

<li id="cite_note-FukudaTerlaky-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-FukudaTerlaky_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FukudaTerlaky_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FukudaTerlaky_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFukudaTerlaky1997" class="citation journal cs1"><a href="Komei_Fukuda" title="Komei Fukuda">Fukuda, Komei</a>; <a href="Tam%C3%A1s_Terlaky" title="Tamás Terlaky">Terlaky, Tamás</a> (1997). Thomas M. Liebling; Dominique de Werra (eds.). "Criss-cross methods: A fresh view on pivot algorithms". <i>Mathematical Programming, Series B</i>. <b>79</b> (<span class="nowrap">1–</span>3): <span class="nowrap">369–</span>395. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.9373">10.1.1.36.9373</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02614325">10.1007/BF02614325</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1464775">1464775</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2794181">2794181</a>.</cite></span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFBorgwardt1987">Borgwardt (1987)</a></span>

</li>

</li>

<li id="cite_note-Murty-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Murty_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMurty1983">Murty (1983)</a></span>

</li>

<li id="cite_note-PS-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-PS_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPapadimitriouSteiglitz">Papadimitriou & Steiglitz</a></span>

</li>

<li id="cite_note-Roos-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Roos_20-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoos1990" class="citation journal cs1">Roos, C. (1990). "An exponential example for Terlaky's pivoting rule for the criss-cross simplex method". <i>Mathematical Programming</i>. Series A. <b>46</b> (1): <span class="nowrap">79–</span>84. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01585729">10.1007/BF01585729</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1045573">1045573</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:33463483">33463483</a>.</cite></span>

</li>

<li id="cite_note-Strang-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Strang_21-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFStrang1987" class="citation journal cs1"><a href="Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (1 June 1987). "Karmarkar's algorithm and its place in applied mathematics". <i><a href="The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a></i>. <b>9</b> (2): <span class="nowrap">4–</span>10. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03025891">10.1007/BF03025891</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0343-6993">0343-6993</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0883185">0883185</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123541868">123541868</a>.</cite></span>

</li>

<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFVaidya1987" class="citation conference cs1">Vaidya, Pravin M. (1987). <i>An algorithm for linear programming which requires <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {O}(((m+n)n^{2}+(m+n)^{1.5}n)L)}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {O}(((m+n)n^{2}+(m+n)^{1.5}n)L)}</annotation>

</semantics>

</math></span><img src="./7a3195ee797e060dbc7fd9f2f903fcb0d9a70318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.162ex; height:3.176ex;" alt="{\displaystyle {O}(((m+n)n^{2}+(m+n)^{1.5}n)L)}" loading="lazy"></span> arithmetic operations</i>. 28th Annual IEEE Symposium on Foundations of Computer Science. FOCS.</cite></span>

</li>

<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFVaidya1989" class="citation conference cs1">Vaidya, Pravin M. (1989). "Speeding-up linear programming using fast matrix multiplication". <i>30th Annual Symposium on Foundations of Computer Science</i>. 30th Annual Symposium on Foundations of Computer Science. FOCS. pp. <span class="nowrap">332–</span>337. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FSFCS.1989.63499">10.1109/SFCS.1989.63499</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8186-1982-1</bdi>.</cite></span>

</li>

<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeeSidford2015" class="citation conference cs1">Lee, Yin-Tat; Sidford, Aaron (2015). <i>Efficient inverse maintenance and faster algorithms for linear programming</i>. FOCS '15 Foundations of Computer Science. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1503.01752">1503.01752</a></span>.</cite></span>

</li>

<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite id="CITEREFCohenLeeSong2018" class="citation conference cs1">Cohen, Michael B.; Lee, Yin-Tat; Song, Zhao (2018). <i>Solving Linear Programs in the Current Matrix Multiplication Time</i>. 51st Annual ACM Symposium on the Theory of Computing. STOC'19. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1810.07896">1810.07896</a></span>.</cite></span>

</li>

<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeeSongZhang2019" class="citation conference cs1">Lee, Yin-Tat; Song, Zhao; Zhang, Qiuyi (2019). <i>Solving Empirical Risk Minimization in the Current Matrix Multiplication Time</i>. Conference on Learning Theory. COLT'19. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1905.04447">1905.04447</a></span>.</cite></span>

</li>

<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><cite id="CITEREFJiangSongWeinsteinZhang2020" class="citation conference cs1">Jiang, Shunhua; Song, Zhao; Weinstein, Omri; Zhang, Hengjie (2020). <i>Faster Dynamic Matrix Inverse for Faster LPs</i>. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2004.07470">2004.07470</a></span>.</cite></span>

</li>

<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite id="CITEREFIllésTerlaky2002" class="citation journal cs1">Illés, Tibor; Terlaky, Tamás (2002). <a rel="nofollow" class="external text" href="https://strathprints.strath.ac.uk/9200/">"Pivot versus interior point methods: Pros and cons"</a>. <i>European Journal of Operational Research</i>. <b>140</b> (2): 170. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.646.3539">10.1.1.646.3539</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-2217%2802%2900061-9">10.1016/S0377-2217(02)00061-9</a>.</cite></span>

</li>

<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFAnstreicherTerlaky1994" class="citation journal cs1">Anstreicher, Kurt M.; Terlaky, Tamás (1994). <a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.42.3.556">"A Monotonic Build-Up Simplex Algorithm for Linear Programming"</a>. <i>Operations Research</i>. <b>42</b> (3): <span class="nowrap">556–</span>561. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fopre.42.3.556">10.1287/opre.42.3.556</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0030-364X">0030-364X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/171894">171894</a>.</cite></span>

</li>

<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.mit.edu/lpsolve/doc/index.htm">"lp_solve reference guide (5.5.2.5)"</a>. <i>mit.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-10</span></span>.</cite></span>

</li>

<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://lpsolve.sourceforge.net/5.5/XLI.htm">"External Language Interfaces"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">3 December</span> 2021</span>.</cite></span>

</li>

<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://lpsolve.sourceforge.net/5.5/lp_solve.htm">"lp_solve command"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">3 December</span> 2021</span>.</cite></span>

</li>

<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://coral.ie.lehigh.edu/~minto/download.html">"COR@L – Computational Optimization Research At Lehigh"</a>. <i>lehigh.edu</i>.</cite></span>

</li>

<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.in-ter-trans.eu/resources/Zesch_Hellingrath_2010_Integrated+Production-Distribution+Planning.pdf">http://www.in-ter-trans.eu/resources/Zesch_Hellingrath_2010_Integrated+Production-Distribution+Planning.pdf</a> OptimJ used in an optimization model for mixed-model assembly lines, University of Münster</span>

</li>

<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076">http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110629022829/http://www.aaai.org/ocs/index.php/AAAI/AAAI10/paper/viewFile/1769/2076">Archived</a> 2011-06-29 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> OptimJ used in an Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFKantorovich1940" class="citation journal cs1">Kantorovich, L. V. (1940). "Об одном эффективном методе решения некоторых классов экстремальных проблем" [A new method of solving some classes of extremal problems]. <i><a href="Proceedings_of_the_USSR_Academy_of_Sciences" title="Proceedings of the USSR Academy of Sciences">Doklady Akad Sci SSSR</a></i>. <b>28</b>: <span class="nowrap">211–</span>214.</cite></li>

<li>F. L. Hitchcock: <i><a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1941201224">The distribution of a product from several sources to numerous localities</a></i>, Journal of Mathematics and Physics, 20, 1941, 224–230.</li>

<li>G.B Dantzig: <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZpYca36h464C&dq=%22Maximization+of+a+linear+function+of+variables+subject+to+linear+inequalities%22&pg=PA24">Maximization of a linear function of variables subject to linear inequalities</a></i>, 1947. Published pp. 339–347 in T.C. Koopmans (ed.):<i>Activity Analysis of Production and Allocation</i>, New York-London 1951 (Wiley & Chapman-Hall)</li>

<li>J. E. Beasley, editor. <i>Advances in Linear and Integer Programming</i>. Oxford Science, 1996. (Collection of surveys)</li>

<li><cite id="CITEREFBland1977" class="citation journal cs1">Bland, Robert G. (1977). "New Finite Pivoting Rules for the Simplex Method". <i>Mathematics of Operations Research</i>. <b>2</b> (2): <span class="nowrap">103–</span>107. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmoor.2.2.103">10.1287/moor.2.2.103</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3689647">3689647</a>.</cite></li>

<li><cite id="CITEREFBorgwardt1987" class="citation book cs1">Borgwardt, Karl-Heinz (1987). <i>The Simplex Algorithm: A Probabilistic Analysis</i>. Algorithms and Combinatorics. Vol. 1. Springer-Verlag.</cite> (Average behavior on random problems)</li>

<li>Richard W. Cottle, ed. <i>The Basic George B. Dantzig</i>. Stanford Business Books, Stanford University Press, Stanford, California, 2003. (Selected papers by <a href="George_B._Dantzig" class="mw-redirect" title="George B. Dantzig">George B. Dantzig</a>)</li>

<li>George B. Dantzig and Mukund N. Thapa. 1997. <i>Linear programming 1: Introduction</i>. Springer-Verlag.</li>

<li><cite id="CITEREFDantzigThapa2003" class="citation book cs1">Dantzig, George B.; Thapa, Mukund N. (2003). <i>Linear Programming 2: Theory and Extensions</i>. Springer-Verlag.</cite> (Comprehensive, covering e.g. <a href="Simplex_algorithm" title="Simplex algorithm">pivoting</a> and interior-point algorithms, large-scale problems, <a href="Dantzig%E2%80%93Wolfe_decomposition" title="Dantzig–Wolfe decomposition">decomposition following Dantzig–Wolfe</a> and <a href="Benders'_decomposition" class="mw-redirect" title="Benders' decomposition">Benders</a>, and introducing <a href="Stochastic_programming" title="Stochastic programming">stochastic programming</a>.)</li>

<li><cite id="CITEREFEdmondsGiles1977" class="citation book cs1">Edmonds, Jack; Giles, Rick (1977). "A Min-Max Relation for Submodular Functions on Graphs". <i>Studies in Integer Programming</i>. Annals of Discrete Mathematics. Vol. 1. pp. <span class="nowrap">185–</span>204. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0167-5060%2808%2970734-9">10.1016/S0167-5060(08)70734-9</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7204-0765-5</bdi>.</cite></li>

<li><cite id="CITEREFFukudaTerlaky1997" class="citation journal cs1">Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling; Dominique de Werra (eds.). "Criss-cross methods: A fresh view on pivot algorithms". <i>Mathematical Programming, Series B</i>. <b>79</b> (<span class="nowrap">1–</span>3): <span class="nowrap">369–</span>395. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.9373">10.1.1.36.9373</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02614325">10.1007/BF02614325</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1464775">1464775</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2794181">2794181</a>.</cite></li>

<li><cite id="CITEREFGondzioTerlaky1996" class="citation book cs1">Gondzio, Jacek; Terlaky, Tamás (1996). <a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~gondzio/CV/oxford.ps">"3 A computational view of interior point methods"</a>. In J. E. Beasley (ed.). <i>Advances in linear and integer programming</i>. Oxford Lecture Series in Mathematics and its Applications. Vol. 4. New York: Oxford University Press. pp. <span class="nowrap">103–</span>144. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1438311">1438311</a>. <a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~gondzio/CV/oxford.ps">Postscript file at website of Gondzio</a> and <a rel="nofollow" class="external text" href="http://www.cas.mcmaster.ca/~terlaky/files/dut-twi-94-73.ps.gz">at McMaster University website of Terlaky</a>.</cite></li>

<li><cite id="CITEREFMurty1983" class="citation book cs1">Murty, Katta G. (1983). <i>Linear programming</i>. New York: John Wiley & Sons, Inc. pp. xix+482. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-09725-9</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0720547">0720547</a>. (comprehensive reference to classical approaches).</cite></li>

<li>Evar D. Nering and <a href="Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a>, 1993, <i>Linear Programs and Related Problems</i>, Academic Press. (elementary)</li>

<li><cite id="CITEREFPadberg1999" class="citation book cs1">Padberg, M. (1999). <i>Linear Optimization and Extensions, Second Edition</i>. Springer-Verlag.</cite> (carefully written account of primal and dual simplex algorithms and projective algorithms, with an introduction to integer linear programming – featuring the <a href="Traveling_salesman_problem" class="mw-redirect" title="Traveling salesman problem">traveling salesman problem</a> for <a href="Odysseus" title="Odysseus">Odysseus</a>.)</li>

<li><cite id="CITEREFPapadimitriouSteiglitz" class="citation book cs1"><a href="Christos_H._Papadimitriou" class="mw-redirect" title="Christos H. Papadimitriou">Papadimitriou, Christos H.</a>; Steiglitz, Kenneth. <i>Combinatorial Optimization: Algorithms and Complexity</i> (Corrected republication with a new preface ed.). Dover.</cite> (computer science)</li>

<li><cite id="CITEREFTodd2002" class="citation journal cs1">Todd, Michael J. (February 2002). "The many facets of linear programming". <i>Mathematical Programming</i>. <b>91</b> (3): <span class="nowrap">417–</span>436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs101070100261">10.1007/s101070100261</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6464735">6464735</a>.</cite> (Invited survey, from the International Symposium on Mathematical Programming.)</li>

<li><cite id="CITEREFVanderbei2001" class="citation book cs1">Vanderbei, Robert J. (2001). <i>Linear Programming: Foundations and Extensions</i>. Springer Verlag.</cite></li>

<li><cite id="CITEREFVazirani2001" class="citation book cs1"><a href="Vijay_Vazirani" title="Vijay Vazirani">Vazirani, Vijay V.</a> (2001). <i>Approximation Algorithms</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65367-7</bdi>.</cite> (Computer science)</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>Dmitris Alevras and Manfred W. Padberg, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=RAUyB8NDHJwC">Linear Optimization and Extensions: Problems and Solutions</a></i>, Universitext, Springer-Verlag, 2001. (Problems from Padberg with solutions.)</li>

<li><cite id="CITEREFde_Bergvan_KreveldOvermarsSchwarzkopf2000" class="citation book cs1">de Berg, Mark; van Kreveld, Marc; <a href="Mark_Overmars" title="Mark Overmars">Overmars, Mark</a>; Schwarzkopf, Otfried (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/computationalgeo00berg"><i>Computational Geometry</i></a></span> (2nd revised ed.). <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65620-3</bdi>.</cite> Chapter 4: Linear Programming: pp. 63–94. Describes a randomized half-plane intersection algorithm for linear programming.</li>

<li><cite id="CITEREFMichael_R._Garey_and_David_S._Johnson1979" class="citation book cs1"><a href="Michael_R._Garey" class="mw-redirect" title="Michael R. Garey">Michael R. Garey</a> and <a href="David_S._Johnson" title="David S. Johnson">David S. Johnson</a> (1979). <a href="Computers_and_Intractability%3A_A_Guide_to_the_Theory_of_NP-Completeness" class="mw-redirect" title="Computers and Intractability: A Guide to the Theory of NP-Completeness"><i>Computers and Intractability: A Guide to the Theory of NP-Completeness</i></a>. W.H. Freeman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7167-1045-5</bdi>.</cite> A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory)</li>

<li><cite id="CITEREFGärtnerMatoušek2006" class="citation book cs1">Gärtner, Bernd; <a href="Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Matoušek, Jiří</a> (2006). <i>Understanding and Using Linear Programming</i>. Berlin: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-30697-8</bdi>.</cite> (elementary introduction for mathematicians and computer scientists)</li>

<li>Cornelis Roos, Tamás Terlaky, Jean-Philippe Vial, <i>Interior Point Methods for Linear Optimization</i>, Second Edition, Springer-Verlag, 2006. (Graduate level)</li>

<li><cite id="CITEREFAlexander_Schrijver2003" class="citation book cs1">Alexander Schrijver (2003). <i>Combinatorial optimization: polyhedra and efficiency</i>. Springer.</cite></li>

<li>Alexander Schrijver, <i>Theory of Linear and Integer Programming</i>. John Wiley & sons, 1998, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-98232-6</bdi> (mathematical)</li>

<li><cite id="CITEREFGerard_SierksmaYori_Zwols2015" class="citation book cs1">Gerard Sierksma; Yori Zwols (2015). <i>Linear and Integer Optimization: Theory and Practice</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-498-71016-9</bdi>.</cite>; with online solver: <a rel="nofollow" class="external free" href="https://online-optimizer.appspot.com/">https://online-optimizer.appspot.com/</a></li>

<li><cite id="CITEREFGerard_SierksmaDiptesh_Ghosh2010" class="citation book cs1">Gerard Sierksma; Diptesh Ghosh (2010). <i>Networks in Action; Text and Computer Exercises in Network Optimization</i>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-5512-8</bdi>.</cite> (linear optimization modeling)</li>

<li>H. P. Williams, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=YJRh0tOes7UC">Model Building in Mathematical Programming</a></i>, Fifth Edition, 2013. (Modeling)</li>

<li>Stephen J. Wright, 1997, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=oQdBzXhZeUkC">Primal-Dual Interior-Point Methods</a></i>, SIAM. (Graduate level)</li>

<li><a href="Yinyu_Ye" title="Yinyu Ye">Yinyu Ye</a>, 1997, <i>Interior Point Algorithms: Theory and Analysis</i>, Wiley. (Advanced graduate-level)</li>

<li><a href="G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a>, Chapters 1–3 and 6–7 in <i>Lectures on Polytopes</i>, Springer-Verlag, New York, 1994. (Geometry)</li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<ul><li><a rel="nofollow" class="external text" href="http://people.brunel.ac.uk/~mastjjb/jeb/or/lp.html">Guidance On Formulating LP Problems</a></li>

<li><a rel="nofollow" class="external text" href="http://glossary.computing.society.informs.org/">Mathematical Programming Glossary</a></li>

<li><a rel="nofollow" class="external text" href="https://lpsolve.sourceforge.net/4.0/LinearProgrammingFAQ.htm">The Linear Programming FAQ</a></li>

<li><a rel="nofollow" class="external text" href="http://plato.asu.edu/bench.html">Benchmarks For Optimisation Software</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics381" style="padding:3px"><table class="nowraplinks hlist mw-collapsible uncollapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="Iterative_method" title="Iterative method">methods</a>, and <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a></li>

<li><a href="Powell's_method" title="Powell's method">Powell's method</a></li>

<li><a href="Line_search" title="Line search">Line search</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li>

<li><a href="Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Trust_region" title="Trust region">Trust region</a></li>

<li><a href="Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li>

<li><a href="Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li>

<li><a href="Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li>

<li><a href="Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li>

<li><a href="Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient</a></li>

<li><a href="Mirror_descent" title="Mirror descent">Mirror</a></li>

<li><a href="Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li>

<li><a href="Powell's_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li>

<li><a href="Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrier_function" title="Barrier function">Barrier methods</a></li>

<li><a href="Penalty_method" title="Penalty method">Penalty methods</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li>

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li>

<li><a href="Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"> and<br><a href="Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink-fragment" href="#Interior_point">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_scaling" title="Affine scaling">Affine scaling</a></li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li>

<li><a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis-</a><a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li>

<li><a href="Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li>

<li><a href="Active-set_method" title="Active-set method">Active-set method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a>

<ul><li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul>

</div></td></tr></tbody></table><div>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dinic's_algorithm" title="Dinic's algorithm">Dinic</a></li>

<li><a href="Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li>

<li><a href="Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li>

<li><a href="Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metaheuristics381" style="font-size:114%;margin:0 4em"><a href="Metaheuristic" title="Metaheuristic">Metaheuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Hill_climbing" title="Hill climbing">Hill climbing</a></li>

<li><a href="Local_search_(optimization)" title="Local search (optimization)">Local search</a></li>

<li><a href="Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li>

<li><a href="Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li>

<li><a href="Tabu_search" title="Tabu search">Tabu search</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Complementarity_problems_and_algorithms137" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Complementarity_problems_and_algorithms137" style="font-size:114%;margin:0 4em"><a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)"> Complementarity problems and algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Complementarity Problems</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul>

<li><a href="Quadratic_programming" title="Quadratic programming"> Quadratic programming (QP)</a></li>

<li><a href="Linear_complementarity_problem" title="Linear complementarity problem"> Linear complementarity problem (LCP)</a></li>

<li><a href="Mixed_linear_complementarity_problem" title="Mixed linear complementarity problem"> Mixed linear (MLCP)</a></li>

<li><a href="Mixed_complementarity_problem" title="Mixed complementarity problem"> Mixed (MCP)</a></li>

<li><a href="Nonlinear_complementarity_problem" title="Nonlinear complementarity problem"> Nonlinear (NCP)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis</a>-<a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex</a> (<a href="George_Dantzig" title="George Dantzig">Dantzig</a>)</li>

<li><a href="Revised_simplex_algorithm" class="mw-redirect" title="Revised simplex algorithm">Revised simplex</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Lemke</a></li></ul>

</div></td></tr></tbody></table></div>

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