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<title>Mathematical optimization</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Mathematical optimization</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Mathematical programming" redirects here. For the peer-reviewed journal, see <a href="Mathematical_Programming" title="Mathematical Programming">Mathematical Programming</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Optimization" and "Optimum" redirect here. For other uses, see <a href="Optimization_(disambiguation)" class="mw-disambig" title="Optimization (disambiguation)">Optimization (disambiguation)</a> and <a href="Optimum_(disambiguation)" class="mw-disambig" title="Optimum (disambiguation)">Optimum (disambiguation)</a>.</div>

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<p><b>Mathematical optimization</b> (alternatively spelled <i>optimisation</i>) or <b>mathematical programming</b> is the selection of a best element, with regard to some criteria, from some set of available alternatives.<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is generally divided into two subfields: <a href="Discrete_optimization" title="Discrete optimization">discrete optimization</a> and <a href="Continuous_optimization" title="Continuous optimization">continuous optimization</a>. Optimization problems arise in all quantitative disciplines from <a href="Computer_science" title="Computer science">computer science</a> and <a href="Engineering" title="Engineering">engineering</a><sup id="cite_ref-edo2021_3-0" class="reference"><a href="#cite_note-edo2021-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> to <a href="Operations_research" title="Operations research">operations research</a> and <a href="Economics" title="Economics">economics</a>, and the development of solution methods has been of interest in <a href="Mathematics" title="Mathematics">mathematics</a> for centuries.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>In the more general approach, an <a href="Optimization_problem" title="Optimization problem">optimization problem</a> consists of <a href="Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximizing or minimizing</a> a <a href="Function_of_a_real_variable" title="Function of a real variable">real function</a> by systematically choosing <a href="Argument_of_a_function" title="Argument of a function">input</a> values from within an allowed set and computing the <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Optimization_problems">Optimization problems</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Optimization_problem" title="Optimization problem">Optimization problem</a></div>

<p>Optimization problems can be divided into two categories, depending on whether the <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a> are <a href="Continuous_variable" class="mw-redirect" title="Continuous variable">continuous</a> or <a href="Discrete_variable" class="mw-redirect" title="Discrete variable">discrete</a>:

</p>

<ul><li>An optimization problem with discrete variables is known as a <i><a href="Discrete_optimization" title="Discrete optimization">discrete optimization</a></i>, in which an <a href="Mathematical_object" title="Mathematical object">object</a> such as an <a href="Integer" title="Integer">integer</a>, <a href="Permutation" title="Permutation">permutation</a> or <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> must be found from a <a href="Countable_set" title="Countable set">countable set</a>.</li>

<li>A problem with continuous variables is known as a <i><a href="Continuous_optimization" title="Continuous optimization">continuous optimization</a></i>, in which optimal arguments from a continuous set must be found. They can include <a href="Constrained_optimization" title="Constrained optimization">constrained problems</a> and multimodal problems.</li></ul>

<p>An optimization problem can be represented in the following way:

</p>

<dl><dd><i>Given:</i> a <a href="Function_(mathematics)" title="Function (mathematics)">function</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 f:A\rightarrow \mathbb {R} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle f:A\rightarrow \mathbb {R} }</annotation>

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</math></span><img src="./5736c080ddbf7c51d3504dbf211a8e848c0ffd91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.251ex; height:2.509ex;" alt="{\displaystyle f:A\rightarrow \mathbb {R} }" loading="lazy"></span> from some <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">A</span> to the <a href="Real_number" title="Real number">real numbers</a></dd>

<dd><i>Sought:</i> an element <span class="texhtml"><b>x</b><sub>0</sub> ∈ <i>A</i></span> such that <span class="texhtml"><i>f</i>(<b>x</b><sub>0</sub>) ≤ <i>f</i>(<b>x</b>)</span> for all <span class="texhtml"><b>x</b> ∈ <i>A</i></span> ("minimization") or such that <span class="texhtml"><i>f</i>(<b>x</b><sub>0</sub>) ≥ <i>f</i>(<b>x</b>)</span> for all <span class="texhtml"><b>x</b> ∈ <i>A</i></span> ("maximization").</dd></dl>

<p>Such a formulation is called an <b><a href="Optimization_problem" title="Optimization problem">optimization problem</a></b> or a <b>mathematical programming problem</b> (a term not directly related to <a href="Computer_programming" title="Computer programming">computer programming</a>, but still in use for example in <a href="Linear_programming" title="Linear programming">linear programming</a> – see <a href="#History">History</a> below). Many real-world and theoretical problems may be modeled in this general framework.

</p><p>Since the following is valid:

</p>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}</annotation>

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</math></span><img src="./b16621e3e03e9543826d663fced8f3f167561a0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.178ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}" loading="lazy"></span></dd></dl>

<p>it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too.

</p><p>Problems formulated using this technique in the fields of <a href="Physics" title="Physics">physics</a> may refer to the technique as <i><a href="Energy" title="Energy">energy</a> minimization</i>,<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> speaking of the value of the function <span class="texhtml mvar" style="font-style:italic;">f</span> as representing the energy of the <a href="System" title="System">system</a> being <a href="Mathematical_model" title="Mathematical model">modeled</a>. In <a href="Machine_learning" title="Machine learning">machine learning</a>, it is always necessary to continuously evaluate the quality of a data model by using a <a href="Loss_function" title="Loss function">cost function</a> where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.

</p><p>Typically, <span class="texhtml mvar" style="font-style:italic;">A</span> is some <a href="Subset" title="Subset">subset</a> of the <a href="Euclidean_space" title="Euclidean space">Euclidean space</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 \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

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

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</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>, often specified by a set of <i><a href="Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a></i>, equalities or inequalities that the members of <span class="texhtml mvar" style="font-style:italic;">A</span> have to satisfy. The <a href="Domain_of_a_function" title="Domain of a function">domain</a> <span class="texhtml mvar" style="font-style:italic;">A</span> of <span class="texhtml mvar" style="font-style:italic;">f</span> is called the <i>search space</i> or the <i>choice set</i>, while the elements of <span class="texhtml mvar" style="font-style:italic;">A</span> are called <i><a href="Candidate_solution" class="mw-redirect" title="Candidate solution">candidate solutions</a></i> or <i>feasible solutions</i>.

</p><p>The function <span class="texhtml mvar" style="font-style:italic;">f</span> is variously called an <i>objective function</i>, <i>criterion function</i>, <i><a href="Loss_function" title="Loss function">loss function</a></i>, <i>cost function</i> (minimization),<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <i>utility function</i> or <i>fitness function</i> (maximization), or, in certain fields, an <i>energy function</i> or <i>energy <a href="Functional_(mathematics)" title="Functional (mathematics)">functional</a></i>. A feasible solution that minimizes (or maximizes) the objective function is called an <i>optimal solution</i>.

</p><p>In mathematics, conventional optimization problems are usually stated in terms of minimization.

</p><p>A <i>local minimum</i> <span class="texhtml"><b>x</b>*</span> is defined as an element for which there exists some <span class="texhtml"><i>δ</i> > 0</span> such that

</p>

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<mtext>where</mtext>

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</mrow>

<msup>

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</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}</annotation>

</semantics>

</math></span><img src="./b2c45ce12a77abe61366d96dcffae4378f357976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.134ex; height:2.843ex;" alt="{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}" loading="lazy"></span></dd></dl>

<p>the expression <span class="texhtml"><i>f</i>(<b>x</b>*) ≤ <i>f</i>(<b>x</b>)</span> holds;

</p><p>that is to say, on some region around <span class="texhtml"><b>x</b>*</span> all of the function values are greater than or equal to the value at that element.

Local maxima are defined similarly.

</p><p>While a local minimum is at least as good as any nearby elements, a <a href="Global_minimum" class="mw-redirect" title="Global minimum">global minimum</a> is at least as good as every feasible element.

Generally, unless the objective function is <a href="Convex_function" title="Convex function">convex</a> in a minimization problem, there may be several local minima.

In a <a href="Convex_optimization" title="Convex optimization">convex problem</a>, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.

</p><p>A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. <a href="Global_optimization" title="Global optimization">Global optimization</a> is the branch of <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a> and <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.

</p>

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

<p>Optimization problems are often expressed with special notation. Here are some examples:

</p>

<div class="mw-heading mw-heading3"><h3 id="Minimum_and_maximum_value_of_a_function">Minimum and maximum value of a function</h3></div>

<p>Consider the following notation:

</p>

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</munder>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}</annotation>

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</math></span><img src="./e63529e0c09654712ebdefef0445a12017794b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.424ex; height:4.343ex;" alt="{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}" loading="lazy"></span></dd></dl>

<p>This denotes the minimum <a href="Value_(mathematics)" title="Value (mathematics)">value</a> of the objective function <span class="texhtml"><i>x</i><sup>2</sup> + 1</span>, when choosing <span class="texhtml mvar" style="font-style:italic;">x</span> from the set of <a href="Real_number" title="Real number">real numbers</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 \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>. The minimum value in this case is 1, occurring at <span class="texhtml"><i>x</i> = 0</span>.

</p><p>Similarly, the notation

</p>

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<annotation encoding="application/x-tex">{\displaystyle \max _{x\in \mathbb {R} }\;2x}</annotation>

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</math></span><img src="./a128d2c279975694f4c98533554a03f500780804.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.85ex; height:4.009ex;" alt="{\displaystyle \max _{x\in \mathbb {R} }\;2x}" loading="lazy"></span></dd></dl>

<p>asks for the maximum value of the objective function <span class="texhtml">2<i>x</i></span>, where <span class="texhtml mvar" style="font-style:italic;">x</span> may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "<a href="Infinity" title="Infinity">infinity</a>" or "<a href="Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a>".

</p>

<div class="mw-heading mw-heading3"><h3 id="Optimal_input_arguments">Optimal input arguments</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Arg_max" title="Arg max">Arg max</a></div>

<p>Consider the following notation:

</p>

</mrow>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}</annotation>

</semantics>

</math></span><img src="./39c071258fcecb43aaf25920b9833589bc35036c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.292ex; height:5.343ex;" alt="{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}" loading="lazy"></span></dd></dl>

<p>or equivalently

</p>

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<msup>

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<mtext>subject to:</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}</annotation>

</semantics>

</math></span><img src="./a2ecc7504e271936684860c5d738740d0841edac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.837ex; height:4.676ex;" alt="{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}" loading="lazy"></span></dd></dl>

<p>This represents the value (or values) of the <a href="Argument_of_a_function" title="Argument of a function">argument</a> <span class="texhtml mvar" style="font-style:italic;">x</span> in the <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">(−∞,−1]</span> that minimizes (or minimize) the objective function <span class="texhtml"><i>x</i><sup>2</sup> + 1</span> (the actual minimum value of that function is not what the problem asks for). In this case, the answer is <span class="texhtml"><i>x</i> = −1</span>, since <span class="texhtml"><i>x</i> = 0</span> is infeasible, that is, it does not belong to the <a href="Feasible_set" class="mw-redirect" title="Feasible set">feasible set</a>.

</p><p>Similarly,

</p>

</mrow>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}</annotation>

</semantics>

</math></span><img src="./ee3de833a31bfb33700e63b8d5df04565e899915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.196ex; height:4.343ex;" alt="{\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}" loading="lazy"></span></dd></dl>

<p>or equivalently

</p>

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<mrow>

</mrow>

</munder>

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<mtext>subject to:</mtext>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}</annotation>

</semantics>

</math></span><img src="./29cd276141e8e134eb156b2fa537ced38c14a5b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.398ex; height:4.676ex;" alt="{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}" loading="lazy"></span></dd></dl>

<p>represents the <span class="texhtml">{<i>x</i>, <i>y</i>}</span> pair (or pairs) that maximizes (or maximize) the value of the objective function <span class="texhtml"><i>x</i> cos <i>y</i></span>, with the added constraint that <span class="texhtml mvar" style="font-style:italic;">x</span> lie in the interval <span class="texhtml">[−5,5]</span> (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form <span class="texhtml">{5, 2<i>k</i><span class="texhtml mvar" style="font-style:italic;">π</span>}</span> and <span class="texhtml">{−5, (2<i>k</i> + 1)<span class="texhtml mvar" style="font-style:italic;">π</span>}</span>, where <span class="texhtml mvar" style="font-style:italic;">k</span> ranges over all <a href="Integer" title="Integer">integers</a>.

</p><p>Operators <span class="texhtml">arg min</span> and <span class="texhtml">arg max</span> are sometimes also written as <span class="texhtml">argmin</span> and <span class="texhtml">argmax</span>, and stand for <i>argument of the minimum</i> and <i>argument of the maximum</i>.

</p>

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

<p><a href="Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a> and <a href="Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> found calculus-based formulae for identifying optima, while <a href="Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> proposed iterative methods for moving towards an optimum.

</p><p>The term "<a href="Linear_programming" title="Linear programming">linear programming</a>" for certain optimization cases was due to <a href="George_Dantzig" title="George Dantzig">George B. Dantzig</a>, although much of the theory had been introduced by <a href="Leonid_Kantorovich" title="Leonid Kantorovich">Leonid Kantorovich</a> in 1939. (<i>Programming</i> in this context does not refer to <a href="Computer_programming" title="Computer programming">computer programming</a>, but comes from the use of <i>program</i> by the <a href="United_States" title="United States">United States</a> military to refer to proposed training and <a href="Logistics" title="Logistics">logistics</a> schedules, which were the problems Dantzig studied at that time.) Dantzig published the <a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a> in 1947, and also <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> and other researchers worked on the theoretical aspects of linear programming (like the theory of <a href="Linear_programming#Duality" title="Linear programming">duality</a>) around the same time.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Other notable researchers in mathematical optimization include the following:

</p>

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<ul><li><a href="Richard_Bellman" class="mw-redirect" title="Richard Bellman">Richard Bellman</a></li>

<li><a href="Dimitri_Bertsekas" title="Dimitri Bertsekas">Dimitri Bertsekas</a></li>

<li><a href="Michel_Bierlaire" title="Michel Bierlaire">Michel Bierlaire</a></li>

<li><a href="Stephen_P._Boyd" title="Stephen P. Boyd">Stephen P. Boyd</a></li>

<li><a href="Roger_Fletcher_(mathematician)" title="Roger Fletcher (mathematician)">Roger Fletcher</a></li>

<li><a href="Martin_Gr%C3%B6tschel" title="Martin Grötschel">Martin Grötschel</a></li>

<li><a href="Ronald_A._Howard" title="Ronald A. Howard">Ronald A. Howard</a></li>

<li><a href="Fritz_John" title="Fritz John">Fritz John</a></li>

<li><a href="Narendra_Karmarkar" title="Narendra Karmarkar">Narendra Karmarkar</a></li>

<li><a href="William_Karush" title="William Karush">William Karush</a></li>

<li><a href="Leonid_Khachiyan" title="Leonid Khachiyan">Leonid Khachiyan</a></li>

<li><a href="Bernard_Koopman" title="Bernard Koopman">Bernard Koopman</a></li>

<li><a href="Harold_Kuhn" class="mw-redirect" title="Harold Kuhn">Harold Kuhn</a></li>

<li><a href="L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a></li>

<li><a href="David_Luenberger" title="David Luenberger">David Luenberger</a></li>

<li><a href="Arkadi_Nemirovski" title="Arkadi Nemirovski">Arkadi Nemirovski</a></li>

<li><a href="Yurii_Nesterov" title="Yurii Nesterov">Yurii Nesterov</a></li>

<li><a href="Lev_Pontryagin" title="Lev Pontryagin">Lev Pontryagin</a></li>

<li><a href="R._Tyrrell_Rockafellar" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a></li>

<li><a href="Albert_W._Tucker" title="Albert W. Tucker">Albert Tucker</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Major_subfields">Major subfields</h2></div>

<ul><li><a href="Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a> studies the case when the objective function is <a href="Convex_function" title="Convex function">convex</a> (minimization) or <a href="Concave_function" title="Concave function">concave</a> (maximization) and the constraint set is <a href="Convex_set" title="Convex set">convex</a>. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.

<ul><li><a href="Linear_programming" title="Linear programming">Linear programming</a> (LP), a type of convex programming, studies the case in which the objective function <i>f</i> is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a <a href="Polyhedron" title="Polyhedron">polyhedron</a> or a <a href="Polytope" title="Polytope">polytope</a> if it is <a href="Bounded_set" title="Bounded set">bounded</a>.</li>

<li><a href="Second-order_cone_programming" title="Second-order cone programming">Second-order cone programming</a> (SOCP) is a convex program, and includes certain types of quadratic programs.</li>

<li><a href="Semidefinite_programming" title="Semidefinite programming">Semidefinite programming</a> (SDP) is a subfield of convex optimization where the underlying variables are <a href="Semidefinite" class="mw-redirect" title="Semidefinite">semidefinite</a> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>. It is a generalization of linear and convex quadratic programming.</li>

<li><a href="Conic_programming" class="mw-redirect" title="Conic programming">Conic programming</a> is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.</li>

<li><a href="Geometric_programming" title="Geometric programming">Geometric programming</a> is a technique whereby objective and inequality constraints expressed as <a href="Posynomials" class="mw-redirect" title="Posynomials">posynomials</a> and equality constraints as <a href="Monomials" class="mw-redirect" title="Monomials">monomials</a> can be transformed into a convex program.</li></ul></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a> studies linear programs in which some or all variables are constrained to take on <a href="Integer" title="Integer">integer</a> values. This is not convex, and in general much more difficult than regular linear programming.</li>

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a> allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.</li>

<li><a href="Fractional_programming" title="Fractional programming">Fractional programming</a> studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.</li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a> studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.</li>

<li><a href="Stochastic_programming" title="Stochastic programming">Stochastic programming</a> studies the case in which some of the constraints or parameters depend on <a href="Random_variable" title="Random variable">random variables</a>.</li>

<li><a href="Robust_optimization" title="Robust optimization">Robust optimization</a> is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.</li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a> is concerned with problems where the set of feasible solutions is discrete or can be reduced to a <a href="Discrete_mathematics" title="Discrete mathematics">discrete</a> one.</li>

<li><a href="Stochastic_optimization" title="Stochastic optimization">Stochastic optimization</a> is used with random (noisy) function measurements or random inputs in the search process.</li>

<li><a href="Infinite-dimensional_optimization" title="Infinite-dimensional optimization">Infinite-dimensional optimization</a> studies the case when the set of feasible solutions is a subset of an infinite-<a href="Dimension" title="Dimension">dimensional</a> space, such as a space of functions.</li>

<li><a href="Heuristic_(computer_science)" title="Heuristic (computer science)">Heuristics</a> and <a href="Metaheuristic" title="Metaheuristic">metaheuristics</a> make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.</li>

<li><a href="Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a> studies the case in which the objective function <i>f</i> is constant (this is used in <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, particularly in <a href="Automated_reasoning" title="Automated reasoning">automated reasoning</a>).

<ul><li><a href="Constraint_programming" title="Constraint programming">Constraint programming</a> is a programming paradigm wherein relations between variables are stated in the form of constraints.</li></ul></li>

<li>Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.</li>

<li><a href="Space_mapping" title="Space mapping">Space mapping</a> is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or <a href="Surrogate_model" title="Surrogate model">surrogate model</a>.</li></ul>

<p>In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

</p>

<ul><li><a href="Calculus_of_variations" title="Calculus of variations">Calculus of variations</a> is concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.</li>

<li><a href="Optimal_control" title="Optimal control">Optimal control</a> theory is a generalization of the calculus of variations which introduces control policies.</li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a> is the approach to solve the <a href="Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a> problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the <a href="Bellman_equation" title="Bellman equation">Bellman equation</a>.</li>

<li><a href="Mathematical_programming_with_equilibrium_constraints" title="Mathematical programming with equilibrium constraints">Mathematical programming with equilibrium constraints</a> is where the constraints include <a href="Variational_inequalities" class="mw-redirect" title="Variational inequalities">variational inequalities</a> or <a href="Complementarity_theory" title="Complementarity theory">complementarities</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Multi-objective_optimization">Multi-objective optimization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multi-objective_optimization" title="Multi-objective optimization">Multi-objective optimization</a></div>

<p>Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the <a href="Pareto_set" class="mw-redirect" title="Pareto set">Pareto set</a>. The curve created plotting weight against stiffness of the best designs is known as the <a href="Pareto_frontier" class="mw-redirect" title="Pareto frontier">Pareto frontier</a>.

</p><p>A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.

</p><p>The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.

</p><p>Multi-objective optimization problems have been generalized further into <a href="Vector_optimization" title="Vector optimization">vector optimization</a> problems where the (partial) ordering is no longer given by the Pareto ordering.

</p>

<div class="mw-heading mw-heading3"><h3 id="Multi-modal_or_global_optimization">Multi-modal or global optimization</h3></div>

<p>Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

</p><p>Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm.

</p><p>Common approaches to <a href="Global_optimization" title="Global optimization">global optimization</a> problems, where multiple local extrema may be present include <a href="Evolutionary_algorithm" title="Evolutionary algorithm">evolutionary algorithms</a>, <a href="Bayesian_optimization" title="Bayesian optimization">Bayesian optimization</a> and <a href="Simulated_annealing" title="Simulated annealing">simulated annealing</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Classification_of_critical_points_and_extrema">Classification of critical points and extrema</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Feasibility_problem">Feasibility problem</h3></div>

<p>The <i><a href="Satisfiability_problem" class="mw-redirect" title="Satisfiability problem">satisfiability problem</a></i>, also called the <i>feasibility problem</i>, is just the problem of finding any <a href="Feasible_solution" class="mw-redirect" title="Feasible solution">feasible solution</a> at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.

</p><p>Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to <a href="Relaxation_(approximation)" title="Relaxation (approximation)">relax</a> the feasibility conditions using a <a href="Slack_variable" title="Slack variable">slack variable</a>; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.

</p>

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

<p>The <a href="Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a> of <a href="Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.

</p>

<div class="mw-heading mw-heading3"><h3 id="Necessary_conditions_for_optimality">Necessary conditions for optimality</h3></div>

<p><a href="Fermat's_theorem_(stationary_points)" class="mw-redirect" title="Fermat's theorem (stationary points)">One of Fermat's theorems</a> states that optima of unconstrained problems are found at <a href="Stationary_point" title="Stationary point">stationary points</a>, where the first derivative or the gradient of the objective function is zero (see <a href="First_derivative_test" class="mw-redirect" title="First derivative test">first derivative test</a>). More generally, they may be found at <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a>, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.

</p><p>Optima of equality-constrained problems can be found by the <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> method. The optima of problems with equality and/or inequality constraints can be found using the '<a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a>'.

</p>

<div class="mw-heading mw-heading3"><h3 id="Sufficient_conditions_for_optimality">Sufficient conditions for optimality</h3></div>

<p>While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a>) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the <a href="Hessian_matrix#Bordered_Hessian" title="Hessian matrix">bordered Hessian</a> in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see '<a href="Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a>'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.

</p>

<div class="mw-heading mw-heading3"><h3 id="Sensitivity_and_continuity_of_optima">Sensitivity and continuity of optima</h3></div>

<p>The <a href="Envelope_theorem" title="Envelope theorem">envelope theorem</a> describes how the value of an optimal solution changes when an underlying <a href="Parameter" title="Parameter">parameter</a> changes. The process of computing this change is called <a href="Comparative_statics" title="Comparative statics">comparative statics</a>.

</p><p>The <a href="Maximum_theorem" title="Maximum theorem">maximum theorem</a> of <a href="Claude_Berge" title="Claude Berge">Claude Berge</a> (1963) describes the continuity of an optimal solution as a function of underlying parameters.

</p>

<div class="mw-heading mw-heading3"><h3 id="Calculus_of_optimization">Calculus of optimization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">Critical point (mathematics)</a>, <a href="Differential_calculus" title="Differential calculus">Differential calculus</a>, <a href="Gradient" title="Gradient">Gradient</a>, <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a>, <a href="Definite_matrix" title="Definite matrix">Definite matrix</a>, <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a>, <a href="Rademacher's_theorem" title="Rademacher's theorem">Rademacher's theorem</a>, <a href="Convex_function" title="Convex function">Convex function</a>, and <a href="Convex_analysis" title="Convex analysis">Convex analysis</a></div>

<p>For unconstrained problems with twice-differentiable functions, some <a href="Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> can be found by finding the points where the <a href="Gradient" title="Gradient">gradient</a> of the objective function is zero (that is, the stationary points). More generally, a zero <a href="Subgradient" class="mw-redirect" title="Subgradient">subgradient</a> certifies that a local minimum has been found for <a href="Convex_optimization" title="Convex optimization">minimization problems with convex</a> <a href="Convex_function" title="Convex function">functions</a> and other <a href="Rademacher's_theorem" title="Rademacher's theorem">locally</a> <a href="Lipschitz_function" class="mw-redirect" title="Lipschitz function">Lipschitz functions</a>, which meet in loss function minimization of the neural network. The positive-negative momentum estimation lets to avoid the local minimum and converges at the objective function global minimum.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p>Further, critical points can be classified using the <a href="Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">definiteness</a> of the <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a>: If the Hessian is <i>positive</i> definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of <a href="Saddle_point" title="Saddle point">saddle point</a>.

</p><p>Constrained problems can often be transformed into unconstrained problems with the help of <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multipliers</a>. <a href="Lagrangian_relaxation" title="Lagrangian relaxation">Lagrangian relaxation</a> can also provide approximate solutions to difficult constrained problems.

</p><p>When the objective function is a <a href="Convex_function" title="Convex function">convex function</a>, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as <a href="Interior-point_method" title="Interior-point method">interior-point methods</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Global_convergence">Global convergence</h3></div>

<p>More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on <a href="Line_search" title="Line search">line searches</a>, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses <a href="Trust_region" title="Trust region">trust regions</a>. Both line searches and trust regions are used in modern methods of <a href="Subgradient_method" title="Subgradient method">non-differentiable optimization</a>. Usually, a global optimizer is much slower than advanced local optimizers (such as <a href="BFGS_method" class="mw-redirect" title="BFGS method">BFGS</a>), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.

</p>

<div class="mw-heading mw-heading2"><h2 id="Computational_optimization_techniques">Computational optimization techniques</h2></div>

<p>To solve problems, researchers may use <a href="Algorithm" title="Algorithm">algorithms</a> that terminate in a finite number of steps, or <a href="Iterative_method" title="Iterative method">iterative methods</a> that converge to a solution (on some specified class of problems), or <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a> that may provide approximate solutions to some problems (although their iterates need not converge).

</p>

<div class="mw-heading mw-heading3"><h3 id="Optimization_algorithms">Optimization algorithms</h3></div>

<div role="note" class="hatnote navigation-not-searchable">For a more comprehensive list, see <a href="List_of_optimization_algorithms" class="mw-redirect" title="List of optimization algorithms">List of optimization algorithms</a>.</div>

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a> of <a href="George_Dantzig" title="George Dantzig">George Dantzig</a>, designed for <a href="Linear_programming" title="Linear programming">linear programming</a></li>

<li>Extensions of the simplex algorithm, designed for <a href="Quadratic_programming" title="Quadratic programming">quadratic programming</a> and for <a href="Linear-fractional_programming" title="Linear-fractional programming">linear-fractional programming</a></li>

<li>Variants of the simplex algorithm that are especially suited for <a href="Flow_network" title="Flow network">network optimization</a></li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial algorithms</a></li>

<li><a href="Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization algorithms</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Iterative_methods">Iterative methods</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Iterative_method" title="Iterative method">Iterative method</a></div>

<p>The <a href="Iterative_methods" class="mw-redirect" title="Iterative methods">iterative methods</a> used to solve problems of <a href="Nonlinear_programming" title="Nonlinear programming">nonlinear programming</a> differ according to whether they <a href="Subroutine" class="mw-redirect" title="Subroutine">evaluate</a> <a href="Hessian_matrix" title="Hessian matrix">Hessians</a>, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.

</p><p>One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.

</p>

<ul><li>Methods that evaluate Hessians (or approximate Hessians, using <a href="Finite_difference" title="Finite difference">finite differences</a>):

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

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a>: A Newton-based method for small-medium scale <i>constrained</i> problems. Some versions can handle large-dimensional problems.</li>

<li><a href="Interior_point_methods" class="mw-redirect" title="Interior point methods">Interior point methods</a>: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.</li></ul></li>

<li>Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):

<ul><li><a href="Coordinate_descent" title="Coordinate descent">Coordinate descent</a> methods: Algorithms which update a single coordinate in each iteration</li>

<li><a href="Conjugate_gradient_method" title="Conjugate gradient method">Conjugate gradient methods</a>: <a href="Iterative_method" title="Iterative method">Iterative methods</a> for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)</li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient descent</a> (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.</li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient methods</a>: An iterative method for large <a href="Rademacher's_theorem" title="Rademacher's theorem">locally</a> <a href="Lipschitz_continuity" title="Lipschitz continuity">Lipschitz functions</a> using <a href="Subgradient" class="mw-redirect" title="Subgradient">generalized gradients</a>. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.</li>

<li>Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for <a href="Convex_optimization" title="Convex optimization">convex minimization</a> problems (similar to conjugate gradient methods).</li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid method</a>: An iterative method for small problems with <a href="Quasiconvex_function" title="Quasiconvex function">quasiconvex</a> objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.</li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Conditional gradient method (Frank–Wolfe)</a> for approximate minimization of specially structured problems with <a href="Linear_constraints" class="mw-redirect" title="Linear constraints">linear constraints</a>, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).</li>

<li><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton methods</a>: Iterative methods for medium-large problems (e.g. N<1000).</li>

<li><a href="Simultaneous_perturbation_stochastic_approximation" title="Simultaneous perturbation stochastic approximation">Simultaneous perturbation stochastic approximation</a> (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.</li></ul></li>

<li>Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.

<ul><li><a href="Interpolation" title="Interpolation">Interpolation</a> methods</li>

<li><a href="Pattern_search_(optimization)" title="Pattern search (optimization)">Pattern search</a> methods, which have better convergence properties than the <a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead heuristic (with simplices)</a>, which is listed below.</li>

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

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">Heuristic algorithm</a></div>

<p>Besides (finitely terminating) <a href="Algorithm" title="Algorithm">algorithms</a> and (convergent) <a href="Iterative_method" title="Iterative method">iterative methods</a>, there are <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a>. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:

</p>

<ul><li><a href="Differential_evolution" title="Differential evolution">Differential evolution</a></li>

<li><a href="Dynamic_relaxation" title="Dynamic relaxation">Dynamic relaxation</a></li>

<li><a href="Evolutionary_algorithms" class="mw-redirect" title="Evolutionary algorithms">Evolutionary algorithms</a></li>

<li><a href="Genetic_algorithms" class="mw-redirect" title="Genetic algorithms">Genetic algorithms</a></li>

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

<li><a href="Memetic_algorithm" title="Memetic algorithm">Memetic algorithm</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead simplicial heuristic</a>: A popular heuristic for approximate minimization (without calling gradients)</li>

<li><a href="Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a></li>

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

<li><a href="Stochastic_tunneling" title="Stochastic tunneling">Stochastic tunneling</a></li>

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

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

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

<p>Problems in <a href="Rigid_body_dynamics" title="Rigid body dynamics">rigid body dynamics</a> (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> on a constraint manifold;<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a <a href="Linear_complementarity_problem" title="Linear complementarity problem">linear complementarity problem</a>, which can also be viewed as a QP (quadratic programming) problem.

</p><p>Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the <a href="Engineering_optimization" title="Engineering optimization">engineering optimization</a>, and another recent and growing subset of this field is <a href="Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">multidisciplinary design optimization</a>, which, while useful in many problems, has in particular been applied to <a href="Aerospace_engineering" title="Aerospace engineering">aerospace engineering</a> problems.

</p><p>This approach may be applied in cosmology and astrophysics.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Economics_and_finance">Economics and finance</h3></div>

<p><a href="Economics" title="Economics">Economics</a> is closely enough linked to optimization of <a href="Agent_(economics)" title="Agent (economics)">agents</a> that an influential definition relatedly describes economics <i>qua</i> science as the "study of human behavior as a relationship between ends and <a href="Scarce" class="mw-redirect" title="Scarce">scarce</a> means" with alternative uses.<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> Modern optimization theory includes traditional optimization theory but also overlaps with <a href="Game_theory" title="Game theory">game theory</a> and the study of economic <a href="Equilibrium_(economics)" class="mw-redirect" title="Equilibrium (economics)">equilibria</a>. The <i><a href="Journal_of_Economic_Literature" title="Journal of Economic Literature">Journal of Economic Literature</a></i> <a href="JEL_classification_codes" class="mw-redirect" title="JEL classification codes">codes</a> classify mathematical programming, optimization techniques, and related topics under <a href="JEL_classification_codes#Mathematical_and_quantitative_methods_JEL:_C_Subcategories" class="mw-redirect" title="JEL classification codes">JEL:C61-C63</a>.

</p><p>In microeconomics, the <a href="Utility_maximization_problem" title="Utility maximization problem">utility maximization problem</a> and its <a href="Dual_problem" class="mw-redirect" title="Dual problem">dual problem</a>, the <a href="Expenditure_minimization_problem" title="Expenditure minimization problem">expenditure minimization problem</a>, are economic optimization problems. Insofar as they behave consistently, <a href="Consumer" title="Consumer">consumers</a> are assumed to maximize their <a href="Utility" title="Utility">utility</a>, while <a href="Firm" class="mw-redirect" title="Firm">firms</a> are usually assumed to maximize their <a href="Profit_(economics)" title="Profit (economics)">profit</a>. Also, agents are often modeled as being <a href="Risk_aversion" title="Risk aversion">risk-averse</a>, thereby preferring to avoid risk. <a href="Asset_pricing" title="Asset pricing">Asset prices</a> are also modeled using optimization theory, though the underlying mathematics relies on optimizing <a href="Stochastic_process" title="Stochastic process">stochastic processes</a> rather than on static optimization. <a href="International_trade_theory" title="International trade theory">International trade theory</a> also uses optimization to explain trade patterns between nations. The optimization of <a href="Portfolio_(finance)" title="Portfolio (finance)">portfolios</a> is an example of multi-objective optimization in economics.

</p><p>Since the 1970s, economists have modeled dynamic decisions over time using <a href="Control_theory" title="Control theory">control theory</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, dynamic <a href="Search_theory" title="Search theory">search models</a> are used to study <a href="Labor_economics" class="mw-redirect" title="Labor economics">labor-market behavior</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> A crucial distinction is between deterministic and stochastic models.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="Macroeconomics" title="Macroeconomics">Macroeconomists</a> build <a href="Dynamic_stochastic_general_equilibrium" title="Dynamic stochastic general equilibrium">dynamic stochastic general equilibrium (DSGE)</a> models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Electrical_engineering">Electrical engineering</h3></div>

<p>Some common applications of optimization techniques in <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a> include <a href="Active_filter" title="Active filter">active filter</a> design,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> stray field reduction in superconducting magnetic energy storage systems, <a href="Space_mapping" title="Space mapping">space mapping</a> design of <a href="Microwave" title="Microwave">microwave</a> structures,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> handset antennas,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical <a href="Surrogate_model" title="Surrogate model">surrogate model</a> and <a href="Space_mapping" title="Space mapping">space mapping</a> methodologies since the discovery of <a href="Space_mapping" title="Space mapping">space mapping</a> in 1993.<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><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> Optimization techniques are also used in <a href="Power-flow_analysis" class="mw-redirect" title="Power-flow analysis">power-flow analysis</a>.<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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Civil_engineering">Civil engineering</h3></div>

<p>Optimization has been widely used in civil engineering. <a href="Construction_management" title="Construction management">Construction management</a> and <a href="Transportation_engineering" title="Transportation engineering">transportation engineering</a> are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,<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> <a href="Resource_leveling" title="Resource leveling">resource leveling</a>,<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><sup id="cite_ref-:0_27-0" class="reference"><a href="#cite_note-:0-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> <a href="Hydrological_optimization" title="Hydrological optimization">water resource allocation</a>, <a href="Traffic" title="Traffic">traffic</a> management<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> and schedule optimization.

</p>

<div class="mw-heading mw-heading3"><h3 id="Operations_research">Operations research</h3></div>

<p>Another field that uses optimization techniques extensively is <a href="Operations_research" title="Operations research">operations research</a>.<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> Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses <a href="Stochastic_programming" title="Stochastic programming">stochastic programming</a> to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and <a href="Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a> methods.

</p>

<div class="mw-heading mw-heading3"><h3 id="Control_engineering">Control engineering</h3></div>

<p>Mathematical optimization is used in much modern controller design. High-level controllers such as <a href="Model_predictive_control" title="Model predictive control">model predictive control</a> (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.

</p>

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

<p>Optimization techniques are regularly used in <a href="Geophysics" title="Geophysics">geophysical</a> parameter estimation problems. Given a set of geophysical measurements, e.g. <a href="Seismology" title="Seismology">seismic recordings</a>, it is common to solve for the <a href="Mineral_physics" title="Mineral physics">physical properties</a> and <a href="Structure_of_the_earth" class="mw-redirect" title="Structure of the earth">geometrical shapes</a> of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.

</p>

<div class="mw-heading mw-heading3"><h3 id="Molecular_modeling">Molecular modeling</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Molecular_modeling" class="mw-redirect" title="Molecular modeling">Molecular modeling</a></div>

<p>Nonlinear optimization methods are widely used in <a href="Conformational_analysis" class="mw-redirect" title="Conformational analysis">conformational analysis</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Computational_systems_biology">Computational systems biology</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Computational_systems_biology" class="mw-redirect" title="Computational systems biology">Computational systems biology</a></div>

<p>Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.<sup id="cite_ref-Papoutsakis_1984_30-0" class="reference"><a href="#cite_note-Papoutsakis_1984-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> <a href="Linear_programming" title="Linear programming">Linear programming</a> has been applied to calculate the maximal possible yields of fermentation products,<sup id="cite_ref-Papoutsakis_1984_30-1" class="reference"><a href="#cite_note-Papoutsakis_1984-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> and to infer gene regulatory networks from multiple microarray datasets<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> as well as transcriptional regulatory networks from high-throughput data.<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> <a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a> has been used to analyze energy metabolism<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> and has been applied to metabolic engineering and parameter estimation in biochemical pathways.<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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Machine_learning">Machine learning</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Machine_learning#Optimization" title="Machine learning">Machine learning § Optimization</a></div>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="List_of_optimization_software" title="List of optimization software">List of optimization software</a></div>

<ul><li><a href="Brachistochrone_curve" title="Brachistochrone curve">Brachistochrone curve</a></li>

<li><a href="Curve_fitting" title="Curve fitting">Curve fitting</a></li>

<li><a href="Deterministic_global_optimization" title="Deterministic global optimization">Deterministic global optimization</a></li>

<li><a href="Goal_programming" title="Goal programming">Goal programming</a></li>

<li><a href="List_of_publications_in_mathematics#Optimization" title="List of publications in mathematics">Important publications in optimization</a></li>

<li><a href="Least_squares" title="Least squares">Least squares</a></li>

<li><a href="Mathematical_Optimization_Society" title="Mathematical Optimization Society">Mathematical Optimization Society</a> (formerly Mathematical Programming Society)</li>

<li>Mathematical optimization algorithms</li>

<li>Mathematical optimization software</li>

<li><a href="Process_optimization" title="Process optimization">Process optimization</a></li>

<li><a href="Simulation-based_optimization" title="Simulation-based optimization">Simulation-based optimization</a></li>

<li><a href="Test_functions_for_optimization" title="Test functions for optimization">Test functions for optimization</a></li>

<li><a href="Vehicle_routing_problem" title="Vehicle routing problem">Vehicle routing problem</a></li></ul>

</div>

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

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<li id="cite_note-Papoutsakis_1984-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Papoutsakis_1984_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Papoutsakis_1984_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFPapoutsakis1984" class="citation journal cs1">Papoutsakis, Eleftherios Terry (February 1984). "Equations and calculations for fermentations of butyric acid bacteria". <i>Biotechnology and Bioengineering</i>. <b>26</b> (2): <span class="nowrap">174–</span>187. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1984BiotB..26..174P">1984BiotB..26..174P</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbit.260260210">10.1002/bit.260260210</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/0006-3592">0006-3592</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18551704">18551704</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:25023799">25023799</a>.</cite></span>

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<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><cite id="CITEREFWangJoshiZhangXu2006" class="citation journal cs1">Wang, Yong; Joshi, Trupti; Zhang, Xiang-Sun; Xu, Dong; Chen, Luonan (2006-07-24). "Inferring gene regulatory networks from multiple microarray datasets". <i>Bioinformatics</i>. <b>22</b> (19): <span class="nowrap">2413–</span>2420. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtl396">10.1093/bioinformatics/btl396</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/1460-2059">1460-2059</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16864593">16864593</a>.</cite></span>

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<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><cite id="CITEREFWangWangZhangChen2007" class="citation journal cs1">Wang, Rui-Sheng; Wang, Yong; Zhang, Xiang-Sun; Chen, Luonan (2007-09-22). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtm465">"Inferring transcriptional regulatory networks from high-throughput data"</a>. <i>Bioinformatics</i>. <b>23</b> (22): <span class="nowrap">3056–</span>3064. <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/10.1093%2Fbioinformatics%2Fbtm465">10.1093/bioinformatics/btm465</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/1460-2059">1460-2059</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17890736">17890736</a>.</cite></span>

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<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><cite id="CITEREFVoPaul_LeePalsson2007" class="citation journal cs1">Vo, Thuy D.; Paul Lee, W.N.; Palsson, Bernhard O. (May 2007). "Systems analysis of energy metabolism elucidates the affected respiratory chain complex in Leigh's syndrome". <i>Molecular Genetics and Metabolism</i>. <b>91</b> (1): <span class="nowrap">15–</span>22. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ymgme.2007.01.012">10.1016/j.ymgme.2007.01.012</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/1096-7192">1096-7192</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17336115">17336115</a>.</cite></span>

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<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><cite id="CITEREFMendesKell1998" class="citation journal cs1"><a href="Pedro_Pedrosa_Mendes" class="mw-redirect" title="Pedro Pedrosa Mendes">Mendes, P.</a>; Kell, D. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2F14.10.869">"Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation"</a>. <i>Bioinformatics</i>. <b>14</b> (10): <span class="nowrap">869–</span>883. <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/10.1093%2Fbioinformatics%2F14.10.869">10.1093/bioinformatics/14.10.869</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/1367-4803">1367-4803</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9927716">9927716</a>.</cite></span>

</li>

</ol></div></div>

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

<ul><li><cite class="citation book cs1"><a href="Stephen_P._Boyd" title="Stephen P. Boyd">Boyd, Stephen P.</a>; Vandenberghe, Lieven (2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/"><i>Convex Optimization</i></a>. Cambridge: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-83378-7</bdi>.</cite></li>

<li><cite class="citation book cs1">Gill, P. E.; Murray, W.; <a href="Margaret_H._Wright" title="Margaret H. Wright">Wright, M. H.</a> (1982). <i>Practical Optimization</i>. London: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-283952-8</bdi>.</cite></li>

<li><cite class="citation book cs1"><a href="Jon_Lee_(mathematician)" title="Jon Lee (mathematician)">Lee, Jon</a> (2004). <i>A First Course in Combinatorial Optimization</i>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-01012-8</bdi>.</cite></li>

<li><cite class="citation book cs1"><a href="Jorge_Nocedal" title="Jorge Nocedal">Nocedal, Jorge</a>; Wright, Stephen J. (2006). <a rel="nofollow" class="external text" href="http://www.ece.northwestern.edu/~nocedal/book/num-opt.html"><i>Numerical Optimization</i></a> (2nd ed.). Berlin: Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-30303-0</bdi>.</cite></li>

<li>G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (eds.): <i>Optimization</i>, Elsevier, (1989).</li>

<li>Stanislav Walukiewicz：<i>Integer Programming</i>, Springer，ISBN 978-9048140688, (1990).</li>

<li>R. Fletcher: <i>Practical Methods of Optimization</i>, 2nd Ed., Wiley, (2000).</li>

<li>Panos M. Pardalos：<i>Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems</i>, Springer，ISBN 978-1-44194829-8, (2000).</li>

<li>Xiaoqi Yang, K. L. Teo, Lou Caccetta (Eds.)：<i>Optimization Methods and Applications</i>，Springer, ISBN 978-0-79236866-3, (2001).</li>

<li>Panos M. Pardalos, and Mauricio G. C. Resende(Eds.)：<i>Handbook of Applied Optimization</i>、Oxford Univ Pr on Demand, ISBN 978-0-19512594-8, (2002).</li>

<li>Wil Michiels, Emile Aarts, and Jan Korst: <i>Theoretical Aspects of Local Search</i>, Springer, ISBN 978-3-64207148-5, (2006).</li>

<li>Der-San Chen, Robert G. Batson, and Yu Dang： <i>Applied Integer Programming: Modeling and Solution</i>，Wiley，ISBN 978-0-47037306-4, (2010).</li>

<li>Mykel J. Kochenderfer and Tim A. Wheeler: <i>Algorithms for Optimization</i>, The MIT Press, ISBN 978-0-26203942-0, (2019).</li>

<li>Vladislav Bukshtynov: <i>Optimization: Success in Practice</i>, CRC Press (Taylor & Francis), ISBN 978-1-03222947-8, (2023) .</li>

<li>Rosario Toscano: <i>Solving Optimization Problems with the Heuristic Kalman Algorithm: New Stochastic Methods</i>, Springer, ISBN 978-3-031-52458-5 (2024).</li>

<li>Immanuel M. Bomze, Tibor Csendes, Reiner Horst and Panos M. Pardalos: <i>Developments in Global Optimization</i>, Kluwer Academic, ISBN 978-1-4419-4768-0 (2010).</li></ul>

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

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</div>

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://plato.asu.edu/guide.html">"Decision Tree for Optimization Software"</a>.</cite> Links to optimization source codes</li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~neum/glopt.html">"Global optimization"</a>.</cite></li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://see.stanford.edu/Course/EE364A">"EE364a: Convex Optimization I"</a>. <i>Course from Stanford University</i>.</cite></li>

<li><cite id="CITEREFVaroquaux" class="citation web cs1">Varoquaux, Gaël. <a rel="nofollow" class="external text" href="https://scipy-lectures.org/advanced/mathematical_optimization/index.html">"Mathematical Optimization: Finding Minima of Functions"</a>.</cite></li></ul>

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</style><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em">: <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%"><a href="Linear_programming" title="Linear programming">Linear</a> 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 href="Linear_programming#Interior_point" title="Linear programming">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>

<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="Major_mathematics_areas1069" style="padding:3px"><table class="nowraplinks hlist 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="Major_mathematics_areas1069" style="font-size:114%;margin:0 4em">Major <a href="Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a>

<ul><li><a href="Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li>

<li><a href="Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li>

<li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations</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="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Set_theory" title="Set theory">Set theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Algebra" title="Algebra">Algebra</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="Abstract_algebra" title="Abstract algebra">Abstract</a></li>

<li><a href="Commutative_algebra" title="Commutative algebra">Commutative</a></li>

<li><a href="Elementary_algebra" title="Elementary algebra">Elementary</a></li>

<li><a href="Group_theory" title="Group theory">Group theory</a></li>

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

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li>

<li><a href="Universal_algebra" title="Universal algebra">Universal</a></li>

<li><a href="Homological_algebra" title="Homological algebra">Homological</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</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="Calculus" title="Calculus">Calculus</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Complex_analysis" title="Complex analysis">Complex analysis</a></li>

<li><a href="Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete_mathematics" title="Discrete mathematics">Discrete</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="Combinatorics" title="Combinatorics">Combinatorics</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li>

<li><a href="Order_theory" title="Order theory">Order theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Geometry" title="Geometry">Geometry</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="Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li>

<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>

<li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li>

<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Number_theory" title="Number theory">Number theory</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="Arithmetic" title="Arithmetic">Arithmetic</a></li>

<li><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li>

<li><a href="Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li>

<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Topology" title="Topology">Topology</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="General_topology" title="General topology">General</a></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a></li>

<li><a href="Geometric_topology" title="Geometric topology">Geometric</a></li>

<li><a href="Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Applied_mathematics" title="Applied mathematics">Applied</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="Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li>

<li><a href="Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li>

<li><a href="Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Systems_science" title="Systems science">Systems science</a>

<ul><li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computational_mathematics" title="Computational mathematics">Computational</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="Computer_science" title="Computer science">Computer science</a></li>

<li><a href="Theory_of_computation" title="Theory of computation">Theory of computation</a></li>

<li><a href="Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li>

<li><a href="Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</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="Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a>

<ul><li><a href="List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li>

<li><a href="Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li>

<li><a href="List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li>

<li><a href="Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li>

<li><a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li>

<li><a href="Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw external" title="commons:Category:Mathematics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Systems_engineering118" style="padding:3px"><table class="nowraplinks hlist 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="Systems_engineering118" style="font-size:114%;margin:0 4em"><a href="Systems_engineering" title="Systems engineering">Systems engineering</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subfields</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="Aerospace_engineering" title="Aerospace engineering">Aerospace engineering</a></li>

<li><a href="Biological_systems_engineering" title="Biological systems engineering">Biological systems engineering</a></li>

<li><a href="Cognitive_systems_engineering" title="Cognitive systems engineering">Cognitive systems engineering</a></li>

<li><a href="Configuration_management" title="Configuration management">Configuration management</a></li>

<li><a href="Earth_systems_engineering_and_management" title="Earth systems engineering and management">Earth systems engineering and management</a></li>

<li><a href="Electrical_engineering" title="Electrical engineering">Electrical engineering</a></li>

<li><a href="Enterprise_systems_engineering" title="Enterprise systems engineering">Enterprise systems engineering</a></li>

<li><a href="Health_systems_engineering" title="Health systems engineering">Health systems engineering</a></li>

<li><a href="Performance_engineering" title="Performance engineering">Performance engineering</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability engineering</a></li>

<li><a href="Safety_engineering" title="Safety engineering">Safety engineering</a></li>

<li><a href="Stanford_torus" title="Stanford torus">Sociocultural Systems Engineering</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Processes</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="Requirements_engineering" title="Requirements engineering">Requirements engineering</a></li>

<li><a href="Functional_Analysis_and_Allocation" class="mw-redirect" title="Functional Analysis and Allocation">Functional Analysis and Allocation</a></li>

<li><a href="System_integration" title="System integration">System integration</a></li>

<li><a href="Verification_and_validation" title="Verification and validation">Verification and validation</a></li>

<li><a href="Design_review" title="Design review">Design review</a></li>

<li><a href="System_of_systems_engineering" title="System of systems engineering">System of systems engineering</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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="Business_process" title="Business process">Business process</a></li>

<li><a href="Fault_tolerance" title="Fault tolerance">Fault tolerance</a></li>

<li><a href="System" title="System">System</a></li>

<li><a href="System_lifecycle" class="mw-redirect" title="System lifecycle">System lifecycle</a></li>

<li><a href="V-model" title="V-model">V-Model</a></li>

<li><a href="Systems_development_life_cycle" title="Systems development life cycle">Systems development life cycle</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</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="Decision-making" title="Decision-making">Decision-making</a></li>

<li><a href="Function_model" title="Function model">Function modelling</a></li>

<li><a href="Quality_function_deployment" title="Quality function deployment">Quality function deployment</a></li>

<li><a href="Spare_part" title="Spare part">Spare part</a></li>

<li><a href="System_dynamics" title="System dynamics">System dynamics</a></li>

<li><a href="Systems_Modeling_Language" class="mw-redirect" title="Systems Modeling Language">Systems Modeling Language</a></li>

<li><a href="Systems_analysis" title="Systems analysis">Systems analysis</a></li>

<li><a href="Systems_modeling" title="Systems modeling">Systems modeling</a></li>

<li><a href="Work_breakdown_structure" title="Work breakdown structure">Work breakdown structure</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</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="James_S._Albus" title="James S. Albus">James S. Albus</a></li>

<li><a href="Ruzena_Bajcsy" title="Ruzena Bajcsy">Ruzena Bajcsy</a></li>

<li><a href="Benjamin_S._Blanchard" title="Benjamin S. Blanchard">Benjamin S. Blanchard</a></li>

<li><a href="Wernher_von_Braun" title="Wernher von Braun">Wernher von Braun</a></li>

<li><a href="Kathleen_Carley" title="Kathleen Carley">Kathleen Carley</a></li>

<li><a href="Harold_Chestnut" title="Harold Chestnut">Harold Chestnut</a></li>

<li><a href="Wolt_Fabrycky" title="Wolt Fabrycky">Wolt Fabrycky</a></li>

<li><a href="Barbara_Grosz" class="mw-redirect" title="Barbara Grosz">Barbara Grosz</a></li>

<li><a href="Arthur_David_Hall_III" title="Arthur David Hall III">Arthur David Hall III</a></li>

<li><a href="Derek_Hitchins" title="Derek Hitchins">Derek Hitchins</a></li>

<li><a href="Robert_E._Machol" title="Robert E. Machol">Robert E. Machol</a></li>

<li><a href="Radhika_Nagpal" title="Radhika Nagpal">Radhika Nagpal</a></li>

<li><a href="Simon_Ramo" title="Simon Ramo">Simon Ramo</a></li>

<li><a href="Joseph_Francis_Shea" title="Joseph Francis Shea">Joseph Francis Shea</a></li>

<li><a href="Katia_Sycara" title="Katia Sycara">Katia Sycara</a></li>

<li><a href="Manuela_M._Veloso" title="Manuela M. Veloso">Manuela M. Veloso</a></li>

<li><a href="John_N._Warfield" title="John N. Warfield">John N. Warfield</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related fields</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="Control_engineering" title="Control engineering">Control engineering</a></li>

<li><a href="Computer_engineering" title="Computer engineering">Computer engineering</a></li>

<li><a href="Industrial_engineering" title="Industrial engineering">Industrial engineering</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li>

<li><a href="Project_management" title="Project management">Project management</a></li>

<li><a href="Quality_management" title="Quality management">Quality management</a></li>

<li><a href="Risk_management" title="Risk management">Risk management</a></li>

<li><a href="Software_engineering" title="Software engineering">Software engineering</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li>Category</li></ul>

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

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