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<title>Optimization problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Optimization problem</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Mathematical_optimization" title="Mathematical optimization">Mathematical optimization</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Engineering" title="Engineering">engineering</a>, <a href="Computer_science" title="Computer science">computer science</a> and <a href="Economics" title="Economics">economics</a>, an <b>optimization problem</b> is the <a href="Computational_problem" title="Computational problem">problem</a> of finding the <i>best</i> solution from all <a href="Feasible_solution" class="mw-redirect" title="Feasible solution">feasible solutions</a>.

</p><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>:

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<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 an optimal value from a <a href="Continuous_function" title="Continuous function">continuous function</a> must be found. They can include <a href="Constrained_optimization" title="Constrained optimization">constrained problems</a> and multimodal problems.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Search_space">Search space</h2></div>

<p>In the context of an optimization problem, the <b>search space</b> refers to the set of all possible points or solutions that satisfy the problem's constraints, targets, or goals.<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> These points represent the feasible solutions that can be evaluated to find the optimal solution according to the objective function. The search space is often defined by the domain of the function being optimized, encompassing all valid inputs that meet the problem's requirements.<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>

</p><p>The search space can vary significantly in size and complexity depending on the problem. For example, in a continuous optimization problem, the search space might be a multidimensional real-valued domain defined by bounds or constraints. In a discrete optimization problem, such as combinatorial optimization, the search space could consist of a finite set of permutations, combinations, or configurations.

</p><p>In some contexts, the term <i>search space</i> may also refer to the optimization of the domain itself, such as determining the most appropriate set of variables or parameters to define the problem. Understanding and effectively navigating the search space is crucial for designing efficient algorithms, as it directly influences the computational complexity and the likelihood of finding an optimal solution.

</p>

<div class="mw-heading mw-heading2"><h2 id="Continuous_optimization_problem">Continuous optimization problem</h2></div>

<p>The <i><a href="Canonical_form" title="Canonical form">standard form</a></i> of a <a href="Continuity_(mathematics)" class="mw-redirect" title="Continuity (mathematics)">continuous</a> optimization problem is<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<mi>minimize</mi>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\\&&&h_{j}(x)=0,\quad j=1,\dots ,p\end{aligned}}}</annotation>

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where

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<ul><li><span class="texhtml"><i>f</i> : <a href="Euclidean_space" title="Euclidean space">ℝ<sup><i>n</i></sup></a> → <a href="Real_numbers" class="mw-redirect" title="Real numbers">ℝ</a></span> is the <i><a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a></i> to be minimized over the <span class="texhtml mvar" style="font-style:italic;">n</span>-variable vector <span class="texhtml mvar" style="font-style:italic;">x</span>,</li>

<li><span class="texhtml"><i>g<sub>i</sub></i>(<i>x</i>) ≤ 0</span> are called <i>inequality <a href="Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a></i></li>

<li><span class="texhtml"><i>h<sub>j</sub></i>(<i>x</i>) = 0</span> are called <i>equality constraints</i>, and</li>

<p>If <span class="texhtml"><i>m</i> = <i>p</i> = 0</span>, the problem is an unconstrained optimization problem. By convention, the standard form defines a <b>minimization problem</b>. A <b>maximization problem</b> can be treated by <a href="Additive_inverse" title="Additive inverse">negating</a> the objective function.

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<div class="mw-heading mw-heading2"><h2 id="Combinatorial_optimization_problem">Combinatorial optimization problem</h2></div>

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

<p>Formally, a <a href="Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> problem <span class="texhtml mvar" style="font-style:italic;">A</span> is a quadruple <span class="texhtml">(<i>I</i>, <i>f</i>, <i>m</i>, <i>g</i>)</span>, where

</p>

<ul><li><span class="texhtml">I</span> is a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of instances;</li>

<li>given an instance <span class="texhtml"><i>x</i> ∈ <i>I</i></span>, <span class="texhtml"><i>f</i>(<i>x</i>)</span> is the set of feasible solutions;</li>

<li>given an instance <span class="texhtml mvar" style="font-style:italic;">x</span> and a feasible solution <span class="texhtml mvar" style="font-style:italic;">y</span> of <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml"><i>m</i>(<i>x</i>, <i>y</i>)</span> denotes the <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> of <span class="texhtml mvar" style="font-style:italic;">y</span>, which is usually a <a href="Positive_(mathematics)" class="mw-redirect" title="Positive (mathematics)">positive</a> <a href="Real_number" title="Real number">real</a>.</li>

<li><span class="texhtml mvar" style="font-style:italic;">g</span> is the goal function, and is either <span class="texhtml"><a href="Minimum_(mathematics)" class="mw-redirect" title="Minimum (mathematics)">min</a></span> or <span class="texhtml"><a href="Maximum_(mathematics)" class="mw-redirect" title="Maximum (mathematics)">max</a></span>.</li></ul>

<p>The goal is then to find for some instance <span class="texhtml mvar" style="font-style:italic;">x</span> an <i>optimal solution</i>, that is, a feasible solution <span class="texhtml mvar" style="font-style:italic;">y</span> with

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<annotation encoding="application/x-tex">{\displaystyle m(x,y)=g\left\{m(x,y'):y'\in f(x)\right\}.}</annotation>

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</p><p>For each combinatorial optimization problem, there is a corresponding <a href="Decision_problem" title="Decision problem">decision problem</a> that asks whether there is a feasible solution for some particular measure <span class="texhtml"><i>m</i><sub>0</sub></span>. For example, if there is a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> <span class="texhtml mvar" style="font-style:italic;">G</span> which contains vertices <span class="texhtml mvar" style="font-style:italic;">u</span> and <span class="texhtml mvar" style="font-style:italic;">v</span>, an optimization problem might be "find a path from <span class="texhtml mvar" style="font-style:italic;">u</span> to <span class="texhtml mvar" style="font-style:italic;">v</span> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <span class="texhtml mvar" style="font-style:italic;">u</span> to <span class="texhtml mvar" style="font-style:italic;">v</span> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

</p><p>In the field of <a href="Approximation_algorithm" title="Approximation algorithm">approximation algorithms</a>, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<sup id="cite_ref-Ausiello03_4-0" class="reference"><a href="#cite_note-Ausiello03-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Counting_problem_(complexity)" title="Counting problem (complexity)">Counting problem (complexity)</a> – Type of computational problem</li>

<li><a href="Design_Optimization" class="mw-redirect" title="Design Optimization">Design Optimization</a></li>

<li><a href="Ekeland's_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li>

<li><a href="Function_problem" title="Function problem">Function problem</a> – Type of computational problem</li>

<li><a href="Glove_problem" title="Glove problem">Glove problem</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a> – Discipline concerning the application of advanced analytical methods</li>

<li><a href="Satisficing" title="Satisficing">Satisficing</a> – Cognitive heuristic of searching for an acceptable decision − the optimum need not be found, just a "good enough" solution.</li>

<li><a href="Search_problem" title="Search problem">Search problem</a> – Class of computational problems</li>

<li><a href="Semi-infinite_programming" title="Semi-infinite programming">Semi-infinite programming</a></li></ul>

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://courses.cs.washington.edu/courses/cse473/06sp/GeneticAlgDemo/searchs.html">"Search Space"</a>. <i>courses.cs.washington.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-05-10</span></span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.lesswrong.com/w/search-space">"Search Space - LessWrong"</a>. <i>www.lesswrong.com</i>. 2020-09-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2025-05-10</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBoydVandenberghe2004" class="citation book cs1">Boyd, Stephen P.; Vandenberghe, Lieven (2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=143"><i>Convex Optimization</i></a> <span class="cs1-format">(pdf)</span>. Cambridge University Press. p. 129. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83378-3</bdi>.</cite></span>

</li>

<li id="cite_note-Ausiello03-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ausiello03_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFAusiello2003" class="citation cs2">Ausiello, Giorgio; et al. (2003), <i>Complexity and Approximation</i> (Corrected ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65431-5</bdi></cite></span>

</li>

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

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

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.ipctech.com/how-traffic-shaping-optimizes-network-bandwidth">"How Traffic Shaping Optimizes Network Bandwidth"</a>. <i>IPC</i>. 12 July 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">13 February</span> 2017</span>.</cite></li></ul>

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</style><div id="Convex_analysis_and_variational_analysis174" style="font-size:114%;margin:0 4em"><a href="Convex_analysis" title="Convex analysis">Convex analysis</a> and <a href="Variational_analysis" title="Variational analysis">variational analysis</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic 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="Convex_combination" title="Convex combination">Convex combination</a></li>

<li><a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Convex_set" title="Convex set">Convex set</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_convexity_topics" title="List of convexity topics">Topics (list)</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="Choquet_theory" title="Choquet theory">Choquet theory</a></li>

<li><a href="Convex_geometry" title="Convex geometry">Convex geometry</a></li>

<li><a href="Convex_metric_space" title="Convex metric space">Convex metric space</a></li>

<li><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></li>

<li><a href="Duality_(optimization)" title="Duality (optimization)">Duality</a></li>

<li><a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex topological vector space</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</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="Convex_conjugate" title="Convex conjugate">Convex conjugate</a></li>

<li><a href="Concave_function" title="Concave function">Concave</a></li>

<li>(<a href="Closed_convex_function" title="Closed convex function">Closed</a></li>

<li><a href="Logarithmically_convex_function" title="Logarithmically convex function">Logarithmically</a></li>

<li><a href="Proper_convex_function" title="Proper convex function">Proper</a></li>

<li><a href="Pseudoconvex_function" title="Pseudoconvex function">Pseudo-</a></li>

<li><a href="Quasiconvex_function" title="Quasiconvex function">Quasi-</a>) <a href="Convex_function" title="Convex function">Convex function</a></li>

<li><a href="Invex_function" title="Invex function">Invex function</a></li>

<li><a href="Legendre_transformation" title="Legendre transformation">Legendre transformation</a></li>

<li><a href="Semi-continuity" title="Semi-continuity">Semi-continuity</a></li>

<li><a href="Subderivative" title="Subderivative">Subderivative</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</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="Carath%C3%A9odory's_theorem_(convex_hull)" title="Carathéodory's theorem (convex hull)">Carathéodory's theorem</a></li>

<li><a href="Ekeland's_variational_principle" title="Ekeland's variational principle">Ekeland's variational principle</a></li>

<li><a href="Fenchel%E2%80%93Moreau_theorem" title="Fenchel–Moreau theorem">Fenchel–Moreau theorem</a></li>

<li><a href="Fenchel-Young_inequality" class="mw-redirect" title="Fenchel-Young inequality">Fenchel-Young inequality</a></li>

<li><a href="Jensen's_inequality" title="Jensen's inequality">Jensen's inequality</a></li>

<li><a href="Hermite%E2%80%93Hadamard_inequality" title="Hermite–Hadamard inequality">Hermite–Hadamard inequality</a></li>

<li><a href="Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman theorem</a></li>

<li><a href="Mazur's_lemma" title="Mazur's lemma">Mazur's lemma</a></li>

<li><a href="Shapley%E2%80%93Folkman_lemma" title="Shapley–Folkman lemma">Shapley–Folkman lemma</a></li>

<li><a href="Ursescu_theorem#Robinson–Ursescu_theorem" title="Ursescu theorem">Robinson–Ursescu</a></li>

<li><a href="Ursescu_theorem#Simons'_theorem" title="Ursescu theorem">Simons</a></li>

<li><a href="Ursescu_theorem" title="Ursescu theorem">Ursescu</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets</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="Convex_hull" title="Convex hull">Convex hull</a></li>

<li>(<a href="Orthogonally_convex_set" class="mw-redirect" title="Orthogonally convex set">Orthogonally</a>, <a href="Pseudoconvexity" title="Pseudoconvexity">Pseudo-</a>) <a href="Convex_set" title="Convex set">Convex set</a></li>

<li><a href="Effective_domain" title="Effective domain">Effective domain</a></li>

<li><a href="Epigraph_(mathematics)" title="Epigraph (mathematics)">Epigraph</a></li>

<li><a href="Hypograph_(mathematics)" title="Hypograph (mathematics)">Hypograph</a></li>

<li><a href="John_ellipsoid" title="John ellipsoid">John ellipsoid</a></li>

<li><a href="Radial_set" title="Radial set">Radial set</a>/<a href="Algebraic_interior" title="Algebraic interior">Algebraic interior</a></li>

<li><a href="Zonotope" class="mw-redirect" title="Zonotope">Zonotope</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Series</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="Convex_series#Types_of_subsets" title="Convex series">Convex series related</a> (<a href="Convex_series#Types_of_subsets" title="Convex series">(cs, lcs)-closed</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(cs, bcs)-complete</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(lower) ideally convex</a>, <a href="Convex_series#Types_of_subsets" title="Convex series">(H<i>x</i>)</a>, and <a href="Convex_series#Types_of_subsets" title="Convex series">(Hw<i>x</i>)</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Duality_(optimization)" title="Duality (optimization)">Duality</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="Dual_system" title="Dual system">Dual system</a></li>

<li><a href="Duality_gap" title="Duality gap">Duality gap</a></li>

<li><a href="Strong_duality" title="Strong duality">Strong duality</a></li>

<li><a href="Weak_duality" title="Weak duality">Weak duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications and related</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="Convexity_in_economics" title="Convexity in economics">Convexity in economics</a></li></ul>

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

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