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<title>Constraint (mathematics)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Constraint (mathematics)</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For constraints in Hamiltonian mechanics, see <a href="Constraint_(classical_mechanics)" class="mw-redirect" title="Constraint (classical mechanics)">constraint (classical mechanics)</a>, <a href="First_class_constraint" class="mw-redirect" title="First class constraint">first class constraint</a>, <a href="Primary_constraint" title="Primary constraint">primary constraint</a>, and <a href="Holonomic_constraint" class="mw-redirect" title="Holonomic constraint">holonomic constraint</a>.</div>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>constraint</b> is a condition of an <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a> problem that the solution must satisfy. There are several types of constraints—primarily <a href="Equality_(mathematics)" title="Equality (mathematics)">equality</a> constraints, <a href="Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a> constraints, and <a href="Integer_programming" title="Integer programming">integer constraints</a>. The set of <a href="Candidate_solution" class="mw-redirect" title="Candidate solution">candidate solutions</a> that satisfy all constraints is called the <a href="Feasible_set" class="mw-redirect" title="Feasible set">feasible set</a>.<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>

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

<p>The following is a simple optimization problem:

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<annotation encoding="application/x-tex">{\displaystyle \min f(\mathbf {x} )=x_{1}^{2}+x_{2}^{4}}</annotation>

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</math></span><img src="./abf2aa7b3052af741fae0e359e5144695956b6c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:3.176ex;" alt="{\displaystyle \min f(\mathbf {x} )=x_{1}^{2}+x_{2}^{4}}" loading="lazy"></span></dd></dl>

<p>subject to

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

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</math></span><img src="./ab1b9b1f483f7025385bf7073c0463cfd57fecf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{1}\geq 1}" loading="lazy"></span></dd></dl>

<p>and

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

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</math></span><img src="./3e13651ca80d3aa6e15f3a4e59e740449b35ea39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.292ex; height:2.509ex;" alt="{\displaystyle x_{2}=1,}" loading="lazy"></span></dd></dl>

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

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

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</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> denotes the vector (<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>).

</p><p>In this example, the first line defines the function to be minimized (called the <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a>, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are <a href="Hard_constraint" class="mw-redirect" title="Hard constraint">hard constraints</a>, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions.

</p><p>Without the constraints, the solution would be (0,0), where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )}">

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

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</math></span><img src="./e41ea95e6949bf4cef6426116364ba87e0fdcd60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.499ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} )}" loading="lazy"></span> has the lowest value. But this solution does not satisfy the constraints. The solution of the <a href="Constrained_optimization" title="Constrained optimization">constrained optimization</a> problem stated above is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =(1,1)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =(1,1)}</annotation>

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</math></span><img src="./2a7403d874e944599b1c85b49b3e9612557e20bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.678ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =(1,1)}" loading="lazy"></span>, which is the point with the smallest value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )}">

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

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

<ul><li>If an inequality constraint holds with <i>equality</i> at the optimal point, the constraint is said to be <b><style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style><span class="vanchor"><span class="vanchor-text">binding</span></span></b>, as the point <i>cannot</i> be varied in the direction of the constraint even though doing so would improve the value of the objective function.</li>

<li>If an inequality constraint holds as a <i>strict inequality</i> at the optimal point (that is, does not hold with equality), the constraint is said to be <b><span class="vanchor"><span class="vanchor-text">non-binding</span></span></b>, as the point <i>could</i> be varied in the direction of the constraint, although it would not be optimal to do so. Under certain conditions, as for example in convex optimization, if a constraint is non-binding, the optimization problem would have the same solution even in the absence of that constraint.</li>

<li>If a constraint is not satisfied at a given point, the point is said to be <b><a href="Feasible_region" title="Feasible region">infeasible</a></b>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Hard_and_soft_constraints">Hard and soft constraints</h2></div>

<p>If the problem mandates that the constraints be satisfied, as in the above discussion, the constraints are sometimes referred to as <i>hard constraints</i>. However, in some problems, called <a href="Constraint_satisfaction_problem#Flexible_CSPs" title="Constraint satisfaction problem">flexible constraint satisfaction problems</a>, it is preferred but not required that certain constraints be satisfied; such non-mandatory constraints are known as <i><a href="Constraint_optimization" class="mw-redirect" title="Constraint optimization">soft constraints</a></i>. Soft constraints arise in, for example, <a href="Preference-based_planning" title="Preference-based planning">preference-based planning</a>. In a <a href="MAX-CSP" class="mw-redirect" title="MAX-CSP">MAX-CSP</a> problem, a number of constraints are allowed to be violated, and the quality of a solution is measured by the number of satisfied constraints.

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<div class="mw-heading mw-heading2"><h2 id="Global_constraints">Global constraints</h2></div>

<p>Global constraints<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> are constraints representing a specific relation on a number of variables, taken altogether. Some of them, such as the <a rel="nofollow" class="external text" href="https://sofdem.github.io/gccat/gccat/Calldifferent.html"><code>alldifferent</code></a> constraint, can be rewritten as a conjunction of atomic constraints in a simpler language: the <code>alldifferent</code> constraint holds on <i>n</i> variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}...x_{n}}">

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

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</math></span><img src="./bdde0534ab35272277fd8b11ae834225da6fc438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.034ex; height:2.009ex;" alt="{\displaystyle x_{1}...x_{n}}" loading="lazy"></span>, and is satisfied if the variables take values which are pairwise different. It is semantically equivalent to the conjunction of inequalities <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\neq x_{2},x_{1}\neq x_{3}...,x_{2}\neq x_{3},x_{2}\neq x_{4}...x_{n-1}\neq x_{n}}">

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

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</math></span><img src="./3991ce3942aedfa252c77ddd91a825e07f8e8a35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.066ex; height:2.676ex;" alt="{\displaystyle x_{1}\neq x_{2},x_{1}\neq x_{3}...,x_{2}\neq x_{3},x_{2}\neq x_{4}...x_{n-1}\neq x_{n}}" loading="lazy"></span>. Other global constraints extend the expressivity of the constraint framework. In this case, they usually capture a typical structure of combinatorial problems. For instance, the <code><a href="Regular_constraint" title="Regular constraint">regular</a></code> constraint expresses that a sequence of variables is accepted by a <a href="Deterministic_finite_automaton" title="Deterministic finite automaton">deterministic finite automaton</a>.

</p><p>Global constraints are used<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> to simplify the modeling of <a href="Constraint_satisfaction_problems" class="mw-redirect" title="Constraint satisfaction problems">constraint satisfaction problems</a>, to extend the expressivity of constraint languages, and also to improve the <a href="Constraint_programming" title="Constraint programming">constraint resolution</a>: indeed, by considering the variables altogether, infeasible situations can be seen earlier in the solving process. Many of the global constraints are referenced into an <a rel="nofollow" class="external text" href="https://sofdem.github.io/gccat/">online catalog.</a>

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<ul><li><a href="Constraint_algebra" title="Constraint algebra">Constraint algebra</a></li>

<li><a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a></li>

<li><a href="Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a></li>

<li><a href="Level_set" title="Level set">Level set</a></li>

<li><a href="Linear_programming" title="Linear programming">Linear programming</a></li>

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

<li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Satisfiability_modulo_theories" title="Satisfiability modulo theories">Satisfiability modulo theories</a></li></ul>

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

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFTakayama1985" class="citation book cs1">Takayama, Akira (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalecon00taka"><i>Mathematical Economics</i></a></span> (2nd ed.). New York: Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalecon00taka/page/61">61</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-31498-4</bdi>.</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 id="CITEREFRossiVan_BeekWalsh2006" class="citation book cs1">Rossi, Francesca; Van Beek, Peter; Walsh, Toby (2006). "7". <i>Handbook of constraint programming</i> (1st ed.). Amsterdam: Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080463643</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/162587579">162587579</a>.</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="CITEREFRossi2003" class="citation book cs1">Rossi, Francesca (2003). <i>Principles and Practice of Constraint Programming CP 2003 00 : 9th International Conference, CP 2003, Kinsale, Ireland, September 29 October 3, 2003. Proceedings</i>. Berlin: Springer-Verlag Berlin Heidelberg. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540451938</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/771185146">771185146</a>.</cite></span>

</li>

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

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

<ul><li><cite id="CITEREFBeveridgeSchechter1970" class="citation book cs1">Beveridge, Gordon S. G.; Schechter, Robert S. (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TfhVXlWtOPQC&pg=PA5">"Essential Features in Optimization"</a>. <i>Optimization: Theory and Practice</i>. New York: McGraw-Hill. pp. <span class="nowrap">5–</span>8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-005128-3</bdi>.</cite></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="http://www.neos-guide.org/non-lp-faq">Nonlinear programming FAQ</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191030134942/https://neos-guide.org/non-lp-faq">Archived</a> 2019-10-30 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="http://glossary.computing.society.informs.org/">Mathematical Programming Glossary</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100328165516/http://glossary.computing.society.informs.org/">Archived</a> 2010-03-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul></div><div><div class="zim-footer">

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