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<title>Sequential quadratic programming</title>

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

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<p><b>Sequential quadratic programming</b> (<b>SQP</b>) is an <a href="Iterative_method" title="Iterative method">iterative method</a> for <a href="Nonlinear_programming" title="Nonlinear programming">constrained nonlinear optimization</a>, also known as Lagrange-Newton method. SQP methods are used on <a href="Mathematics" title="Mathematics">mathematical</a> problems for which the <a href="Objective_function" class="mw-redirect" title="Objective function">objective function</a> and the constraints are twice <a href="Continuously_differentiable" class="mw-redirect" title="Continuously differentiable">continuously differentiable</a>, but not necessarily convex.

</p><p>SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to <a href="Newton's_method" title="Newton's method">Newton's method</a> for finding a point where the gradient of the objective vanishes. If the problem has only equality constraints, then the method is equivalent to applying <a href="Newton's_method" title="Newton's method">Newton's method</a> to the first-order optimality conditions, or <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a>, of the problem.

</p>

<div class="mw-heading mw-heading2"><h2 id="Algorithm_basics">Algorithm basics</h2></div>

<p>Consider a <a href="Nonlinear_programming" title="Nonlinear programming">nonlinear programming</a> problem of the form:

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<mtext>subject to</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}\min \limits _{x}&f(x)\\{\mbox{subject to}}&h(x)\geq 0\\&g(x)=0.\end{array}}}</annotation>

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</math></span><img src="./cb693c90b2dfbf7f4941794e017b03307b58c5b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.542ex; margin-bottom: -0.296ex; width:22.067ex; height:10.843ex;" alt="{\displaystyle {\begin{array}{rl}\min \limits _{x}&f(x)\\{\mbox{subject to}}&h(x)\geq 0\\&g(x)=0.\end{array}}}" 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 x\in \mathbb {R} ^{n}}">

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

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</math></span><img src="./c520ee2cb6ccf8a93c89a8c58a8378796bd52e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" loading="lazy"></span>, <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:\mathbb {R} ^{n}\rightarrow \mathbb {R} }">

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

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</math></span><img src="./ab54a3a84df609b7ac88c7f37cc10cbd5e5761fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }" loading="lazy"></span>, <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 h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m_{I}}}">

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<annotation encoding="application/x-tex">{\displaystyle h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m_{I}}}</annotation>

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</math></span><img src="./ddf29acb1430681fa9cdc60ce5d73577a82d2bc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.977ex; height:2.343ex;" alt="{\displaystyle h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m_{I}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m_{E}}}">

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<annotation encoding="application/x-tex">{\displaystyle g:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m_{E}}}</annotation>

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</p><p>The <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrangian</a> for this problem is<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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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}(x,\lambda ,\sigma )=f(x)+\lambda h(x)+\sigma g(x),}</annotation>

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</math></span><img src="./bf5a116bd3533934e83477f14bbfb3ab4319417b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.757ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}(x,\lambda ,\sigma )=f(x)+\lambda h(x)+\sigma g(x),}" 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 \lambda }">

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

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</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> are <a href="Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a>.

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<div class="mw-heading mw-heading3"><h3 id="The_equality_constrained_case">The equality constrained case</h3></div>

<p>If the problem does not have inequality constraints (that 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 m_{I}=0}">

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</math></span><img src="./58c9e1602f091e34f47f771900d07750b3227672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.362ex; height:2.509ex;" alt="{\displaystyle m_{I}=0}" loading="lazy"></span>), the first-order optimality conditions (aka <a href="Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">KKT conditions</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 \nabla {\mathcal {L}}(x,\sigma )=0}">

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<annotation encoding="application/x-tex">{\displaystyle \nabla {\mathcal {L}}(x,\sigma )=0}</annotation>

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</math></span><img src="./4f411056698a9a9883a8e03cf1589d33cdd27a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.303ex; height:2.843ex;" alt="{\displaystyle \nabla {\mathcal {L}}(x,\sigma )=0}" loading="lazy"></span> are a set of nonlinear equations that may be iteratively solved with <a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's Method</a>. Newton's method linearizes the KKT conditions at the current iterate <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 \left[x_{k},\sigma _{k}\right]^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle \left[x_{k},\sigma _{k}\right]^{T}}</annotation>

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</math></span><img src="./c10d6e703307d42b5f2d1af8c33b212ab3767667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.551ex; height:3.343ex;" alt="{\displaystyle \left[x_{k},\sigma _{k}\right]^{T}}" loading="lazy"></span>,

which provides the following expression for the Newton step <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 \left[d_{x},d_{\sigma }\right]^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle \left[d_{x},d_{\sigma }\right]^{T}}</annotation>

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</math></span><img src="./7927d0bf47d81a4c60731edab9884aa46016553b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.48ex; height:3.343ex;" alt="{\displaystyle \left[d_{x},d_{\sigma }\right]^{T}}" loading="lazy"></span>:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}d_{x}\\d_{\sigma }\end{bmatrix}}=-[\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})]^{-1}\nabla _{x}{\mathcal {L}}(x_{k},\sigma _{k})=-{\begin{bmatrix}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})&\nabla g(x_{k},\sigma _{k})\\\nabla g^{T}(x_{k},\sigma _{k})&0\end{bmatrix}}^{-1}{\begin{bmatrix}\nabla f(x_{k})+\sigma _{k}\nabla g(x_{k})\\g(x_{k})\end{bmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}d_{x}\\d_{\sigma }\end{bmatrix}}=-[\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})]^{-1}\nabla _{x}{\mathcal {L}}(x_{k},\sigma _{k})=-{\begin{bmatrix}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})&\nabla g(x_{k},\sigma _{k})\\\nabla g^{T}(x_{k},\sigma _{k})&0\end{bmatrix}}^{-1}{\begin{bmatrix}\nabla f(x_{k})+\sigma _{k}\nabla g(x_{k})\\g(x_{k})\end{bmatrix}}}</annotation>

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</math></span><img src="./6594296e6c90bd8497b32853e8e533ae7bb7157f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:99.854ex; height:7.009ex;" alt="{\displaystyle {\begin{bmatrix}d_{x}\\d_{\sigma }\end{bmatrix}}=-[\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})]^{-1}\nabla _{x}{\mathcal {L}}(x_{k},\sigma _{k})=-{\begin{bmatrix}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})&\nabla g(x_{k},\sigma _{k})\\\nabla g^{T}(x_{k},\sigma _{k})&0\end{bmatrix}}^{-1}{\begin{bmatrix}\nabla f(x_{k})+\sigma _{k}\nabla g(x_{k})\\g(x_{k})\end{bmatrix}}}" loading="lazy"></span>,

</p><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 \nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})}">

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</math></span><img src="./6ddd6dd7a31f484ba29af8a8d730b615ca72ba3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.33ex; height:3.009ex;" alt="{\displaystyle \nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})}" loading="lazy"></span> denotes the <a href="Hessian_matrix" title="Hessian matrix">Hessian matrix</a> of the Lagrangian, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{x}}">

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</math></span><img src="./70babe4f029cd1ab41fd5d912577b01e87017088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.381ex; height:2.509ex;" alt="{\displaystyle d_{\sigma }}" loading="lazy"></span> are the primal and dual displacements, respectively. Note that the Lagrangian Hessian is not explicitly inverted and a linear system is solved instead.

</p><p>When the Lagrangian Hessian <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 \nabla ^{2}{\mathcal {L}}(x_{k},\sigma _{k})}">

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<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}{\mathcal {L}}(x_{k},\sigma _{k})}</annotation>

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</math></span><img src="./cd5bf89f995f244c20dee60581d786ee0758b066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.272ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}{\mathcal {L}}(x_{k},\sigma _{k})}" loading="lazy"></span> is not <a href="Definite_matrix" title="Definite matrix">positive definite</a>, the Newton step may not exist or it may characterize a stationary point that is not a local minimum (but rather, a local maximum or a saddle point). In this case, the Lagrangian Hessian must be regularized, for example one can add a multiple of the identity to it such that the resulting matrix is positive definite.

</p><p>An alternative view for obtaining the primal-dual displacements is to construct and solve a local quadratic model of the original problem at the current iterate:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}\min \limits _{d_{x}}&f(x_{k})+\nabla f(x_{k})^{T}d_{x}+{\frac {1}{2}}d_{x}^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})d_{x}\\\mathrm {s.t.} &g(x_{k})+\nabla g(x_{k})^{T}d_{x}=0.\end{array}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}\min \limits _{d_{x}}&f(x_{k})+\nabla f(x_{k})^{T}d_{x}+{\frac {1}{2}}d_{x}^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})d_{x}\\\mathrm {s.t.} &g(x_{k})+\nabla g(x_{k})^{T}d_{x}=0.\end{array}}}</annotation>

</semantics>

</math></span><img src="./181451ab797142e8fda9bfc4985f72973ffa9627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:49.326ex; height:8.176ex;" alt="{\displaystyle {\begin{array}{rl}\min \limits _{d_{x}}&f(x_{k})+\nabla f(x_{k})^{T}d_{x}+{\frac {1}{2}}d_{x}^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\sigma _{k})d_{x}\\\mathrm {s.t.} &g(x_{k})+\nabla g(x_{k})^{T}d_{x}=0.\end{array}}}" loading="lazy"></span>

</p><p>The optimality conditions of this <a href="Quadratic_programming" title="Quadratic programming">quadratic problem</a> correspond to the linearized KKT conditions of the original problem.

Note that the term <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(x_{k})}">

<msub>

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

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</math></span><img src="./20d432970858c21563481c08f9c2bf50dffbeb06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle f(x_{k})}" loading="lazy"></span> in the expression above may be left out, since it is constant under the <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 \min \limits _{d}}">

</mrow>

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

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</math></span><img src="./e8220df6974561029c6513beac1d2567bc7b8761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.875ex; height:3.843ex;" alt="{\displaystyle \min \limits _{d}}" loading="lazy"></span> operator.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_inequality_constrained_case">The inequality constrained case</h3></div>

<p>In the presence of inequality constraints (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{I}>0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle m_{I}>0}</annotation>

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</math></span><img src="./76bef649b5515ada43e2d3d90665457d6c8f7f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.362ex; height:2.509ex;" alt="{\displaystyle m_{I}>0}" loading="lazy"></span>), we can naturally extend the definition of the local quadratic model introduced in the previous section:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\frac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &h(x_{k})+\nabla h(x_{k})^{T}d\geq 0\\&g(x_{k})+\nabla g(x_{k})^{T}d=0.\end{array}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\frac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &h(x_{k})+\nabla h(x_{k})^{T}d\geq 0\\&g(x_{k})+\nabla g(x_{k})^{T}d=0.\end{array}}}</annotation>

</semantics>

</math></span><img src="./56ed051403b7958a6b7d1da894fb13b0fafca0c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:50.473ex; height:11.509ex;" alt="{\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\frac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &h(x_{k})+\nabla h(x_{k})^{T}d\geq 0\\&g(x_{k})+\nabla g(x_{k})^{T}d=0.\end{array}}}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="The_SQP_algorithm">The SQP algorithm</h3></div>

<p>The SQP algorithm starts from the initial iterate <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_{0},\lambda _{0},\sigma _{0})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (x_{0},\lambda _{0},\sigma _{0})}</annotation>

</semantics>

</math></span><img src="./bb4f6b670867132c9f5bf4ef03f6db945f9ad459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.052ex; height:2.843ex;" alt="{\displaystyle (x_{0},\lambda _{0},\sigma _{0})}" loading="lazy"></span>. At each iteration, the QP subproblem is built and solved; the resulting Newton step direction <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [d_{x},d_{\lambda },d_{\sigma }]^{T}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle [d_{x},d_{\lambda },d_{\sigma }]^{T}}</annotation>

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</math></span><img src="./50c7e19eb8415cd34567d914135fc2ce314c369b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.913ex; height:3.176ex;" alt="{\displaystyle [d_{x},d_{\lambda },d_{\sigma }]^{T}}" loading="lazy"></span> is used to update current iterate:

</p><p><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 \left[x_{k+1},\lambda _{k+1},\sigma _{k+1}\right]^{T}=\left[x_{k},\lambda _{k},\sigma _{k}\right]^{T}+[d_{x},d_{\lambda },d_{\sigma }]^{T}.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \left[x_{k+1},\lambda _{k+1},\sigma _{k+1}\right]^{T}=\left[x_{k},\lambda _{k},\sigma _{k}\right]^{T}+[d_{x},d_{\lambda },d_{\sigma }]^{T}.}</annotation>

</semantics>

</math></span><img src="./7644096148d471e14401a7224b4d29d2bd399f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.859ex; height:3.343ex;" alt="{\displaystyle \left[x_{k+1},\lambda _{k+1},\sigma _{k+1}\right]^{T}=\left[x_{k},\lambda _{k},\sigma _{k}\right]^{T}+[d_{x},d_{\lambda },d_{\sigma }]^{T}.}" loading="lazy"></span>

</p><p>This process is repeated for <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 k=0,1,2,\ldots }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6d4b81efe1ce928c56c7b143cb2591f2100246c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.622ex; height:2.509ex;" alt="{\displaystyle k=0,1,2,\ldots }" loading="lazy"></span> until some convergence criterion is satisfied.

</p>

<div class="mw-heading mw-heading2"><h2 id="Practical_implementations">Practical implementations</h2></div>

<p>Practical implementations of the SQP algorithm are significantly more complex than its basic version above. To adapt SQP for real-world applications, the following challenges must be addressed:

</p>

<ul><li>The possibility of an infeasible QP subproblem.</li>

<li>QP subproblem yielding a bad step: one that either fails to reduce the target or increases constraints violation.</li>

<li>Breakdown of iterations due to significant deviation of the target/constraints from their quadratic/linear models.</li></ul>

<p>To overcome these challenges, various strategies are typically employed:

</p>

<ul><li>Use of merit functions, which assess progress towards a constrained solution, non-monotonic steps or filter methods.</li>

<li>Trust region or line search methods to manage deviations between the quadratic model and the actual target.</li>

<li>Special feasibility restoration phases to handle infeasible subproblems, or the use of L1-penalized subproblems to gradually decrease infeasibility</li></ul>

<p>These strategies can be combined in numerous ways, resulting in a diverse range of SQP methods.

</p>

<div class="mw-heading mw-heading2"><h2 id="Alternative_approaches">Alternative approaches</h2></div>

<ul><li><a href="Sequential_linear_programming" class="mw-redirect" title="Sequential linear programming">Sequential linear programming</a></li>

<li><a href="Sequential_linear-quadratic_programming" title="Sequential linear-quadratic programming">Sequential linear-quadratic programming</a></li>

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

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

<p>SQP methods have been implemented in well known numerical environments such as <a href="MATLAB" title="MATLAB">MATLAB</a> and <a href="GNU_Octave" title="GNU Octave">GNU Octave</a>. There also exist numerous software libraries, including open source:

</p>

<ul><li><a href="SciPy" title="SciPy">SciPy</a> (de facto standard for scientific Python) has scipy.optimize.minimize(method='SLSQP') solver.</li>

<li><a rel="nofollow" class="external text" href="https://nlopt.readthedocs.io/en/latest/">NLopt</a> (C/C++ implementation, with numerous interfaces including Julia, Python, R, MATLAB/Octave), implemented by Dieter Kraft as part of a package for optimal control, and modified by S. G. Johnson.<sup id="cite_ref-Kraft_2-0" class="reference"><a href="#cite_note-Kraft-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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></li>

<li><a href="ALGLIB" title="ALGLIB">ALGLIB</a> SQP solver (C++, C#, Java, Python API)</li>

<li><a rel="nofollow" class="external text" href="https://docs.acados.org/index.html">acados</a> (C with interfaces to Python, MATLAB, Simulink, Octave) implements a SQP method tailored to the problem structure arising in <a href="Optimal_control" title="Optimal control">optimal control</a>, but tackles also general nonlinear programs.</li></ul>

<p>and commercial

</p>

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

<li><a href="KNITRO" class="mw-redirect" title="KNITRO">KNITRO</a><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> (C, C++, C#, Java, Python, Julia, Fortran)</li>

<li><a href="NPSOL" title="NPSOL">NPSOL</a> (Fortran)</li>

<li><a href="SNOPT" title="SNOPT">SNOPT</a> (Fortran)</li>

<li><a href="NLPQL" class="mw-redirect" title="NLPQL">NLPQL</a> (Fortran)</li>

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

<li><a href="SuanShu_numerical_library" title="SuanShu numerical library">SuanShu</a> (Java)</li></ul>

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

<li><a href="Secant_method" title="Secant method">Secant method</a></li>

<li><a href="Model_predictive_control" title="Model predictive control">Model Predictive Control</a></li></ul>

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

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</style><div class="reflist">

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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}}

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

</style><cite id="CITEREFJorge_Nocedal_and_Stephen_J._Wright2006" class="citation book cs1">Jorge Nocedal and Stephen J. Wright (2006). <a rel="nofollow" class="external text" href="http://www.ece.northwestern.edu/~nocedal/book/num-opt.html"><i>Numerical Optimization</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-30303-1</bdi>.</cite></span>

</li>

<li id="cite_note-Kraft-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kraft_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKraft1994" class="citation journal cs1">Kraft, Dieter (Sep 1994). <a rel="nofollow" class="external text" href="https://github.com/scipy/scipy/blob/master/scipy/optimize/slsqp/slsqp_optmz.f">"Algorithm 733: TOMP–Fortran modules for optimal control calculations"</a>. <i>ACM Transactions on Mathematical Software</i>. <b>20</b> (3): <span class="nowrap">262–</span>281. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.512.2567">10.1.1.512.2567</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F192115.192124">10.1145/192115.192124</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:16077051">16077051</a><span class="reference-accessdate">. Retrieved <span class="nowrap">1 February</span> 2019</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://nlopt.readthedocs.io/en/latest/NLopt_Algorithms/#slsqp">"NLopt Algorithms: SLSQP"</a>. <i>Read the Docs</i>. July 1988<span class="reference-accessdate">. Retrieved <span class="nowrap">1 February</span> 2019</span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.artelys.com/tools/knitro_doc/2_userGuide/algorithms.html">KNITRO User Guide: Algorithms</a></span>

</li>

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

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

<ul><li><cite id="CITEREFBonnansGilbertLemaréchalSagastizábal2006" class="citation book cs1">Bonnans, J. Frédéric; Gilbert, J. Charles; <a href="Claude_Lemar%C3%A9chal" title="Claude Lemaréchal">Lemaréchal, Claude</a>; <a href="Claudia_Sagastiz%C3%A1bal" title="Claudia Sagastizábal">Sagastizábal, Claudia A.</a> (2006). <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/applications/book/978-3-540-35445-1"><i>Numerical optimization: Theoretical and practical aspects</i></a>. Universitext (Second revised ed. of translation of 1997 French ed.). Berlin: Springer-Verlag. pp. xiv+490. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-35447-5">10.1007/978-3-540-35447-5</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-35445-1</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2265882">2265882</a>.</cite></li>

<li><cite id="CITEREFJorge_Nocedal_and_Stephen_J._Wright2006" class="citation book cs1">Jorge Nocedal and Stephen J. Wright (2006). <a rel="nofollow" class="external text" href="http://www.ece.northwestern.edu/~nocedal/book/num-opt.html"><i>Numerical Optimization</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-30303-1</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="https://neos-guide.org/guide/algorithms/sqp/">Sequential Quadratic Programming at NEOS guide</a></li></ul>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed 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="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>

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

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

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

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

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

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

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

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

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

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

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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