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<span class="path"><a href="index.html">FOSSEE Optimization Toolbox</a> >> <a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a> > fmincon</span>
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<div class="refnamediv"><h1 class="refname">fmincon</h1>
<p class="refpurpose">Solves a multi-variable constrainted optimization problem</p></div>
<div class="refsynopsisdiv"><h3 class="title">Calling Sequence</h3>
<div class="synopsis"><pre><span class="default">xopt</span><span class="default"> = </span><span class="functionid">fmincon</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">fmincon</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">fmincon</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">fmincon</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">nlc</span><span class="default">)</span>
<span class="default">xopt</span><span class="default"> = </span><span class="functionid">fmincon</span><span class="default">(</span><span class="default">f</span><span class="default">,</span><span class="default">x0</span><span class="default">,</span><span class="default">A</span><span class="default">,</span><span class="default">b</span><span class="default">,</span><span class="default">Aeq</span><span class="default">,</span><span class="default">beq</span><span class="default">,</span><span class="default">lb</span><span class="default">,</span><span class="default">ub</span><span class="default">,</span><span class="default">nlc</span><span class="default">,</span><span class="default">options</span><span class="default">)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">] = </span><span class="functionid">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">]= </span><span class="functionid">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">]= </span><span class="functionid">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">,</span><span class="default">lambda</span><span class="default">]=</span><span class="functionid">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">,</span><span class="default">lambda</span><span class="default">,</span><span class="default">gradient</span><span class="default">]=</span><span class="functionid">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">xopt</span><span class="default">,</span><span class="default">fopt</span><span class="default">,</span><span class="default">exitflag</span><span class="default">,</span><span class="default">output</span><span class="default">,</span><span class="default">lambda</span><span class="default">,</span><span class="default">gradient</span><span class="default">,</span><span class="default">hessian</span><span class="default">]=</span><span class="functionid">fmincon</span><span class="default">(.....)</span></pre></div></div>
<div class="refsection"><h3 class="title">Parameters</h3>
<dl><dt><span class="term">f :</span>
<dd><p class="para">a function, representing the objective function of the problem</p></dd></dt>
<dt><span class="term">x0 :</span>
<dd><p class="para">a vector of doubles, containing the starting values of variables of size (1 X n) or (n X 1) where 'n' is the number of Variables</p></dd></dt>
<dt><span class="term">A :</span>
<dd><p class="para">a matrix of doubles, containing the coefficients of linear inequality constraints of size (m X n) where 'm' is the number of linear inequality constraints</p></dd></dt>
<dt><span class="term">b :</span>
<dd><p class="para">a vector of doubles, related to 'A' and containing the the Right hand side equation of the linear inequality constraints of size (m X 1)</p></dd></dt>
<dt><span class="term">Aeq :</span>
<dd><p class="para">a matrix of doubles, containing the coefficients of linear equality constraints of size (m1 X n) where 'm1' is the number of linear equality constraints</p></dd></dt>
<dt><span class="term">beq :</span>
<dd><p class="para">a vector of doubles, related to 'Aeq' and containing the the Right hand side equation of the linear equality constraints of size (m1 X 1)</p></dd></dt>
<dt><span class="term">lb :</span>
<dd><p class="para">a vector of doubles, containing the lower bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of Variables</p></dd></dt>
<dt><span class="term">ub :</span>
<dd><p class="para">a vector of doubles, containing the upper bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of Variables</p></dd></dt>
<dt><span class="term">nlc :</span>
<dd><p class="para">a function, representing the Non-linear Constraints functions(both Equality and Inequality) of the problem. It is declared in such a way that non-linear inequality constraints are defined first as a single row vector (c), followed by non-linear equality constraints as another single row vector (ceq). Refer Example for definition of Constraint function.</p></dd></dt>
<dt><span class="term">options :</span>
<dd><p class="para">a list, containing the option for user to specify. See below for details.</p></dd></dt>
<dt><span class="term">xopt :</span>
<dd><p class="para">a vector of doubles, cointating the computed solution of the optimization problem</p></dd></dt>
<dt><span class="term">fopt :</span>
<dd><p class="para">a scalar of double, containing the the function value at x</p></dd></dt>
<dt><span class="term">exitflag :</span>
<dd><p class="para">a scalar of integer, containing the flag which denotes the reason for termination of algorithm. See below for details.</p></dd></dt>
<dt><span class="term">output :</span>
<dd><p class="para">a structure, containing the information about the optimization. See below for details.</p></dd></dt>
<dt><span class="term">lambda :</span>
<dd><p class="para">a structure, containing the Lagrange multipliers of lower bound, upper bound and constraints at the optimized point. See below for details.</p></dd></dt>
<dt><span class="term">gradient :</span>
<dd><p class="para">a vector of doubles, containing the Objective's gradient of the solution.</p></dd></dt>
<dt><span class="term">hessian :</span>
<dd><p class="para">a matrix of doubles, containing the Lagrangian's hessian of the solution.</p></dd></dt></dl></div>
<div class="refsection"><h3 class="title">Description</h3>
<p class="para">Search the minimum of a constrained optimization problem specified by :
Find the minimum of f(x) such that</p>
<p class="para"><span><img src='./_LaTeX_fmincon.xml_1.png' style='position:relative;top:63px;width:221px;height:134px'/></span></p>
<p class="para">The routine calls Ipopt for solving the Constrained Optimization problem, Ipopt is a library written in C++.</p>
<p class="para">The options allows the user to set various parameters of the Optimization problem.
It should be defined as type "list" and contains the following fields.
<ul class="itemizedlist"><li>Syntax : options= list("MaxIter", [---], "CpuTime", [---], "GradObj", ---, "Hessian", ---, "GradCon", ---);</li>
<li>MaxIter : a Scalar, containing the Maximum Number of Iteration that the solver should take.</li>
<li>CpuTime : a Scalar, containing the Maximum amount of CPU Time that the solver should take.</li>
<li>GradObj : a function, representing the gradient function of the Objective in Vector Form.</li>
<li>Hessian : a function, representing the hessian function of the Lagrange in Symmetric Matrix Form with Input parameters x, Objective factor and Lambda. Refer Example for definition of Lagrangian Hessian function.</li>
<li>GradCon : a function, representing the gradient of the Non-Linear Constraints (both Equality and Inequality) of the problem. It is declared in such a way that gradient of non-linear inequality constraints are defined first as a separate Matrix (cg of size m2 X n or as an empty), followed by gradient of non-linear equality constraints as a separate Matrix (ceqg of size m2 X n or as an empty) where m2 & m3 are number of non-linear inequality and equality constraints respectively.</li>
<li>Default Values : options = list("MaxIter", [3000], "CpuTime", [600]);</li></ul></p>
<p class="para">The exitflag allows to know the status of the optimization which is given back by Ipopt.
<ul class="itemizedlist"><li>exitflag=0 : Optimal Solution Found</li>
<li>exitflag=1 : Maximum Number of Iterations Exceeded. Output may not be optimal.</li>
<li>exitflag=2 : Maximum amount of CPU Time exceeded. Output may not be optimal.</li>
<li>exitflag=3 : Stop at Tiny Step.</li>
<li>exitflag=4 : Solved To Acceptable Level.</li>
<li>exitflag=5 : Converged to a point of local infeasibility.</li></ul></p>
<p class="para">For more details on exitflag see the ipopt documentation, go to http://www.coin-or.org/Ipopt/documentation/</p>
<p class="para">The output data structure contains detailed informations about the optimization process.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>output.Iterations: The number of iterations performed during the search</li>
<li>output.Cpu_Time: The total cpu-time spend during the search</li>
<li>output.Objective_Evaluation: The number of Objective Evaluations performed during the search</li>
<li>output.Dual_Infeasibility: The Dual Infeasiblity of the final soution</li>
<li>output.Message: The output message for the problem</li></ul></p>
<p class="para">The lambda data structure contains the Lagrange multipliers at the end
of optimization. In the current version the values are returned only when the the solution is optimal.
It has type "struct" and contains the following fields.
<ul class="itemizedlist"><li>lambda.lower: The Lagrange multipliers for the lower bound constraints.</li>
<li>lambda.upper: The Lagrange multipliers for the upper bound constraints.</li>
<li>lambda.eqlin: The Lagrange multipliers for the linear equality constraints.</li>
<li>lambda.ineqlin: The Lagrange multipliers for the linear inequality constraints.</li>
<li>lambda.eqnonlin: The Lagrange multipliers for the non-linear equality constraints.</li>
<li>lambda.ineqnonlin: The Lagrange multipliers for the non-linear inequality constraints.</li></ul></p>
<p class="para"></p></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Find x in R^2 such that it minimizes:</span>
<span class="scilabcomment">//f(x)= -x1 -x2/3</span>
<span class="scilabcomment">//x0=[0,0]</span>
<span class="scilabcomment">//constraint-1 (c1): x1 + x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 2</span>
<span class="scilabcomment">//constraint-2 (c2): x1 + x2/4 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//constraint-3 (c3): x1 - x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 2</span>
<span class="scilabcomment">//constraint-4 (c4): -x1/4 - x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//constraint-5 (c5): -x1 - x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= -1</span>
<span class="scilabcomment">//constraint-6 (c6): -x1 + x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 2</span>
<span class="scilabcomment">//constraint-7 (c7): x1 + x2 = 2</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilaboperator">-</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">-</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">/</span><span class="scilabnumber">3</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Starting point, linear constraints and variable bounds</span>
<span class="scilabid">x0</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span> <span class="scilabdefault">,</span> <span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span> <span class="scilabdefault">;</span> <span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilaboperator">/</span><span class="scilabnumber">4</span> <span class="scilabdefault">;</span> <span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span> <span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilaboperator">/</span><span class="scilabnumber">4</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span> <span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span> <span class="scilabdefault">;</span> <span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilabnumber">2</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">Aeq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">nlc</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Gradient of objective function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">fGrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilaboperator">/</span><span class="scilabnumber">3</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Hessian of lagrangian</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">lHess</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">obj</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabinputoutputargs">obj</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">GradObj</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">fGrad</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">Hessian</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">lHess</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Calling Ipopt</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabdefault">,</span><span class="scilabid">grad</span><span class="scilabdefault">,</span><span class="scilabid">hessian</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">fmincon</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span><span class="scilabdefault">,</span> <span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabdefault">,</span><span class="scilabid">nlc</span><span class="scilabdefault">,</span><span class="scilabid">options</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Find x in R^3 such that it minimizes:</span>
<span class="scilabcomment">//f(x)= x1*x2 + x2*x3</span>
<span class="scilabcomment">//x0=[0.1 , 0.1 , 0.1]</span>
<span class="scilabcomment">//constraint-1 (c1): x1^2 - x2^2 + x3^2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 2</span>
<span class="scilabcomment">//constraint-2 (c2): x1^2 + x2^2 + x3^2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 10</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Starting point, linear constraints and variable bounds</span>
<span class="scilabid">x0</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0.1</span> <span class="scilabdefault">,</span> <span class="scilabnumber">0.1</span> <span class="scilabdefault">,</span> <span class="scilabnumber">0.1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">Aeq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Nonlinear constraints</span>
<span class="scilabfkeyword">function</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">c</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">ceq</span><span class="scilabopenclose">]</span><span class="scilaboperator">=</span><span class="scilabfunctionid">nlc</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">c</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">-</span> <span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">-</span> <span class="scilabnumber">2</span> <span class="scilabdefault">,</span> <span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span> <span class="scilaboperator">-</span> <span class="scilabnumber">10</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">ceq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Gradient of objective function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">fGrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Hessian of the Lagrange Function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">lHess</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">obj</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabinputoutputargs">obj</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Gradient of Non-Linear Constraints</span>
<span class="scilabfkeyword">function</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">cg</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">ceqg</span><span class="scilabopenclose">]</span><span class="scilaboperator">=</span><span class="scilabfunctionid">cGrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">cg</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span> <span class="scilabdefault">,</span> <span class="scilaboperator">-</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span> <span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span> <span class="scilabdefault">;</span> <span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span> <span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span> <span class="scilabdefault">,</span> <span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">ceqg</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">GradObj</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">fGrad</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">Hessian</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">lHess</span><span class="scilabdefault">,</span><span class="scilabstring">"</span><span class="scilabstring">GradCon</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">cGrad</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Calling Ipopt</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">fmincon</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span><span class="scilabdefault">,</span> <span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabdefault">,</span><span class="scilabfunctionid">nlc</span><span class="scilabdefault">,</span><span class="scilabid">options</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//The below problem is an unbounded problem:</span>
<span class="scilabcomment">//Find x in R^3 such that it minimizes:</span>
<span class="scilabcomment">//f(x)= -(x1^2 + x2^2 + x3^2)</span>
<span class="scilabcomment">//x0=[0.1 , 0.1 , 0.1]</span>
<span class="scilabcomment">// x1 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 0</span>
<span class="scilabcomment">// x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 0</span>
<span class="scilabcomment">// x3 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 0</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilaboperator">-</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Starting point, linear constraints and variable bounds</span>
<span class="scilabid">x0</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0.1</span> <span class="scilabdefault">,</span> <span class="scilabnumber">0.1</span> <span class="scilabdefault">,</span> <span class="scilabnumber">0.1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">Aeq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">500</span><span class="scilabopenclose">]</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Calling Ipopt</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabdefault">,</span><span class="scilabid">grad</span><span class="scilabdefault">,</span><span class="scilabid">hessian</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">fmincon</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span><span class="scilabdefault">,</span> <span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabid">options</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Examples</h3>
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//The below problem is an infeasible problem:</span>
<span class="scilabcomment">//Find x in R^3 such that in minimizes:</span>
<span class="scilabcomment">//f(x)=x1*x2 + x2*x3</span>
<span class="scilabcomment">//x0=[1,1,1]</span>
<span class="scilabcomment">//constraint-1 (c1): x1^2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//constraint-2 (c2): x1^2 + x2^2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//constraint-3 (c3): x3^2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//constraint-4 (c4): x1^3 = 0.5</span>
<span class="scilabcomment">//constraint-5 (c5): x2^2 + x3^2 = 0.75</span>
<span class="scilabcomment">// 0 </span><span class="scilabcomment"><</span><span class="scilabcomment">= x1 </span><span class="scilabcomment"><</span><span class="scilabcomment">=0.6</span>
<span class="scilabcomment">// 0.2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= x2 </span><span class="scilabcomment"><</span><span class="scilabcomment">= inf</span>
<span class="scilabcomment">// -inf </span><span class="scilabcomment"><</span><span class="scilabcomment">= x3 </span><span class="scilabcomment"><</span><span class="scilabcomment">= 1</span>
<span class="scilabcomment">//Objective function to be minimised</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">f</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Starting point, linear constraints and variable bounds</span>
<span class="scilabid">x0</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">A</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">b</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">Aeq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">beq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">lb</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span> <span class="scilabnumber">0.2</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabconstants">%inf</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabid">ub</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabnumber">0.6</span> <span class="scilabconstants">%inf</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Nonlinear constraints</span>
<span class="scilabfkeyword">function</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">c</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">ceq</span><span class="scilabopenclose">]</span><span class="scilaboperator">=</span><span class="scilabfunctionid">nlc</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">c</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">ceq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">3</span><span class="scilaboperator">-</span><span class="scilabnumber">0.5</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilaboperator">-</span><span class="scilabnumber">0.75</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Gradient of objective function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">fGrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">+</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Hessian of the Lagrange Function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span><span class="scilabfunctionid">lHess</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">obj</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">y</span><span class="scilaboperator">=</span> <span class="scilabinputoutputargs">obj</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">1</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabspecial">..</span>
<span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span><span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabspecial">..</span>
<span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">4</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">6</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabopenclose">]</span> <span class="scilaboperator">+</span> <span class="scilabinputoutputargs">lambda</span><span class="scilabopenclose">(</span><span class="scilabnumber">5</span><span class="scilabopenclose">)</span><span class="scilaboperator">*</span><span class="scilabopenclose">[</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Gradient of Non-Linear Constraints</span>
<span class="scilabfkeyword">function</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">cg</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">ceqg</span><span class="scilabopenclose">]</span><span class="scilaboperator">=</span><span class="scilabfunctionid">cGrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">cg</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">ceqg</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">3</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">^</span><span class="scilabnumber">2</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">0</span><span class="scilabdefault">;</span><span class="scilabnumber">0</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</span><span class="scilabdefault">,</span><span class="scilabnumber">2</span><span class="scilaboperator">*</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">//Options</span>
<span class="scilabid">options</span><span class="scilaboperator">=</span><a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">CpuTime</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">500</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">GradObj</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">fGrad</span><span class="scilabdefault">,</span> <span class="scilabstring">"</span><span class="scilabstring">Hessian</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">lHess</span><span class="scilabdefault">,</span><span class="scilabstring">"</span><span class="scilabstring">GradCon</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">cGrad</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</span>
<span class="scilabcomment">//Calling Ipopt</span>
<span class="scilabopenclose">[</span><span class="scilabid">x</span><span class="scilabdefault">,</span><span class="scilabid">fval</span><span class="scilabdefault">,</span><span class="scilabid">exitflag</span><span class="scilabdefault">,</span><span class="scilabid">output</span><span class="scilabdefault">,</span><span class="scilabid">lambda</span><span class="scilabdefault">,</span><span class="scilabid">grad</span><span class="scilabdefault">,</span><span class="scilabid">hessian</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span><span class="scilabid">fmincon</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">f</span><span class="scilabdefault">,</span> <span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</span><span class="scilabdefault">,</span><span class="scilabid">Aeq</span><span class="scilabdefault">,</span><span class="scilabid">beq</span><span class="scilabdefault">,</span><span class="scilabid">lb</span><span class="scilabdefault">,</span><span class="scilabid">ub</span><span class="scilabdefault">,</span><span class="scilabfunctionid">nlc</span><span class="scilabdefault">,</span><span class="scilabid">options</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Press ENTER to continue</span></pre></td><td valign="top"><a href="scilab://scilab.execexample/"><img src="ScilabExecute.png" border="0"/></a></td><td valign="top"><a href="scilab://scilab.editexample/"><img src="ScilabEdit.png" border="0"/></a></td><td></td></tr></table></div></div>
<div class="refsection"><h3 class="title">Authors</h3>
<ul class="itemizedlist"><li class="member">R.Vidyadhar , Vignesh Kannan</li></ul></div>
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