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<span class="path"><a href="index.html">Symphony Toolbox</a> >> <a href="section_19f4f1e5726c01d683e8b82be0a7e910.html">Symphony Toolbox</a> > fminimax</span>
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<div class="refnamediv"><h1 class="refname">fminimax</h1>
<p class="refpurpose">Solves minimax constraint problem</p></div>
<div class="refsynopsisdiv"><h3 class="title">Calling Sequence</h3>
<div class="synopsis"><pre><span class="default">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</span><span class="default">,</span><span class="default">x0</span><span class="default">)</span>
<span class="default">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</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">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</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">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</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">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</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">nonlinfun</span><span class="default">)</span>
<span class="default">x</span><span class="default"> = </span><span class="functionid">fminimax</span><span class="default">(</span><span class="default">fun</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">nonlinfun</span><span class="default">,</span><span class="default">options</span><span class="default">)</span>
<span class="default">[</span><span class="default">x</span><span class="default">, </span><span class="default">fval</span><span class="default">] = </span><span class="default">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">x</span><span class="default">, </span><span class="default">fval</span><span class="default">, </span><span class="default">maxfval</span><span class="default">]= </span><span class="default">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">x</span><span class="default">, </span><span class="default">fval</span><span class="default">, </span><span class="default">maxfval</span><span class="default">, </span><span class="default">exitflag</span><span class="default">]= </span><span class="default">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">x</span><span class="default">, </span><span class="default">fval</span><span class="default">, </span><span class="default">maxfval</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">fmincon</span><span class="default">(.....)</span>
<span class="default">[</span><span class="default">x</span><span class="default">, </span><span class="default">fval</span><span class="default">, </span><span class="default">maxfval</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">fmincon</span><span class="default">(.....)</span></pre></div></div>
<div class="refsection"><h3 class="title">Parameters</h3>
<dl><dt><span class="term">fun:</span>
<dd><p class="para">The function to be minimized. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x.</p></dd></dt>
<dt><span class="term">x0:</span>
<dd><p class="para">a nx1 or 1xn matrix of doubles, where n is the number of variables, the initial guess for the optimization algorithm</p></dd></dt>
<dt><span class="term">A:</span>
<dd><p class="para">a nil x n matrix of doubles, where n is the number of variables and nil is the number of linear inequalities. If A==[] and b==[], it is assumed that there is no linear inequality constraints. If (A==[] & b<>[]), fminimax generates an error (the same happens if (A<>[] & b==[]))</p></dd></dt>
<dt><span class="term">b:</span>
<dd><p class="para">a nil x 1 matrix of doubles, where nil is the number of linear inequalities</p></dd></dt>
<dt><span class="term">Aeq:</span>
<dd><p class="para">a nel x n matrix of doubles, where n is the number of variables and nel is the number of linear equalities. If Aeq==[] and beq==[], it is assumed that there is no linear equality constraints. If (Aeq==[] & beq<>[]), fminimax generates an error (the same happens if (Aeq<>[] & beq==[]))</p></dd></dt>
<dt><span class="term">beq:</span>
<dd><p class="para">a nel x 1 matrix of doubles, where nel is the number of linear equalities</p></dd></dt>
<dt><span class="term">lb:</span>
<dd><p class="para">a nx1 or 1xn matrix of doubles, where n is the number of variables. The lower bound for x. If lb==[], then the lower bound is automatically set to -inf</p></dd></dt>
<dt><span class="term">ub:</span>
<dd><p class="para">a nx1 or 1xn matrix of doubles, where n is the number of variables. The upper bound for x. If ub==[], then the upper bound is automatically set to +inf</p></dd></dt>
<dt><span class="term">nonlinfun:</span>
<dd><p class="para">function that computes the nonlinear inequality constraints c(x) <= 0 and nonlinear equality constraints ceq(x) = 0.</p></dd></dt>
<dt><span class="term">x:</span>
<dd><p class="para">a nx1 matrix of doubles, the computed solution of the optimization problem</p></dd></dt>
<dt><span class="term">fval:</span>
<dd><p class="para">a vector of doubles, the value of fun at x</p></dd></dt>
<dt><span class="term">maxfval:</span>
<dd><p class="para">a 1x1 matrix of doubles, the maximum value in vector fval</p></dd></dt>
<dt><span class="term">exitflag:</span>
<dd><p class="para">a 1x1 matrix of floating point integers, the exit status</p></dd></dt>
<dt><span class="term">output:</span>
<dd><p class="para">a struct, the details of the optimization process</p></dd></dt>
<dt><span class="term">lambda:</span>
<dd><p class="para">a struct, the Lagrange multipliers at optimum</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></dl></div>
<div class="refsection"><h3 class="title">Description</h3>
<p class="para">fminimax minimizes the worst-case (largest) value of a set of multivariable functions, starting at an initial estimate. This is generally referred to as the minimax problem.</p>
<p class="para"><span><img src='./_LaTeX_fminimax.xml_1.png' style='position:relative;top:51px;width:427px;height:110px'/></span></p>
<p class="para">Currently, fminimax calls fmincon which uses the ip-opt algorithm.</p>
<p class="para">max-min problems can also be solved with fminimax, using the identity</p>
<p class="para"><span><img src='./_LaTeX_fminimax.xml_2.png' style='position:relative;top:6px;width:281px;height:20px'/></span></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", ---, "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>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 objective function must have header :
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabid">F</span> <span class="scilaboperator">=</span> <span class="scilabid">fun</span><span class="scilabopenclose">(</span><span class="scilabid">x</span><span class="scilabopenclose">)</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>
where x is a n x 1 matrix of doubles and F is a m x 1 matrix of doubles where m is the total number of objective functions inside F.
On input, the variable x contains the current point and, on output, the variable F must contain the objective function values.</p>
<p class="para">By default, the gradient options for fminimax are turned off and and fmincon does the gradient opproximation of minmaxObjfun. In case the GradObj option is off and GradConstr option is on, fminimax approximates minmaxObjfun gradient using numderivative toolbox.</p>
<p class="para">If we can provide exact gradients, we should do so since it improves the convergence speed of the optimization algorithm.</p>
<p class="para">Furthermore, we must enable the "GradObj" option with the statement :
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabid">minimaxOptions</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="scilabid">fGrad</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</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>
This will let fminimax know that the exact gradient of the objective function is known, so that it can change the calling sequence to the objective function. Note that, fGrad should be mentioned in the form of N x n where n is the number of variables, N is the number of functions in objective function.</p>
<p class="para">The constraint function must have header :
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabopenclose">[</span><span class="scilabid">c</span><span class="scilabdefault">,</span> <span class="scilabid">ceq</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">confun</span><span class="scilabopenclose">(</span><span class="scilabid">x</span><span class="scilabopenclose">)</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>
where x is a n x 1 matrix of dominmaxUbles, c is a 1 x nni matrix of doubles and ceq is a 1 x nne matrix of doubles (nni : number of nonlinear inequality constraints, nne : number of nonlinear equality constraints).
On input, the variable x contains the current point and, on output, the variable c must contain the nonlinear inequality constraints and ceq must contain the nonlinear equality constraints.</p>
<p class="para">By default, the gradient options for fminimax are turned off and and fmincon does the gradient opproximation of confun. In case the GradObj option is on and GradCons option is off, fminimax approximates confun gradient using numderivative toolbox.</p>
<p class="para">If we can provide exact gradients, we should do so since it improves the convergence speed of the optimization algorithm.</p>
<p class="para">Furthermore, we must enable the "GradCon" option with the statement :
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabid">minimaxOptions</span> <span class="scilaboperator">=</span> <a class="scilabcommand" href="scilab://list">list</a><span class="scilabopenclose">(</span><span class="scilabstring">"</span><span class="scilabstring">GradCon</span><span class="scilabstring">"</span><span class="scilabdefault">,</span><span class="scilabid">confunGrad</span><span class="scilabopenclose">)</span><span class="scilabdefault">;</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>
This will let fminimax know that the exact gradient of the objective function is known, so that it can change the calling sequence to the objective function.</p>
<p class="para">The constraint derivative function must have header :
<div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabopenclose">[</span><span class="scilabid">dc</span><span class="scilabdefault">,</span><span class="scilabid">dceq</span><span class="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">confungrad</span><span class="scilabopenclose">(</span><span class="scilabid">x</span><span class="scilabopenclose">)</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>
where dc is a nni x n matrix of doubles and dceq is a nne x n matrix of doubles.</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></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">// A basic case :</span>
<span class="scilabcomment">// we provide only the objective function and the nonlinear constraint</span>
<span class="scilabcomment">// function</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">f</span><span class="scilaboperator">=</span><span class="scilabfunctionid">myfun</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">f</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">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">48</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">40</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">304</span><span class="scilabdefault">;</span> <span class="scilabcomment">//Objectives</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</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="scilabnumber">2</span> <span class="scilaboperator">-</span> <span class="scilabnumber">3</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="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</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">3</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">18</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">4</span><span class="scilabopenclose">)</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="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">5</span><span class="scilabopenclose">)</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">8</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">// The initial guess</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="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">// The expected solution : only 4 digits are guaranteed</span>
<span class="scilabcomment">//xopt = [4 4]</span>
<span class="scilabcomment">//fopt = [0 -64 -2 -8 0]</span>
<span class="scilabid">maxfopt</span> <span class="scilaboperator">=</span> <span class="scilabnumber">0</span>
<span class="scilabcomment">// Run fminimax</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">fopt</span><span class="scilabdefault">,</span><span class="scilabid">maxfval</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="scilabopenclose">]</span> <span class="scilaboperator">=</span> <span class="scilabid">fminimax</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">myfun</span><span class="scilabdefault">,</span> <span class="scilabid">x0</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">// A case where we provide the gradient of the objective</span>
<span class="scilabcomment">// functions and the Jacobian matrix of the constraints.</span>
<span class="scilabcomment">// The objective function and its gradient</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">f</span><span class="scilaboperator">=</span><span class="scilabfunctionid">myfun</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">f</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">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">48</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">40</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">304</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">2</span><span class="scilabopenclose">)</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="scilabnumber">2</span> <span class="scilaboperator">-</span> <span class="scilabnumber">3</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="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">3</span><span class="scilabopenclose">)</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">3</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">18</span><span class="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">4</span><span class="scilabopenclose">)</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="scilabdefault">;</span>
<span class="scilabinputoutputargs">f</span><span class="scilabopenclose">(</span><span class="scilabnumber">5</span><span class="scilabopenclose">)</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">8</span><span class="scilabdefault">;</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">// Defining gradient of myfun</span>
<span class="scilabfkeyword">function</span> <span class="scilabinputoutputargs">G</span><span class="scilaboperator">=</span><span class="scilabfunctionid">myfungrad</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabinputoutputargs">G</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span> <span class="scilabnumber">4</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">48</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">1</span><span class="scilabopenclose">)</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">1</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="scilaboperator">-</span> <span class="scilabnumber">40</span><span class="scilabdefault">,</span> <span class="scilaboperator">-</span><span class="scilabnumber">6</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">3</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="scilabdefault">;</span> <span class="scilabopenclose">]</span><span class="scilaboperator">'</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">// The nonlinear constraints and the Jacobian</span>
<span class="scilabcomment">// matrix of the 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">confun</span><span class="scilabopenclose">(</span><span class="scilabinputoutputargs">x</span><span class="scilabopenclose">)</span>
<span class="scilabcomment">// Inequality constraints</span>
<span class="scilabinputoutputargs">c</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1.5</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">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="scilabdefault">,</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">10</span><span class="scilabopenclose">]</span>
<span class="scilabcomment">// No nonlinear equality constraints</span>
<span class="scilabinputoutputargs">ceq</span><span class="scilaboperator">=</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">// Defining gradient of confungrad</span>
<span class="scilabfkeyword">function</span> <span class="scilabopenclose">[</span><span class="scilabinputoutputargs">DC</span><span class="scilabdefault">, </span><span class="scilabinputoutputargs">DCeq</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="scilabcomment">// DC(:,i) = gradient of the i-th constraint</span>
<span class="scilabcomment">// DC = [</span>
<span class="scilabcomment">// Dc1/Dx1 Dc1/Dx2</span>
<span class="scilabcomment">// Dc2/Dx1 Dc2/Dx2</span>
<span class="scilabcomment">// ]</span>
<span class="scilabinputoutputargs">DC</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="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</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="scilabinputoutputargs">x</span><span class="scilabopenclose">(</span><span class="scilabnumber">1</span><span class="scilabopenclose">)</span><span class="scilaboperator">-</span><span class="scilabnumber">1</span><span class="scilabdefault">,</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="scilabopenclose">]</span><span class="scilaboperator">'</span>
<span class="scilabinputoutputargs">DCeq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilaboperator">'</span>
<span class="scilabfkeyword">endfunction</span>
<span class="scilabcomment">// Test with both gradient of objective and gradient of constraints</span>
<span class="scilabid">minimaxOptions</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">myfungrad</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">// The initial guess</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">10</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span>
<span class="scilabcomment">// The expected solution : only 4 digits are guaranteed</span>
<span class="scilabcomment">//xopt = [0.92791 7.93551]</span>
<span class="scilabcomment">//fopt = [6.73443 -189.778 6.73443 -8.86342 0.86342]</span>
<span class="scilabid">maxfopt</span> <span class="scilaboperator">=</span> <span class="scilabnumber">6.73443</span>
<span class="scilabcomment">// Run fminimax</span>
<span class="scilabopenclose">[</span><span class="scilabid">xopt</span><span class="scilabdefault">,</span><span class="scilabid">fopt</span><span class="scilabdefault">,</span><span class="scilabid">maxfval</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">fminimax</span><span class="scilabopenclose">(</span><span class="scilabfunctionid">myfun</span><span class="scilabdefault">,</span><span class="scilabid">x0</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span><span class="scilabopenclose">[</span><span class="scilabopenclose">]</span><span class="scilabdefault">,</span> <span class="scilabfunctionid">confun</span><span class="scilabdefault">,</span> <span class="scilabid">minimaxOptions</span><span class="scilabopenclose">)</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">Animesh Baranawal</li></ul></div>
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