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diff --git a/help/en_US/scilab_en_US_help/fminimax.html b/help/en_US/scilab_en_US_help/fminimax.html index 22b6ad9..2aa5c38 100644 --- a/help/en_US/scilab_en_US_help/fminimax.html +++ b/help/en_US/scilab_en_US_help/fminimax.html @@ -16,7 +16,7 @@ </td> <td width="40%" class="center"> - <span class="top"><a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a></span> + <span class="top"><a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a></span> </td> <td width="30%" class="next"> @@ -29,7 +29,7 @@ - <span class="path"><a href="index.html">FOSSEE Optimization Toolbox</a> >> <a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a> > fminimax</span> + <span class="path"><a href="index.html">FOSSEE Optimization Toolbox</a> >> <a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a> > fminimax</span> <br /><br /> <div class="refnamediv"><h1 class="refname">fminimax</h1> @@ -49,64 +49,72 @@ <span class="default">[</span><span class="default">xopt</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="functionid">fminimax</span><span class="default">(.....)</span> <span class="default">[</span><span class="default">xopt</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="functionid">fminimax</span><span class="default">(.....)</span></pre></div></div> -<div class="refsection"><h3 class="title">Parameters</h3> +<div class="refsection"><h3 class="title">Input 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> + <dd><p class="para">The function to be minimized. fun is a function that has a vector x as an input argument, and contains the objective functions evaluated at x.</p></dd></dt> <dt><span class="term">x0 :</span> - <dd><p class="para">a vector of double, contains initial guess of variables.</p></dd></dt> + <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 double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt> + <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 double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.</p></dd></dt> + <dd><p class="para">A vector of doubles, related to 'A' and represents the linear coefficients in 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 double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt> + <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 double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.</p></dd></dt> + <dd><p class="para">A vector of double, vector of doubles, related to 'Aeq' and represents the linear coefficients in the equality constraints of size (m1 X 1).</p></dd></dt> <dt><span class="term">lb :</span> - <dd><p class="para">a vector of double, contains lower bounds of the variables.</p></dd></dt> + <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 double, contains upper bounds of the variables.</p></dd></dt> + <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">nonlinfun:</span> - <dd><p class="para">function that computes the nonlinear inequality constraints c⋅x ≤ 0 and nonlinear equality constraints c⋅x = 0.</p></dd></dt> - <dt><span class="term">xopt :</span> - <dd><p class="para">a vector of double, the computed solution of the optimization problem.</p></dd></dt> - <dt><span class="term">fopt :</span> - <dd><p class="para">a double, the value of the function at x.</p></dd></dt> + <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 (c), and the non-linear equality constraints (ceq) are defined as separate single row vectors.</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">Outputs</h3> + <dl><dt><span class="term">xopt :</span> + <dd><p class="para">A vector of doubles, containing the computed solution of the optimization problem.</p></dd></dt> + <dt><span class="term">fval :</span> + <dd><p class="para">A vector of doubles, containing the values of the objective functions at the end of the optimization problem.</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> + <dd><p class="para">A double, representing the maximum value in the vector fval.</p></dd></dt> <dt><span class="term">exitflag :</span> - <dd><p class="para">The exit status. See below for details.</p></dd></dt> + <dd><p class="para">An 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">The structure consist of statistics about the optimization. See below for details.</p></dd></dt> + <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">The structure consist of the Lagrange multipliers at the solution of problem. See below for details.</p></dd></dt></dl></div> + <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></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">fminimax minimizes the worst-case (largest) value of a set of multivariable functions, starting at an initial estimate, a problem 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:318px;height:133px'/></span></p> + <p class="para">Currently, fminimax calls fmincon which uses the Ipopt solver.</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 : + + <p class="para"><h3 class="title">Options</h3> +The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:</p> + <p class="para">options= list("MaxIter", [---], "CpuTime", [---], "GradObj", ---, "GradCon", ---);</p> + <p class="para"><ul class="itemizedlist"><li>MaxIter : A Scalar, specifying the Maximum Number of iterations that the solver should take.</li> +<li>CpuTime : A Scalar, specifying the Maximum amount of CPU Time in seconds 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 &amp; m3 are number of non-linear inequality and equality constraints respectively.</li> +<li>Default Values : options = list("MaxIter", [3000], "CpuTime", [600]);</li></ul> +The default values for the various items are given as:</p> + <p class="para">options = list("MaxIter", [3000], "CpuTime", [600]);</p> + <p class="para">The objective function must have a 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 : + <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 GradCon option is on, fminimax approximates minmaxObjfun gradient using the numderivative toolbox.</p> +<h3 class="title">Syntax</h3> +<p class="para">Some syntactic details about fminimax, including the syntax for the gradient, defining the non-linear constraints, and the constraint derivative function have been provided below:</p> + <p class="para">If the user can provide exact gradients, it should be done, since it improves the convergence speed of the optimization algorithm.</p> + <p class="para">Furthermore, we can 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 : + <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). +Where x is a n x 1 matrix of minimax doubles, 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> @@ -116,35 +124,55 @@ This will let fminimax know that the exact gradient of the objective function is <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. + <p class="para">The exitflag allows the user to know the status of the optimization which is returned by Ipopt. The values it can take and what they indicate is described below: +<ul class="itemizedlist"><li>0 : Optimal Solution Found</li> +<li>1 : Maximum Number of Iterations Exceeded. Output may not be optimal.</li> +<li>2 : Maximum amount of CPU Time exceeded. Output may not be optimal.</li> +<li>3 : Stop at Tiny Step.</li> +<li>4 : Solved To Acceptable Level.</li> +<li>5 : Converged to a point of local infeasibility.</li></ul></p> + <p class="para">For more details on exitflag, see the ipopt documentation which can be found on http://www.coin-or.org/Ipopt/documentation/</p> + <p class="para">The output data structure contains detailed information about the optimization process. +It is of type "struct" and contains the following fields. +<ul class="itemizedlist"><li>output.Iterations: The number of iterations performed.</li> +<li>output.Cpu_Time : The total cpu-time taken.</li> +<li>output.Objective_Evaluation: The number of Objective Evaluations performed.</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> +<li>lambda.ineqnonlin: The Lagrange multipliers for the non-linear inequality constraints.</li></ul></p></div> +<p class="para">A few examples displaying the various functionalities of fminimax have been provided below. You will find a series of problems and the appropriate code snippets to solve them.</p> +<div class="refsection"><h3 class="title">Example</h3> + Here we solve a simple objective function, subjected to no constraints. +<p class="para"><span><img src='./_LaTeX_fminimax.xml_3.png' style='position:relative;top:56px;width:312px;height:148px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 1:</span> +<span class="scilabcomment">// Objective 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">// Run fminimax</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">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></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">Example</h3> + We proceed to add simple linear inequality constraints. -<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> +<p class="para"><span><img src='./_LaTeX_fminimax.xml_4.png' style='position:relative;top:29px;width:225px;height:66px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 2:</span> +<span class="scilabcomment">//Objective 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> @@ -154,18 +182,102 @@ It has type "struct" and contains the following fields. <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="scilabid">xopt</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">4</span> <span class="scilabnumber">4</span><span class="scilabopenclose">]</span> -<span class="scilabid">fopt</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">0</span> <span class="scilaboperator">-</span><span class="scilabnumber">64</span> <span class="scilaboperator">-</span><span class="scilabnumber">2</span> <span class="scilaboperator">-</span><span class="scilabnumber">8</span> <span class="scilabnumber">0</span><span class="scilabopenclose">]</span> -<span class="scilabid">maxfopt</span> <span class="scilaboperator">=</span> <span class="scilabnumber">0</span> +<span class="scilabcomment">//Linear Inequality constraints</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="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">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">// Run fminimax</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">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="scilabdefault">,</span><span class="scilabid">A</span><span class="scilabdefault">,</span><span class="scilabid">b</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">Example</h3> +Here we build up on the previous example by adding linear equality constraints. +We add the following constraints to the problem specified above: +<p class="para"><span><img src='./_LaTeX_fminimax.xml_5.png' style='position:relative;top:17px;width:134px;height:42px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 3:</span> +<span class="scilabcomment">//Objective 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">//Linear Inequality constraints</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="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">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">//We specify the linear equality constraints below.</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="scilaboperator">-</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="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabid">beq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</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="scilabcomment">// Run fminimax</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">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="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="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">Example</h3> +In this example, we proceed to add the upper and lower bounds to the objective function. +<p class="para"><span><img src='./_LaTeX_fminimax.xml_6.png' style='position:relative;top:17px;width:145px;height:42px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 4:</span> +<span class="scilabcomment">//Objective 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">//Linear Inequality constraints</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="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">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">//We specify the linear equality constraints below.</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="scilaboperator">-</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="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabid">beq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</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="scilabcomment">//The upper and lower bounds for the objective function are defined in simple vectors as shown below.</span> +<span class="scilabid">lb</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="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="scilabconstants">%inf</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> <span class="scilabcomment">//</span> <span class="scilabcomment">// Run fminimax</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">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> +<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">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="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="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">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> + +<div class="refsection"><h3 class="title">Example</h3> +Finally, we add the non-linear constraints to the problem. Note that there is a notable difference in the way this is done as compared to defining the linear constraints. + <p class="para"><span><img src='./_LaTeX_fminimax.xml_7.png' style='position:relative;top:22px;width:139px;height:52px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 5:</span> +<span class="scilabcomment">//Objective 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">//Linear Inequality constraints</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="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">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">//We specify the linear equality constraints below.</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="scilaboperator">-</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="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabid">beq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</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="scilabcomment">//The upper and lower bounds for the objective function are defined in simple vectors as shown below.</span> +<span class="scilabid">lb</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="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="scilabconstants">%inf</span><span class="scilabdefault">;</span><span class="scilabnumber">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> <span class="scilabcomment">//</span> +<span class="scilabcomment">//Nonlinear constraints are required to be defined as a single function with the inequality and equality constraints in separate vectors.</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="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">// Run fminimax</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">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="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="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">Example</h3> +<p class="para">We can further enhance the functionality of fminimax by setting input options. We can pre-define the gradient of the objective function and/or the hessian of the lagrange function and thereby improve the speed of computation. This is elaborated on in example 6. We take the following problem, specify the gradients, and the jacobian matrix of the constraints. We also set solver parameters using the options.</p> +<p class="para"><span><img src='./_LaTeX_fminimax.xml_8.png' style='position:relative;top:17px;width:205px;height:42px'/></span></p> + <p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 6: Using the available options</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> @@ -173,6 +285,7 @@ It has type "struct" and contains the following fields. <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> @@ -200,7 +313,7 @@ It has type "struct" and contains the following fields. <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="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">MaxIter</span><span class="scilabstring">"</span><span class="scilabdefault">,</span> <span class="scilabopenclose">[</span><span class="scilabnumber">3000</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">600</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">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> @@ -210,6 +323,30 @@ It has type "struct" and contains the following fields. <span class="scilabcomment">// Run fminimax</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">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">Example</h3> +Infeasible Problems: Find x in R^2 such that it minimizes the objective function used above under the following constraints: +<p class="para"><span><img src='./_LaTeX_fminimax.xml_9.png' style='position:relative;top:19px;width:158px;height:46px'/></span></p> +<p class="para"></p> + <div class="programlisting"><table border="0" width="100%"><tr><td width="98%"><pre class="scilabcode"><span class="scilabcomment">//Example 7:</span> +<span class="scilabcomment">//Objective 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">//Linear Inequality constraints</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="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">1</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">//We specify the linear equality constraints below.</span> +<span class="scilabid">Aeq</span> <span class="scilaboperator">=</span> <span class="scilabopenclose">[</span><span class="scilabnumber">1</span><span class="scilaboperator">/</span><span class="scilabnumber">3</span><span class="scilabdefault">,</span><span class="scilaboperator">-</span><span class="scilabnumber">5</span><span class="scilabdefault">;</span> <span class="scilabnumber">2</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">11</span><span class="scilabdefault">;</span><span class="scilabnumber">8</span><span class="scilabopenclose">]</span><span class="scilabdefault">;</span> +<span class="scilabcomment">// Run fminimax</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">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="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="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> <br /> @@ -223,7 +360,7 @@ It has type "struct" and contains the following fields. </td> <td width="40%" class="center"> - <span class="top"><a href="section_031bbc67ce78762a40093bfdff4eaa3b.html">FOSSEE Optimization Toolbox</a></span> + <span class="top"><a href="section_44e1f57c5225357b5fe53cb5fad967e9.html">FOSSEE Optimization Toolbox</a></span> </td> <td width="30%" class="next"> |