Quickguide MINIMAX Problem

This page is part of the Quickguide Manual. See Quickguide.

The constrained minimax (mima) problem is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{\max r(x)} \\ \mbox{subject to} & x_L & \leq & x & \leq & x_U \\ & b_L & \leq & Ax & \leq & b_U \\ & c_L & \leq & c(x) & \leq & c_U \\\end{array} }

where $x,x_{L},x_{U}\in R^{n}$, $r(x)\in R^{N}$, $c(x),c_{L},c_{U}\in R^{m_{1}}$, $b_{L},b_{U}\in R^{m_{2}}$ and $A\in R^{m_{2}\times n}$.

The following files define a problem in TOMLAB.

File: tomlab/quickguide/mimaQG_r.m, mimaQG_J.m

r:   Residual vector
J:   Jacobian matrix

The following file illustrates how to solve a minimax problem in TOMLAB. Also view the m-files specified above for more information.

File: tomlab/quickguide/minimaxQG.m

Open the file for viewing, and execute minimaxQG in Matlab.

 % minimaxQG is a small example problem for defining and solving
 % minimax programming problems using the TOMLAB format.
 
 Name = 'Madsen-Tinglett 2';
 x_0  = [1;1];        % Initial value
 x_L  = [-10;-10];    % Lower bounds on x
 x_U  = [10;10];      % Upper bounds on x
 
 % Solve the problem min max |r_i(x)|, where i = 1,2,3
 % Solve the problem by eliminating abs, doubling the residuals, reverse sign
 % i.e. min max [r_1; r_2; r_3; -r_1; -r_2; -r_3];
 y = [-1.5; -2.25; -2.625]; % The data values
 y = [y ; -y];           % Eliminate abs, double the residuals, reverse sign
 t = [];                 % No time vector used
 
 %
 % Add the linear constraint -x(1) + x(2) + 2 >= 0
 % Write the constraint as x(1) - x(2) <= 2
 
 % The A matrix could be specified dense or sparse
 % A   = sparse([1 -1]);
 
 A   = [1 -1];
 b_L = -inf;
 b_U = 2;
 
 % Generate the problem structure using the Tomlab Quick format
 %
 % The part in the residuals dependent on x are defined in mima_r.m 
 % The Jacobian is defined in mima_J.m
 
 Prob = clsAssign('mimaQG_r', 'mimaQG_J', [], x_L, x_U, Name, x_0, y, t, ...
                  [],[],[],[],A,b_L,b_U);
 
 % Set the optimal values into the structure (for nice result presenting)
 
 Prob.x_opt=[2.3660254038, 0.3660254038];
 % Optimal residuals:
 r_opt = [0; 0.2009618943;0.375];
 
 % Compute optimal function value:
 Prob.f_opt=max(abs(r_opt));
 
 % Use the standard method in infSolve
 Prob.InfType = 1;
 
 % Get the default solver
 % Solver = GetSolver('con',1,0);
 
 Prob.SolverInf = 'conSolve';
 
 % One may set other solvers:
 %Prob.SolverInf = 'snopt';
 %Prob.SolverInf = 'npsol';
 %Prob.SolverInf = 'minos';
 %Prob.SolverInf = 'conSolve';
 
 % Set print level 2 to get output from PrintResult at the end
 PriLev = 2;
 Prob.PriLevOpt = 0;
 Result  = tomRun('infSolve', Prob, PriLev);