Quickguide MINIMAX Problem

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

The constrained minimax (mima) problem is defined as

${\begin{array}{cccccc}\min \limits _{x}&{\max r(x)}\\{\mbox{subject to}}&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&c_{L}&\leq &c(x)&\leq &c_{U}\\\end{array}}$

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);