Quickguide MINIMAXLIN Problem

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

The linear minimax (mimalin) problem is defined as

${\begin{array}{cccccc}\min \limits _{x}&{\max Dx}\\{\mbox{subject to}}&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$

where $x,x_{L},x_{U}\in R^{n}$, $b_{L},b_{U}\in R^{m_{1}}$, $A\in R^{m_{1}\times n}$ and $D\in R^{m_{2}\times n}$.

The following file illustrates how to solve a linear minimax problem in TOMLAB.

File: tomlab/quickguide/minimaxlinQG.m

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

 % minimaxlinQG is a small example problem for defining and solving
 % linear minimax programming problems using the TOMLAB format.
 
 Name = 'Linear Minimax Test 1';
 x_0  = [1;1;1;1];          % Initial value
 x_L  = [-10;-10;-10;-10];  % Lower bounds on x
 x_U  = [10;10;10;10];      % Upper bounds on x
 
 % Solve the problem min max Dx while eliminating abs for the final two
 % residuals by adding them with reverse signs.
 % i.e. min max [D_1; D_2; D_3; -D_2; -D_3];
 D = [9 8 7 6; -4 5 -6 -2; 3 4 5 -6; 4 -5 6 2; -3 -4 -5 6]; % D Matrix
 
 % 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 0 0]);
 
 A   = [1 -1 0 0];
 b_L = -inf;
 b_U = 2;
 
 c = zeros(4,1); % Dummy objective
 
 % Generate an LP problem using the Tomlab Quick format
 % Use mipAssign if solving a mixed-integer problem
 Prob = lpAssign(c, A, b_L, b_U, x_L, x_U, x_0, Name);
 Prob.QP.D = D;
 
 Prob.f_Low = 0;
 Prob.SolverInf = 'minos';
 
 % One may set other solvers:
 % Prob.SolverInf = 'cplex';
 % Prob.SolverInf = 'xa';
 % Prob.SolverInf = 'snopt';
 % Prob.SolverInf = 'milpSolve';
 
 % Set print level 1 to get output from PrintResult at the end
 PriLev = 1;
 Prob.PriLevOpt = 0;
 
 Result  = tomRun('infLinSolve', Prob, PriLev);