Quickguide LINRAT Problem

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

The linearly constrained linear ratio (LINRAT) problem is defined as

${\begin{array}{cccccc}\min \limits _{x}&{c1x \over c2x}\\{\mbox{subject to}}&x_{L}&\leq &x&\leq &x_{U}\\{}&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$

where $c1,c2,x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $b_{L},b_{U}\in \mathbb {R} ^{m_{2}}$, and $A\in \mathbb {R} ^{m_{1}\times n}$.

The LINRAT solution can be obtained by the use of any suitable linear TOMLAB solver.

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

File: tomlab/quickguide/linratQG.m

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

 % linratQG is a small example problem for defining and solving
 % linear ratio programming problems using the TOMLAB format.
 
 Name = 'Linear Ratio 1';
 x_0  = [2;2;2;2];          % Initial value
 x_L  = [1;2;3;4];          % Lower bounds on x
 x_U  = [100;100;50;50];    % Upper bounds on x
 
 % Define the numerator and denominator for the objective
 c1 = [3;7;9;11]; 
 c2 = [20;15;10;5];
 
 % Add the linear constraint x(1) + x(2) + x(3) + x(4) - 20 <= 0
 % Write the constraint as x(1) + x(2) + x(3) + x(4) <= 20
 
 % The A matrix could be specified dense or sparse
 % A   = sparse([1 1 1 1]);
 
 A   = [1 1 1 1];
 b_L = -inf;
 b_U = 20;
 
 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.c1 = c1;
 Prob.QP.c2 = c2;
 
 Prob.SolverRat = 'minos';
 
 % One may set other solvers:
 % Prob.SolverRat = 'cplex';
 % Prob.SolverRat = 'xa';
 % Prob.SolverRat = 'snopt';
 % Prob.SolverRat = 'milpSolve';
 
 % Set print level 1 to get output from PrintResult at the end
 PriLev = 1;
 Prob.PriLevOpt = 0;
 
 Result  = tomRun('linRatSolve', Prob, PriLev);