Quickguide LINRAT Problem

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

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

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{c1 x \over c2 x} \\ \mbox{subject to} & x_L & \leq & x & \leq & x_U \\ {} & b_L & \leq & Ax & \leq & b_U \\\end{array} }

where Failed to parse (unknown function "\Rdim"): {\displaystyle c1,c2,x,x_L,x_U \in \Rdim{n}} , Failed to parse (unknown function "\Rdim"): {\displaystyle b_L,b_U \in \Rdim{m_2}} , and Failed to parse (unknown function "\Rdim"): {\displaystyle A\in \Rdim{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);