Models Constrained Goal Attainment Problems: goals prob and mco prob

This page is part of TOMLAB Models. See TOMLAB Models.

The TOMLAB bundle testprob provides two sets of problems for constrained goal attainment problems: goals_prob and mco_prob.

An example of a constrained goal attainment problems

The basic structure of a constrained goal attainment problems is the following:

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

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

An example of a problem of this class, (that is also found in the TOMLAB quickguide) is goalsQG:

File: tomlab/quickguide/goalsQG_r.m, goalsQG_J.m, goalsQG_c, goalsQG_dc

r:	Residual  vector
J:	Jacobian  matrix
c: 	Nonlinear constraint vector
dc: 	Nonlinear constraint  gradient matrix

The following file illustrates how to define and solve a problem of this category in TOMLAB.

File: tomlab/quickguide/goalsQG.m

% goalsQG is a small example problem for defining and solving
% multi criteria optimization problems using the TOMLAB format.

Name='EASY-TP355';
% Constrained least squares problem, four quadratic terms and local solutions
% Hock W., Schittkowski K. (1981):
x_0 = zeros(4,1);    % Lower bounds for x.
x_L = zeros(4,1);    % Upper bounds for x.
x_U = 1e5*ones(4,1); % Starting point.
x_min = [];          % For plotting.
x_max = [];          % For plotting.
A   = [1 0 0 0;0 1 0 0];  % Linear constraints.
b_L = [0.1;0.1];          % Lower bounds.
b_U = [0.1;0.1];          % Upper bounds.
c_L = 0;                  % Lower bounds.
c_U = 0;                  % Upper bounds.
y   = zeros(2,1);         % Residuals

Prob = clsAssign('goalsQG_r', 'goalsQG_J', [], x_L, x_U, Name, x_0,...
                 y, [], [], [], [], [],...
                 A, b_L, b_U, 'goalsQG_c', 'goalsQG_dc', [], c_L, c_U,...
                 x_min, x_max);

PriLev = 2;
Result = tomRun('goalSolve', Prob, PriLev);

mco_prob

In glb_prob there are 9 Multi-Criterium unconstrained and constrained nonlinear test problems with up to 10 variables and few constrains. In order to define the problem n and solve it execute the following in Matlab:

Prob	= probInit('mco_prob',n); 
Result  = tomRun('',Prob);

goals_prob

In goals_prob there are 9 constrained goal attainment test problems with sizes to 9 variables and about 10 constrains. In order to define the problem n and solve it execute the following in Matlab:

Prob = probInit('goals_prob',n); 
Result  = goalSolve(Prob,1);