Quickguide GOAL Problem

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

The constrained goal attainment (goal) problem is defined as

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

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

The goal solution can be obtained by the use of any suitable nonlinear TOMLAB solver.

The following files define a problem in TOMLAB.

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 solve a goal attainment problem in TOMLAB. Also view the m-files specified above for more information.

File: tomlab/quickguide/goalsQG.m

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

 % 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);