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16  Constrained Goal Attainment

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

16.1  An example of a constrained goal attainment problems

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



where x,x_L,x_U Rⁿ, r(x) R^N, c(x),c_L,c_U R^m₁, b_L,b_U R^m₂, A R^{m₂ × n}, g R^m, and w 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);

16.2  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);

16.3  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);

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