Models Quadratic Programming Problems: qp prob

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

In qp_prob there are 41 quadratic programming test problems with sizes to nearly 1200 variables and nearly 500 constraints. In order to define the problem n and solve it execute the following in Matlab:

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

The basic structure of a general nonlinear Quadratic Programming problem is:

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

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x\\&\\s/t&{\begin{array}{lcccl}x_{L}&\leq &x&\leq &x_{U}\\b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}\end{array}}$

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c, x, x_L, x_U \in \MATHSET{R}^n} , Failed to parse (unknown function "\MATHSET"): {\displaystyle F \in \MATHSET{R}^{n \times n}} , Failed to parse (unknown function "\MATHSET"): {\displaystyle A \in \MATHSET{R}^{m_1 \times n}} , and Failed to parse (unknown function "\MATHSET"): {\displaystyle b_L,b_U \in \MATHSET{R}^{m_1}} . Equality constraints are defined by setting the lower bound equal to the upper bound, i.e. for constraint $i$: $b_{L}(i)=b_{U}(i)$. Fixed variables are handled the same way.

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

${\begin{array}{ll}\min \limits _{x}&f(x)=4x_{1}^{2}+x_{1}x_{2}+4x_{2}^{2}+3x_{1}-4x_{2}\\&\\s/t&{\begin{array}{lll}x_{1}+x_{2}\leq 5\\x_{1}-x_{2}=0\\x_{1}\geq 0\\x_{2}\geq 0\\\end{array}}\end{array}}$

File: tomlab/quickguide/qpQG.m

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

% qpQG is a small example problem for defining and solving
% quadratic programming problems using the TOMLAB format.

Name  = 'QP Example';
F     = [ 8   1        % Matrix F in 1/2 * x' * F * x + c' * x
          1   8 ];
c     = [ 3  -4 ]';    % Vector c in 1/2 * x' * F * x + c' * x
A     = [ 1   1        % Constraint matrix
          1  -1 ];
b_L   = [-inf  0  ]';  % Lower bounds on the linear constraints
b_U   = [  5   0  ]';  % Upper bounds on the linear constraints
x_L   = [  0   0  ]';  % Lower bounds on the variables
x_U   = [ inf inf ]';  % Upper bounds on the variables
x_0   = [  0   1  ]';  % Starting point
x_min = [-1 -1 ];      % Plot region lower bound parameters
x_max = [ 6  6 ];      % Plot region upper bound parameters

% Assign routine for defining a QP problem.
Prob = qpAssign(F, c, A, b_L, b_U, x_L, x_U, x_0, Name,...
       [], [], [], x_min, x_max);

% Calling driver routine tomRun to run the solver.
% The 1 sets the print level after optimization.

Result = tomRun('qpSolve', Prob, 1);
%Result = tomRun('snopt', Prob, 1);
%Result = tomRun('sqopt', Prob, 1);
%Result = tomRun('cplex', Prob, 1);
%Result = tomRun('knitro', Prob, 1);
%Result = tomRun('conopt', Prob, 1);