Quickguide QP Problem

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

The general formulation in TOMLAB for a quadratic programming problem is:

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

here $c,x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $F\in \mathbb {R} ^{n\times n}$, $A\in \mathbb {R} ^{m_{1}\times n}$, and $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$. 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.

Example problem:

The following file defines this problem in TOMLAB.

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);