Models Linear Semi-Definite Programming Problem with Linear Matrix Inequalities: sdp prob

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

In sdp_prob there is 1 linear semi-definite programming test problem with linear matrix inequalities with 3 variables. In order to define this problem and solve it execute the following in Matlab:

Prob	= probInit('spd_prob',1); 
Result  = tomRun('',Prob);

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

${\begin{array}{rccccl}\min \limits _{x}&{f(x)={c^{T}}x}\\&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&{Q_{0}^{i}+\sum {k=1}{n}Q_{k}^{i}x_{k}\preccurlyeq 0,\qquad i=1,\ldots ,m.}\\\end{array}}$

where $c,x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $A\in \mathbb {R} ^{m_{l}\times n}$, $b_{L},b_{U}\in \mathbb {R} ^{m_{l}}$ and $Q_{k}^{i}$ are symmetric matrices of similar dimensions in each constraint $i$. If there are several LMI constraints, each may have it's own dimension.

The following file is required to define a problem of this category in TOMLAB.

File: tomlab/quickguide/sdpQG.m

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

% sdpQG is a small example problem for defining and solving
% semi definite programming problems with linear matrix
% inequalities using the TOMLAB format.

Name = 'sdp.ps example 2';

% Objective function
c = [1 2 3]';

% Two linear constraints
A =   [ 0 0 1 ; 5 6 0 ];
b_L = [-Inf; -Inf];
b_U = [ 3  ; -3 ];

x_L = -1000*ones(3,1);
x_U =  1000*ones(3,1);

% Two linear matrix inequality constraints. It is OK to give only
% the upper triangular part.
SDP = [];
% First constraint
SDP(1).Q{1} = [2 -1 0 ; 0 2 0 ; 0 0 2];
SDP(1).Q{2} = [2 0 -1 ; 0 2 0 ; 0 0 2];
SDP(1).Qidx = [1; 3];

% Second constraint
SDP(2).Q{1} = diag( [0  1] );
SDP(2).Q{2} = diag( [1 -1] );
SDP(2).Q{3} = diag( [3 -3] );
SDP(2).Qidx = [0; 1; 2];

x_0 = [];

Prob = sdpAssign(c, SDP, A, b_L, b_U, x_L, x_U, x_0, Name);

Result = tomRun('pensdp', Prob, 1);