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

{{Part Of Manual|title=TOMLAB Models|link=[[Models|TOMLAB Models]]}}
In <tt>sdp_prob</tt> 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:

<pre>
Prob	= probInit('spd_prob',1); 
Result  = tomRun('',Prob);
</pre>

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


<math>
\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^{i}_0 + \sum{k=1}{n} Q^{i}_{k}x_{k} \preccurlyeq 0,\qquad i=1,\ldots,m.} \\     \end{array}
</math>

where <math>c, x, x_L, x_U \in \mathbb{R}^n</math>, <math>A \in \mathbb{R}^{m_l \times n}</math>, <math>b_L,b_U \in \mathbb{R}^{m_l}</math> and <math>Q^{i}_{k}</math> are
symmetric matrices of similar dimensions in each constraint <math>i</math>. 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.

<source lang="matlab">
% 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);
</source>