Quickguide SDP Problem

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

The linear semi-definite programming problem with linear matrix inequalities (sdp) is defined as

${\begin{array}{rccccl}\min \limits _{x}&{5}{l}{f(x)={c^{T}}x}&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&{5}{r}{Q_{0}^{i}+\sum {k=1}{n}Q_{k}^{i}x_{k}\preccurlyeq 0,\qquad i=1,\ldots ,m.}\\\end{array}}$

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 defines and solves a problem in TOMLAB.

File: tomlab/quickguide/sdpQG.m

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

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);