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16  SDP Problem

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

where c, x, x_L, x_U Rⁿ, A R^{m_l × n}, b_L,b_U R^m_l and Q_kⁱ 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);

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