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14 Linear SemiDefinite w LMI
In
sdp_prob
there is 1 linear semidefinite 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:


f(x) = c^{T}x 


s/t 
x_{L} 
≤ 
x 
≤ 
x_{U} 

b_{L} 
≤ 
Ax 
≤ 
b_{U} 

Q_{0}^{i} + 

^{n} Q_{k}^{i}x_{k} 0, i=1,…,m. 


(16) 
where
c,
x,
x_{L},
x_{U} R^{n},
A R^{ml
× n},
b_{L},
b_{U} R^{ml} 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);
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