Models Linear Semi-Definite Programming Problem with Linear Matrix Inequalities: sdp prob: Difference between revisions
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<math> | <math> | ||
\begin{array}{rccccl} \min\limits_{x} & | \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> | </math> | ||
where <math>c, x, x_L, x_U \in \ | 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. | symmetric matrices of similar dimensions in each constraint <math>i</math>. If there are several LMI constraints, each may have it's own dimension. | ||
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The following file illustrates how to define and solve a problem of this category in TOMLAB. This problem appears to be infeasible. | 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 | % sdpQG is a small example problem for defining and solving | ||
% semi definite programming problems with linear matrix | % semi definite programming problems with linear matrix | ||
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Result = tomRun('pensdp', Prob, 1); | Result = tomRun('pensdp', Prob, 1); | ||
</ | </source> |
Latest revision as of 18:54, 17 January 2012
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:
where , , and are symmetric matrices of similar dimensions in each constraint . 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);