TomSym example sdp - tomSym SDP demonstration

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

An SDP example, from Wikipedia article about semidefinite programming

Minimize rac subject to [1 rab rac; rab 1 rbc; rac rbc 1] positive semidefinite -0.2 <= rab <= -0.1 0.4 <= rbc <= 0.5

% Variables
toms rab rac rbc

% Objective function
f = rac;

% Constraints
M = [1 rab rac; rab 1 rbc; rac rbc 1]
con = { positiveSemidefinite(M), ...
    -0.2 <= rab <= -0.1, ...
    0.4 <= rbc <=  0.5};

% Initial conditions
x0 = struct('rab',-0.15,'rbc',0.45,'rac',0);

 
M = tomSym(3x3):
 
   [[1 rab rac];[rab 1 rbc];[rac rbc 1]]
 

Compile and solve problem as a NLP

options = struct;
options.type = 'con';
options.name = 'SDP example';
solution = ezsolve(f,con,x0,options);

% Evaluate M using the returned solution
M_opt = subs(M,solution)

Warning: A semidefinite constraint is converted to a nonlinear constraint
involving 9 additional unknowns. 
Time for symbolic processing: 0.064665 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: SDP example                    f_k      -0.977997772830131120
                                       sum(|constr|)      0.000000000010720591
                              f(x_k) + sum(|constr|)     -0.977997772819410470
                                              f(x_0)      0.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv   11 GradEv    9 ConstrEv    9 ConJacEv    9 Iter    7 MinorIter   14
CPU time: 0.015600 sec. Elapsed time: 0.014000 sec. 
M_opt =
    1.0000   -0.2000   -0.9780
   -0.2000    1.0000    0.4000
   -0.9780    0.4000    1.0000

Compile and solve problem as a SDP

options.type = 'sdp';
solution = ezsolve(f,con,x0,options);

% Evaluate M using the returned solution
M_opt = subs(M,solution)

Time for symbolic processing: 0.018392 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: SDP example                    f_k      -0.977997772825720870
                                              f(x_0)      0.000000000000000000

Solver: PENSDP.  EXIT=0.  INFORM=0.
LMI Solver PENSDP 2.2
Solution obtained.

FuncEv   17 GradEv   17 Iter   17 MinorIter   44
Elapsed time: 0.002000 sec. 
M_opt =
    1.0000   -0.2000   -0.9780
   -0.2000    1.0000    0.4000
   -0.9780    0.4000    1.0000

Check the eigenvalues

The eigenvalues should all be positive, even though small negative ones may occur due to tolerances.

eigenvalues = eigs(M_opt)

eigenvalues =
    2.1376
    0.8624
    0.0000