Solutions ofconstraint sos

[Alpha Wagner]

Is there a way to obtain the values of x and y in the example form the yamlip wiki (http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.SumOfSquares)

sdpvar x y lower
p = (1+x*y)^2-x*y+(1-y)^2;
g = [1-x;1+x;1-y;1+y]
[s1,c1] = polynomial([x y],2);
[s2,c2] = polynomial([x y],2);
[s3,c3] = polynomial([x y],2);
[s4,c4] = polynomial([x y],2);
F = [sos(p-lower-[s1 s2 s3 s4]*g), sos(s1), sos(s2), sos(s3), sos(s4)];
solvesos(F,-lower,[],[c1;c2;c3;c4;lower]);

Thanks

[Johan Löfberg]

No, solvesos does not try to extract solutions from tight relaxations.

However, solvemoment, which works in the dual space of solvesos, does so, and in this particular case the first relaxation is tight and a solution can be extracted

[~,extracted] = solvemoment(g>=0,p);

Having said that, the problem is trivial enough for a stadnard b&b LP-relaxation based solver to finish in no time

optimize(g>=0,p,sdpsettings('solver','bmibnb'));

[Alpha Wagner]

Thanks a lot.

What happens if the moment relaxation is not tight? 

Also in how far corresponds the a moment relaxation in the dual space to an sos relaxation in the primal space ?

[Johan Löfberg]

If it isn't tight, it cannot extract a solution, of course, as there is no solution achieving that super-optimal objective value.

The moment relaxation is the dual of the sos, basically.If you look at the semidefinite program that the sos leads to, and then look its dual, you have the model you would have derived if you took the moment route. Effectively, this means that as we can extract a solution from the primal moment matrix, we could extract a solution from the sos also, as sdp solvers compute primal-dual solution. YMIP doesn't do that though. More natural/easy to do it on the moment side.

[Alpha Wagner]

I see thanks.

[Alpha Wagner]

I am usually using mosek within matlab. Can I specify in solvemoment to use a different solver?

Thanks,

Andre

[Johan Löfberg]

of course

>> ops=sdpsettings;ops.moment

ans = 

          order: []

      blockdiag: 0

         solver: ''

         refine: 0

    extractrank: 0

        rceftol: -1

[Alpha Wagner]

Again thanks a lot.

By the way the ref's (Laserre:2001) in the moment tutorial ( http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.MomentRelaxations ) and  (Parrilo:2003) on the sos tutorial ( http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.SumOfSquares ) are not working. 

I get

Warning: fopen(http://users.isy.liu.se/johanl/yalmip/pub/): failed to open stream: HTTP request failed! HTTP/1.1 403 Forbidden in /www/users/rt/johanl/yalmip/cookbook/bibtexref/bibtexref2.php on line 747 Can't open file

Cheers