119 lines
3.0 KiB
Mathematica
119 lines
3.0 KiB
Mathematica
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function x_extract = extractsolution(momentdata,options)
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%EXTRACTSOLUTIONS Tries to extract solutions from moment matrices
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%
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%
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% xoptimal = extractsolution(momentstructure)
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%
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% xoptimal : Extracted solutions
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%
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% momentdata : Problem data, obtained from SOLVEMOMENT (or SOLVESDP)
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% options : Options structure from SDPSETTINGS
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%
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% See also SOLVEMOMENT, SDPSETTINGS
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% Author Johan L<EFBFBD>fberg
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% $Id: extractsolution.m,v 1.1 2006-03-30 13:36:38 joloef Exp $
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moment = momentdata.moment;
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x = momentdata.x;
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monomials = momentdata.monomials;
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n = momentdata.n;
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d = momentdata.d;
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[U,S,V,ranks] = numranks(moment);
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if options.moment.extractrank>0
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% We try a extraction from highest order moment no matter what
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flat = length(moment);
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ranks(flat) = options.moment.extractrank;
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else
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% Find a flat extension
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flat = d+min(find(ranks(1+d:end)-ranks(1:end-d)==0));
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end
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if ~isempty(flat)
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% Find a basis
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r = ranks(flat);
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V = U{flat}(:,1:r)*sqrt(diag(diag(S{flat}(1:r,1:r))));
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if options.moment.rceftol >= 0
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[R,pivot] = rref(V',options.moment.rceftol);R = R';
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else
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% Try to find a reasonable tolerance by avoiding severly badly
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% conditioned R. Hack.., but seem to behave rather robustly.
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cV = cond(V);
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tol = 1e-10;
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[R,pivot] = rref(V',tol);R = R';
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while tol<1 & (cond(R)/cond(cV)>1e4)
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tol = tol*5;
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[R,pivot] = rref(V',tol);R = R';
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end
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end
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% Figure out multiplying matrices using YALMIP code
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w = monomials(pivot);
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for i = 1:n
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xw = x(i)*w;
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k = [];
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for j = 1:length(xw)
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k = [k;find(ismember(xw(j),monomials))];
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end
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N{i} = R(k,:);
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end
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% Things missing in the basis...
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if ~all(cellfun('prodofsize',N)==length(w)^2)
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x_extract = {[]};
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return;
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end
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% Create random convex combination
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rands = rand(n,1);rands = rands/sum(rands);
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M = 0;
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for i = 1:n
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M = M + rands(i)*N{i};
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end
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[Q,T] = schur(M);
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% Extract solution
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for i = 1:r
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for j = 1:n
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x_extract{i}(j,1) = Q(:,i)'*N{j}*Q(:,i);
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end
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end
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% Refine solutions v-Rw = e(x)=0
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if options.moment.refine>0
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xtemp = double(x);
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e = monomials(1:size(R,1))-R*w;
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dedx = jacobian(e,x);
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for j = 1:r
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assign(x,x_extract{j});
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for i = 1:options.moment.refine
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assign(x,double(x)-double(dedx)\double(e));
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end
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x_extract{j} = double(x);
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end
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assign(x,xtemp);
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end
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else
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x_extract = {};
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end
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% Somewhat more stable rank detection
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% looking for sharp drops in singular value
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function [U,S,V,ranks] = numranks(moment)
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for i = 1:length(moment)
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[U{i},S{i},V{i}] = svd(moment{i});
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s = diag(S{i});
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decay = s(2:end)./(eps+s(1:end-1));
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r = min(find(decay<1e-3));
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if isempty(r)
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ranks(i) = rank(moment{i},1e-8);
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else
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ranks(i) = min(r,rank(moment{i},1e-8));
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end
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end
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