%Example 2.3 from E. de Klerk "Aspects of SDP" (p.33)
%
%Strict complementarity does not hold for SDP
%i.e., X + S may have zero eigenvalues

A1 = zeros(3,3); A1(1,1) = 1;
A2 = [0 0 1;0 1 0;1 0 0];
A3 = [0 1 0 ; 1 0 0; 0 0 1];
C = zeros(3,3); C(3,3) = 1; 
b = [1 0 0]';

x = sdpvar(3,1);
LMI = C + A1*x(1) + A2*x(2) + A3*x(3);

F = set(LMI>0);
solvesdp(F,x'*b,sdpsettings('solver','penbmi'));

%The optimal primal variable X
disp('Optimal X:')
disp(dual(F))

%The optimal dual (slack) variable S
disp('Optimal S:')
disp(double(LMI));