Hi, everyone!
I have to minimize this subject function ( norm( S^{b}XS^b, a ) )^a, where a>=1 and is a constant integer, b=(1-a)/(2a), the variable S is a sdp Hermitian matrix, X is a constant but not sdp matrix. It can be proofed that the subject function is convex with respect to S. I try to do this minimization in cvx, but I don’t know how to handle S^bXS^b in cvx. Below is the incomplete code.
m=0.5*[1; 0; 0; 1; 0; 0; 0; 0; 0; 0; 0; 0; 1; 0; 0; -1];
rho = mm’;
v1 = (sqrt(2.)/2.)[0; 1; 0; -1; 0; 0; 0; 0; 0];
v2 = (sqrt(2.)/2.)[0; 0; 1; 0; 0; 0; -1; 0; 0];
rhov=(v1v1’+v2*v2’)/2.0;
X=PartialTranspose(rhov, 2, [3, 3]);
a=2.0;
b=(1-a)/(2a);
cvx_begin sdp quiet
variable S(9, 9) hermitian;
minimize (norm( ,a)^a)
S >= 0 ;
trace(S)==1;
PartialTranspose(S, 2, [3, 3]) >= 0;
cvx_end