
How to prove this value
Max logdet(S)
st.(1)LMI>=0 , LMI=[S,Phi*S,Phi1*S,Phi2*S;(Phi*S)',Sr*S,0,0;(Phi1*S),0,r*S,0;
(Phi2*S)',0,0,r*S];
(2)S>0;
(3)0<r<1;
(4)K*S*K'<=um
S is positive definite matrix variable;Phi,Phi1,Phi2,K,um is fixed.
r is scalar variable.
Because of the product of S and r, I can only maxmize the object by giving r
ie. 20 value between 0 and 1 and run the program in matlab 20 times and find
the largest objective in the 20 cycle. I find there is always a single r such that
the objective is the largest. But I want to proof it in maths , I can not make it
I try to use S prosedure but I can not.
I also try to proof its convexity, but it seems useless.
Do you have any better idea? Thanks very much

Sorry the objective should be Max logdet(S)