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how can I proof this value

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how can I proof this value

Max -logdet(S)
st.(1)LMI>=0 , LMI=[S,Phi*S,Phi1*S,Phi2*S;(Phi*S)',S-r*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)

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