Question of logdet with a complex matrix input

[Lina]

Hi guys,

I met a problem of max min of logdet with a complex matrix variable. The objective is to max min (R_1, R_2).   I show a simple version of my code below:

H_1=(randn(2,4)+j*randn(2,4))/sqrt(2);

H_2=(randn(2,4)+j*randn(2,4))/sqrt(2);

Q_p = sdpvar(4,4,'hermitian','complex');

t=sdpvar(1,1);

Objective=t;

Constraints = [ logdet(eye(N_r)+H_2*Q_p*H_2') - real(trace(H_2*Q_p*H_2'))>t , logdet(eye(N_r)+H_1*Q_p*H_1')- real(trace(H_1*Q_p*H_1'))>t , Q_p>0 , trace(Q_p) <= 10];

sol = optimize(Constraints,-Objective);

It shows the error of:

Undefined operator '>' for input arguments of type 'logdet'.

It should be a convex optimization problem. May I know what I can do to solve the problem. Thanks.

[Johan Löfberg]

logdet is only meant to be used as a term in an objective when using logdet-optimizing SDP solvers.

Maybe it is possible to derive a structured problem though

You have

logdet(I+X) - z >= t

i.e.

det(I+X) >= exp(z  +t)

i.e.

det(I+X)^(1/n) >= (exp(z+t))^(1/n)

i.e.

geomean(I+X) >= exp((z+t)/n)


geomean is a concave operator, and exp is convex, hence the constraint is convex. geomean is SDP representable, and if you use the exponential-cone solver SCS which is interfaced, it supports the intersection of the semidefinite cone and the expoenential cone

[Johan Löfberg]

and note that you cannot use strict inequalities, YALMIP is screaming at you about this.

[Lina]

Thanks for your reply Johan.

I tried this method, and change the constraint into: 

Constraints = [ geomean(eye(N_r)+H_2*Q_p*H_2') >=exp((t+real(trace(H_2*Q_p*H_2')))/N_r) , geomean(eye(N_r)+H_1*Q_p*H_1')>=exp((t+real(trace(H_1*Q_p*H_1')))/N_r) , Q_p>=0 , trace(Q_p) <= 10];

but still shows the error:    

 solvertime: 0

          info: 'No suitable solver'

       problem: -2

    yalmiptime: 0.3800

I installed SDPT3, but it is still unable to solve it.

[Lina]

I added 

Constraints =[Constraints, uncertain(t)];

But still shows the error:

Error using decomposeUncertain (line 21)

There is no uncertainty description in the model.

Error in robustify (line 65)

[UncertainModel,Uncertainty,VariableType,ops] =

decomposeUncertain(F,h,w,ops);

Error in solverobust (line 64)

[F,h,failure] = robustify(varargin{:});

Error in solvesdp (line 157)

        diagnostic =

        solverobust(F(find(~unc_declarations)),h,options,recover(getvariables(sdpvar(F(find(unc_declarations))))));

        

Error in optimize (line 31)

[varargout{1:nargout}] = solvesdp(varargin{:});

Error in main_yalmip_testing (line 84)

optimize(Constraints,-Objective)

May I know whether there is a possible method to solve the problem? Thanks!

[Johan Löfberg]

But I told you that you had to use SCS

[Johan Löfberg]

Why would you do that? It makes no sense to declare t as an uncertain variable

[Lina]

Thanks Johan. It works! The problem I shown above can be solved successfully using "SCS". I am sorry I didn't see your suggestions carefully. 

Actually, the problem I need to solve is max min (R_1, R_2)+R_3. There is an additional logdet in the objective function. 

H_1=(randn(2,4)+j*randn(2,4))/sqrt(2);

H_2=(randn(2,4)+j*randn(2,4))/sqrt(2);

H_3=(randn(2,4)+j*randn(2,4))/sqrt(2);

Q_p = sdpvar(4,4,'hermitian','complex');

t=sdpvar(1,1);

Objective=t+ logdet(eye(N_r)+H_3*Q_p*H_3');

Constraints =[ geomean(eye(N_r)+H_2*Q_p*H_2') >=exp((t+real(trace(H_2*Q_p*H_2')))/N_r) , geomean(eye(N_r)+H_1*Q_p*H_1')>=exp((t+real(trace(H_1*Q_p*H_1')))/N_r) , Q_p>=0 , trace(Q_p) <= 10];

It is still a convex problem. But both "SCS" and "SDPT3" don't work for this problem. Could you please give me any suggestion how to solve it.

Thanks so much.

[Johan Löfberg]

logdet is an operator solely for logdet solvers (i.e., sdpt3). You t owrite it using geomean instead by using det(X) = geomean(X)^n

Objective= t+n*log(geomean(eye(N_r)+H_3*Q_p*H_3'));

[Lina]

It really works! Many Thanks! I do appreciate it!