I want to solve this problem

The constraints are not convex so cvx is inapplicable. (I tried CVX previously.)

I am thinking of using matlab optimization toolbox. I tried the following codes but they failed.

Nt = 4 ;  
M =  64  ; 
Gamma_R = 10^(3/10); 
Gamma_T = 10^(3/10);
gamma_r = 0 ;
gamma_t = 0 ;
noise_variance_dBm = -70;
noise_variance = 10^(-70/10)*1E-3 ;
h_r =(randn(M,1)+1i*randn(M,1))/sqrt(2);
h_t =(randn(M,1)+1i*randn(M,1))/sqrt(2);
h_d =(randn(Nt,1)+1i*randn(Nt,1))/sqrt(2);
Phi_r =(diag(diag(rand(M,M)+1i*rand(M,M))))/sqrt(2); 
Phi_t =(diag(diag(rand(M,M)+1i*rand(M,M))))/sqrt(2); 
for n = 1:length(Nt)
    N = Nt(n);
    G = (randn(M,N)+1i*randn(M,N))/sqrt(2);
end
theta_r = zeros(M,M) ;
theta_t = zeros(M,M) ;
zeta    = zeros(M,M) ;
a_r = [h_r'*Phi_r*G+h_d' , h_r'*Phi_r*G+h_d']*[eye(Nt), zeros(Nt,Nt);zeros(Nt,Nt),zeros(Nt,Nt)]; %.*[2*Nt,2*Nt] ;
a_t = [h_r'*Phi_r*G+h_d' , h_r'*Phi_r*G+h_d']*[zeros(Nt,Nt), zeros(Nt,Nt);zeros(Nt,Nt),eye(Nt)]; %.*[2*Nt,2*Nt] ;
b_r = [h_t'*Phi_t*G , h_t'*Phi_t*G ]*[eye(Nt), zeros(Nt,Nt);zeros(Nt,Nt),zeros(Nt,Nt)] ;
b_t = [h_t'*Phi_t*G , h_t'*Phi_t*G ]*[zeros(Nt,Nt), zeros(Nt,Nt);zeros(Nt,Nt),eye(Nt)] ;
%%
Asc = a_r;
bsc = sqrt(Gamma_R)*sqrt(abs(a_t).^2 + noise_variance^2);
dsc = 0;
gamma = 0 ;
conecons(2) = secondordercone(Asc,bsc,dsc,gamma);

Error using assert
Number of columns in the first argument must equal the number of elements in the third argument.

Error in secondordercone (line 13)
assert(numel(d) == size(A, 2), message('optim:coneprog:SizeMismatchColsOfSocAandD'));

Does anyone have ideas how Matlab toolboxes can be used to solve the problem or how the problem can be solved with matlab based libraries?

[Matt J]

Why secondordercone()? The problem you have posted has a quadratic objective whereas the problem structure assumed by secondordercone() has a linear objective:

[Sitthipong]

Thank you very much. How to solve this problem?

[Torsten]

Use "fmincon".

And use your constraints (to be defined in nonlcon) in the squared form:

(a_r'*w)^2 - gamma_r * ((a_t'*w)^2 + sigma^2) >=0
(b_t'*w)^2 - gamma_t * ((b_r'*w)^2 + sigma^2) >=0

[Sitthipong]

[Walter Roberson]

fmincon can only handle complex values when there are no constraints, if I recall correctly.

[Sitthipong]

Thank you very much. How to solve this problem?

[Torsten]

Include your code - we cannot run a graphics snapshot.

Answer 1

[Alan Weiss]

You can use secondordercone by making a new variable m, a linear objective m, and another second-order cone constraint:

Minimize m such that .

You have to be careful when using complex numbers. Optimization toolbox solvers generally don't work well with complex numbers. Please check that you are satisfying the assumptions of the toolbox.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation