Solver not applicable

[valuestay]

Dear Professor,
This is an optimization problem I encountered in reading literature. The author converted the optimization problem into a convex optimization problem and then solved it with CVX toolbox. I want to use yalmip to reproduce the code, but the error occurred, I don't know how to fix it.
A part of literature:

my code:
test_data.zip

unzip('parameter.zip');
load('parameter.mat');
a_t=(abs(h_d1)+sum(abs(h_1*diag(h_r)),2)).^2;
a_r=(abs(h_d2)+sum(abs(h_2*diag(h_r)),2)).^2;

q=conj(g_1')*diag(g_0);
q_=[q g_d1'];
Q=zeros(M+1,M+1,k);
for i=1:k
    Q(:,:,i)=conj(q_(i,:)')*q_(i,:);
end

theta0_r = sdpvar(M,1,'full','complex');
t0 = sdpvar(1,1);
tk = sdpvar(k,1);
e=sdpvar(k,1);

theta0_r_=[theta0_r;1];
V0=theta0_r_*conj(theta0_r_');
W0=V0*t0;

Constraints1 = [diag(W0)==ones(size(W0,1),1)*t0,...
                           t0+sum(tk)<=T,t0>=0,tk>=0,e>=0];
for i=1:k
    Constraints1=[Constraints1,e(i)<=eta*P*trace(Q(:,:,i)*W0)];
end

Objective1=0;
for i=1:k
    Objective1=Objective1-tk(i)*log(1+e(i)*a_t(i)/tk(i)/sigma2);
end

% options = sdpsettings('solver','SDPT3');
% sol1 = optimize(Constraints1,Objective1,options); 
sol1 = optimize(Constraints1,Objective1); 

if sol1.problem == 0
    % 获取解的值
    R = -value(Objective1);
else
    disp('出错了!');
    sol1.info
    yalmiperror(sol1.problem)
end

error:
I thought that the optimization problem should be solved with SDPT3, but an error was reported :

Solver not applicable (sdpt3 does not support nonconvex quadratic terms in objective)

yalmip automatically assigned fmincon solver to me, and displayed

fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are 
satisfied to within the value of the constraint tolerance.

after iteration, and the obtained solution was incorrect.

[johanlofberg]

You seem to miss the core idea of a semidefinite relaxation, as you simply try to solve the nonconvex problem now, as you introduce W0 as the outer product of two vectors, i.e. a bilinear expression, which then is used in the model creating nonconvex quadratic stuff. The whole idea is that you introduce W0 as a symmetric matrix with a psd constraint, and then remove (i.e. never introduce) the constraint that it is low-rank.

SDPT3 can never be used here as you have a logarithmic operator in the objective. You need an exponential cone solver such as mosek (cvx has some old stuff where it approximates the log stuff, but these days you use an exponential cone solver)

Also, if that objective actually is exponential cone representable, you will have to rewrite it first using operators which are automatically detected as expcone representable. Just being onvex does not suffice (neither for cvx nor yalmip). From a quick glance, if this really is expcone-representable, it is likely some Kullback-Leibler term https://yalmip.github.io/command/kullbackleibler since you have terms of the for x log(1+y/x) = xlog((x+y)/x) = -xlog(x/(x+y)) = -kullbackleibler(x,x+y) or something like that

[johanlofberg]

Show the code

[valuestay]

a_t=(abs(h_d1)+sum(abs(h_1*diag(h_r)),2)).^2;
a_r=(abs(h_d2)+sum(abs(h_2*diag(h_r)),2)).^2;

q=conj(g_1')*diag(g_0);
q_=[q g_d1'];
Q=zeros(M+1,M+1,k);
for i=1:k
    Q(:,:,i)=q_(i,:)'*q_(i,:);
end

theta0_r = sdpvar(M,1,'full','complex');
t0 = sdpvar(1,1);
tk = sdpvar(k,1);
e=sdpvar(k,1);
W0=sdpvar(M+1,M+1,'hermitian','complex');

theta0_r_=[theta0_r;1];
V0=theta0_r_*theta0_r_';

Constraints1 = [W0>=0,W0==V0*t0,diag(W0)==ones(size(W0,1),1)*t0,...
                           t0+sum(tk)<=T,t0>=0,tk>=0,e>=0];
for i=1:k
    Constraints1=[Constraints1,e(i)<=eta*P*trace(Q(:,:,i)*W0)];
end

Objective1=0;
for i=1:k
    Objective1=Objective1+kullbackleibler(tk(i),tk(i)+e(i)*a_t(i)/sigma2);
end

sol1 = optimize(Constraints1,Objective1); 

if sol1.problem == 0
    % 获取解的值
    R = -value(Objective1);
else
    disp('出错了!');
    sol1.info
    yalmiperror(sol1.problem)
end

[johanlofberg]

W0==V0*t0

I cannot repeat what I have said once more...Please read my answers and the paper on what a semidefinite relaxation is

[valuestay]

I apologize for my foolishness.Now I only keep W0, tau_1, tau_k, and e as decision variables, and modify the code to

···
t0 = sdpvar(1,1);
tk = sdpvar(k,1);
e=sdpvar(k,1);
W0=sdpvar(M+1,M+1,'hermitian','complex');

Constraints1 = [W0>=0,diag(W0)==ones(size(W0,1),1)*t0,...
                           t0+sum(tk)<=T,t0>=0,tk>=0,e>=0];
···

[valuestay]

Oh,it worked,mosek was invoked successfully！！！Thank you very much for your patient answer, which helps me a lot.❤