Using SCA and CVX does not converge

[LAI]

Here is my code. I can run the program using the SDPT3 solver, but it does not converge. When I try to use the MOSEK solver, MATLAB tells me that there is no solution.

while 1
cvx_begin

variable comm_rxBeam(FA_num) complex             
variable gamma0
variable zeta(UAV_num,1)
variable eta(UAV_num,1)

 expressions T_off(UAV_num,1) 
 for i = 1:UAV_num
     T_off(i) = computing_offloadingRatio(i)*Data*inv_pos(zeta(i));
  end

 minimize(gamma0)

 subject to

 gamma0 >= 0;
 zeta >= 1e-10;
 eta >= 1e-10;
 
 square_pos(norm(comm_rxBeam)) <= 1;
 
 for i = 1:UAV_num
    T_off(i) <= gamma0;
    zeta(i) <= B*log(1+comm_txPower*eta(i)*comm_referenceGain/(noisepower*d(i)^(alpha)))/log(2);
    eta(i)-2*real(comm_rxBeami'*phase(:,i)*phase(:,i)'*comm_rxBeam)+abs(comm_rxBeami'*phase(:,i))^2 <= 0;
 end

 cvx_end 

 comm_rxBeami = comm_rxBeam;

 iteration = iteration + 1;
 result = [result, gamma0];
 if (length(result)>=10) )
    break; 
 end

end

[Mark_L_Stone]

The result obtained using the SeDuMi solver is inaccurate. The optimal solution is not as good as before the optimization.

Unsafeguarded (no line search or trust region)Successive Convex Approximation (SCA) or generally, alternating variables optimization are unreliable. It might not descend (for minimization problem), i.e., iterates could get worse. It might not converge to anything; and if it does converge, it might not be to a local optimum of the original problem, let alone a global optimum. The solution of successive iterations, and therefore subproblem inputs, can become wilder and wilder, until at some point …

https://x.com/themarklstone/status/1586795881168265216

Don’t apply crude, unsafeguarded (no Trust Region or Line Search) Successive Convex Approximation (SCA) to a new problem … unless your name happens to be Stephen Boyd. There’s a reason high quality non-convex nonlinear optimization solvers are more than 10 lines long.

If you change the starting point and/or the solver you might get lucky. or you might not. You might be better off changing 1e-10 to 0,although I don’t know whether that will make any difference.

[LAI]

Thank you very much for your reply. I replaced the inv_pos function with the quad_over_lin function to achieve SCA convergence, which is quite strange. However, I have encountered a new problem and hope you can help me solve it. Thank you.

I tried to add code to further improve my work, but CVX reported an error. I am sure that this line of code is mathematically correct and very simple.

When I added Code 1, CVX told me:
Disciplined convex programming error:
Invalid numeric values (NaNs) may not be used in CVX expressions.

Error occurs in Eta(i) <= 2*real(comm_rxBeami’*phase(:,i)*phase(:,i)‘*comm_rxBeam)-abs(comm_rxBeami’*phase(:,i))^2;

When I added Code 2, CVX told me there is no solution.
Below is my code.

while 1
cvx_begin
cvx_solver MOSEK

variable comm_rxBeam(num) complex
variable Gamma
variable Zeta(UAV_num,1)
variable Eta(UAV_num,1)
variable computing_allocation(UAV_num,1)

minimize(Gamma)

subject to
Zeta >= 0;
Eta >= 0;
computing_allocation >= 0;
sum(computing_allocation) <= computing_BScapacity;

norm(comm_rxBeam,2) <= 1;
for i=1:UAV_num
    Eta(i) <= 2*real(comm_rxBeami'*phase(:,i)*phase(:,i)'*comm_rxBeam)-abs(comm_rxBeami'*phase(:,i))^2;

    Zeta(i) <= B*log(1+comm_txPower*Eta(i)*comm_referenceGain/(noisepower*d(i)^(alpha)))/log(2);

    Gamma >= quad_over_lin(sqrt(computing_offloadingRatio(i)*computing_sensingData), Zeta(i))...
        %+quad_over_lin(sqrt(offloadingRatio(i)*Data*cyclesPerBit), computing_allocation(i));%尝试添加的代码1
        %+offloadingRatio(i)*Data*cyclesPerBit*inv_pos(computing_allocation(i));%尝试添加的代码2
end
 
cvx_end 

result = [result, Gamma];
if (length(result)>=5) && (abs(result(end)-result(end-1))<1e-5)
    break; 
end
comm_rxBeami = comm_rxBeam;
end

[Mark_L_Stone]

The following error occurs:Disciplined convex programming error: Illegal operation: {real affine} - {invalid}

My guess: on an iteration, CVX did not solve a problem to optimality. Hence, the values populating the CVX variables after cvx_end for that iteration are NaN. When these values are used as input data in the next iteration, CVX will consider them invalid. Is NaN*x^2 a convex expression with respect to a CVX variable x.? You can’t say it is. Neither can CVX. CVX can’t even determine what to do with NaN*x, for which the sign of NaN is irrelevant. Apart from convexity, what problem could it provid…

[LAI]

So your point is that the function 1/x I added has made the problem more complicated. Both of my approaches are correct, but neither has found the optimal solution. Then, is the only thing I can do now to try changing the starting point, or to give up optimizing this “x” within this CVX problem and instead use alternating optimization (I don’t know what trust region should be added for SCA)。

[Mark_L_Stone]

If you want to make SCA or alternating optimization work, you’re pretty much on your own as far as this forum goes. I provided disclaimers, but you’re welcome to try anyway. Good luck.

[LAI]

Okay, thank you. It seems that I can only try to change the initial point.