Using fmincon to maximize a complex scalar function

N = 1;
h = sqrt(0.5)*(randn(1,N)+1i*randn(1,N));
g = sqrt(0.5)*(randn(N,1)+1i*randn(N,1));
phi0 = zeros(N,1);
lb = zeros(N,1);
ub = 2*pi*ones(N,1);
obj = @(phi)-((h*diag(exp(1i*phi))*g));
[sol,fval] = fmincon(obj,phi0,[],[],[],[],lb,ub)

Error using barrier
Objective function is undefined at initial point. Fmincon cannot continue.

Error in fmincon (line 886)
[X,FVAL,EXITFLAG,OUTPUT,LAMBDA,GRAD,HESSIAN] = barrier(funfcn,X,A,B,Aeq,Beq,l,u,confcn,options.HessFcn, ...

phi1 = -(angle(h)+angle(g.'));

If in the above code i am trying to maximize ((h*diag(exp(1i*phi))*g)) where all h,g are complex scalar. Why matlab is giving such error, is it because objective function is complex, or some other reason?

Answer 1

[Walter Roberson]

https://www.mathworks.com/help/optim/ug/fmincon.html#busog7r-fun

Function to minimize, specified as a function handle or function name. fun is a function that accepts a vector or array x and returns a real scalar f, the objective function evaluated at x.

So, yes, the fact that the objective function is complex is a problem. The maximum of complex numbers is not mathematically defined.

Are you trying to find something like the vector that is "most orthagonal" to the inputs ? If so then you should consider the square of the norm.

[Shankul Saini]

Thank you @walter this cleared the doubt

Answer 2

[Matt J]

It is because your objective is returning a complex value at phi0.

>> obj(phi0)

ans =

1.3978 + 1.8395i

[Shankul Saini]

Ok yeah thanks @Matt J for the insight 🙌👍.