optimization toolbox with fmincon

i have a function

f= @(x,y,z) -5-2.*x+4.*x.^3+4.*(1-x.^2).*y+8.*(1-x.^2).*x.*y-12.*(1-x.^2).*x.*y.^2+12.*(1-x.^2).*(1-(abs(y).^2)).*z

with only constraint boundary

lb = [-1 -1 -1]

ub = [1 1 1]

i used fmincon to obtain the minimum f.

[bestxyz, fval] = fmincon(f,xyz0, A, b, Aeq, beq, lb, ub, nonlcon)

The optimization fmincon result in mininum f with the value of x,y,z in real number, but the function f has value of y in complex number through analytical solution. Is it the toolbox is limited to the real root or it can be modified to have the solution with imaginary root?

[John D'Errico]

A search for a minimum is equivalent to a search for the roots of the derivatives of that function. So you can think of this as three nonlinear equations in three unknowns. And that is then equivalent to a higher order polynomial in one variable. So there certainly would be complex roots of the gradient equations.

The problem is, fmincon will not generate a complex valued solution. There are tricks you can do, but there is one glaring problem. What is it?

You place lower and upper bounds on the unknowns. bound constraints make no sense here, IF you wanted to allow complex solutions, since you have placed no constraints on the imaginary part.

Answer 1

[Torsten]

If you want to admit x,y and z to be complex-valued, you also have to change your objective function to be real-valued for complex inputs of the variables (e.g. abs(f) or f*f' or real(f) or imag(f)). What's your choice ?

Here is one example:

f = @(x1,x2,y1,y2,z1,z2) -5-2.*(x1+1i*x2)+4.*(x1+1i*x2).^3+4.*(1-(x1+1i*x2).^2).*(y1+1i*y2)+8.*(1-(x1+1i*x2).^2).*(x1+1i*x2).*(y1+1i*y2)-12.*(1-(x1+1i*x2).^2).*(x1+1i*x2).*(y1+1i*y2).^2+12.*(1-(x1+1i*x2).^2).*(1-(abs((y1+1i*y2)).^2)).*(z1+1i*z2);
g = @(x1,x2,y1,y2,z1,z2) f(x1,x2,y1,y2,z1,z2)*f(x1,x2,y1,y2,z1,z2)';

lb = [-1 -1 -1 -1 -1 -1];
ub = [1 1 1 1 1 1];
xyz0 = zeros(6,1);
[bestxyz, fval] = fmincon(@(x)g(x(1),x(2),x(3),x(4),x(5),x(6)),xyz0, [],[],[],[], lb, ub)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

bestxyz = 6×1

0.0177 -0.0579 0.1697 -0.0297 0.3719 0.0046

fval = 2.2575e-14