Constrained optimization with complex variables

Question

Is there a theory of constrained optimization with complex variables, do you know any textbook on that topic? The typical textbooks on constrained optimization deal with real variables. I actually found some references which loosely explain the procedure of derivating w.r.t. the complex conjugate to obtain optimality conditions but could not find a rigorous mathematical justification. In particular I am interested in using a Lagrangian, and couldn't find any reference on that. Thanks.

Answer 1

I realize this is an old question, but there are several good references on this subject. While the objective function must map to the real line, its argument is free to be complex, which means that its gradient and Hessian can be complex (Hermitian instead of symmetric), however, it is important to recognize that you are not just optimizing $f(z)$ where $f:\mathbb{C}^n\rightarrow\mathbb{R}$ over $z$, but also its complex conjugate $\bar z$, since $f$ cannot be holomorphic and be real-valued.

Constraints are free to be complex $c:\mathbb{C}^n\rightarrow \mathbb{C}^m$, however, the Lagrangian must be real-valued. This approach will lead to exactly the same results as using $z=x+iy$ above.

Optimization over complex variables invokes what is typically called the Wirtinger Calculus (mostly in German references and $\mathbb{CR}$-calculus in in English ones.) A nice clear pedagogical introduction to complex differentiation in with emphasis on optimization can be found here.