I am trying to find the maximum of a hermitian positive definite quadratic form $xQx^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\leq 1$, $i=1,\dots,n$ where $x=(x_1,x_2,\dots,x_n)\in\mathbb{C}^n$.
This is the problem of minimizing a concave function over a convex domain. I have read that this problem is NP-hard but there exist some bounds on the optimum (see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.122.6954&rep=rep1&type=pdf ) . What global optimization technique would your recommend to tackle this problem numerically? Since I am new to the field of optimization, I would appreciate every answer, thanks!
As the cube is convex and compact, it is the convex hull of its extreme points (by Krein-Milman theorem or in case of cube by its simple geometry). As the objective is convex and continuous, it has a global maximizer and one of them is an extreme point of the cube.