I am trying to find the maximum of a hermitian positive definite quadratic form x...

cambrassk3

cambrassk3

Answered

2022-07-03

I am trying to find the maximum of a hermitian positive definite quadratic form x Q x H (where Q = Q H and all eigenvalues of Q are non-negative) over the complex unit cube | x i | 1, i = 1 , , n where x = ( x 1 , x 2 , , x n ) 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. 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!

Answer & Explanation

1s1zubv

1s1zubv

Expert

2022-07-04Added 17 answers

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.

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