On local minimizers of generalized trust-region subproblem

TL;DR

This paper investigates local minimizers of the generalized trust-region subproblem, proving conditions for their existence and complexity, and showing differences between real and complex cases with implications for polynomial-time solvability.

Contribution

It establishes necessary conditions for local nonglobal minimizers of (GT) and demonstrates polynomial-time methods for identifying all such minimizers.

Findings

Two-dimensional (GT) has at most two local nonglobal minimizers.

Necessary conditions include strict complementarity and second-order sufficiency.

No local nonglobal minimizers exist in the complex domain.

Abstract

Generalized trust-region subproblem (GT) is a nonconvex quadratic optimization with a single quadratic constraint. It reduces to the classical trust-region subproblem (T) if the constraint set is a Euclidean ball. (GT) is polynomially solvable based on its inherent hidden convexity. In this paper, we study local minimizers of (GT). Unlike (T) with at most one local nonglobal minimizer, we can prove that two-dimensional (GT) has at most two local nonglobal minimizers, which are shown by example to be attainable. The main contribution of this paper is to prove that, at any local nonglobal minimizer of (GT), not only the strict complementarity condition holds, but also the standard second-order sufficient optimality condition remains necessary. As a corollary, finding all local nonglobal minimizers of (GT) or proving the nonexistence can be done in polynomial time. Finally, for (GT) in complex domain, we prove that there is no local nonglobal minimizer, which demonstrates that real-valued optimization problem may be more difficult to solve than its complex version.