On Local Minimizers of Quadratically Constrained Nonconvex Homogeneous Quadratic Optimization with at Most Two Constraints

TL;DR

This paper investigates the properties of local and global minimizers in nonconvex quadratically constrained quadratic optimization problems with one or two constraints, revealing conditions under which local minimizers are globally optimal and characterizing their Hessian properties.

Contribution

It proves that all local minimizers of the single-constraint case are globally optimal and establishes the necessity of second-order conditions for local non-global minimizers in the two-constraint case.

Findings

Local minimizers of (QQ1) are globally optimal.

(QQ2) can have infinitely many local non-global minimizers.

Second-order conditions are necessary for strict local non-global minimizers.

Abstract

We study nonconvex homogeneous quadratically constrained quadratic optimization with one or two constraints, denoted by (QQ1) and (QQ2), respectively. (QQ2) contains (QQ1), trust region subproblem (TRS) and ellipsoid regularized total least squares problem as special cases. It is known that there is a necessary and sufficient optimality condition for the global minimizer of (QQ2). In this paper, we first show that any local minimizer of (QQ1) is globally optimal. Unlike its special case (TRS) with at most one local non-global minimizer, (QQ2) may have infinitely many local non-global minimizers. At any local non-global minimizer of (QQ2), both linearly independent constraint qualification and strict complementary condition hold, and the Hessian of the Lagrangian has exactly one negative eigenvalue. As a main contribution, we prove that the standard second-order sufficient optimality condition for any strict local non-global minimizer of (QQ2) remains necessary. Applications and the impossibility of further extension are discussed.