New results on the local-nonglobal minimizers of the generalized trust-region subproblem

TL;DR

This paper investigates the properties and bounds of local-nonglobal minimizers in generalized trust-region subproblems, disproves a conjecture, confirms another, and proposes an algorithm with numerical validation.

Contribution

It establishes equivalence of minimizers under joint definiteness, proves bounds on the number of minimizers, and introduces an effective algorithm for finding them.

Findings

Counterexample disproves a previous conjecture.

Maximum of two local-nonglobal minimizers under positive definiteness.

Maximum of one local-nonglobal minimizer under negative definiteness.

Abstract

In this paper, we study the local-nonglobal minimizers of the Generalized Trust-Region subproblem $(GTR)$ and its Equality-constrained version $(GTRE)$. Firstly, the equivalence is established between the local-nonglobal minimizers of both $(GTR)$ and $(GTRE)$ under assumption of the joint definiteness. By the way, a counterexample is presented to disprove a conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Secondly, if the Hessian matrix pair is jointly positive definite, it is proved that each of $(GTR)$ and $(GTRE)$ has at most two local-nonglobal minimizers. This result first confirms the correctness of another conjecture of Song-Wang-Liu-Xia [SIAM J. Optim., 33(2023), pp.267-293]. Thirdly, if the Hessian matrix pair is jointly negative definite, it is verified that each of $(GTR)$ and $(GTRE)$ has at most one local-nonglobal minimizer. In special, if the constraint is reduced to be a ball or a sphere, the above result is just the classical Mart\'{i}nez's. Finally, an algorithm is proposed to find all the local-nonglobal minimizers of $(GTR)$ and $(GTRE)$ or confirm their nonexistence in a tolerance. Preliminary numerical results demonstrate the effectiveness of the algorithm.