Trust-region and $p$-regularized subproblems: local nonglobal minimum is the second smallest objective function value among all first-order stationary points

TL;DR

This paper reveals that the local nonglobal minimizer of trust-region and p-regularized subproblems has the second smallest objective value among all stationary points, linking it to generalized eigenvalue problems.

Contribution

It establishes a novel property of local nonglobal minimizers and connects the problem to generalized eigenvalue computations, extending previous understanding.

Findings

Local nonglobal minimizer has second smallest objective among stationary points.

Extension of property to p-regularized subproblems.

Finding the nonglobal minimizer relates to generalized eigenvalue problems.

Abstract

The local nonglobal minimizer of trust-region subproblem, if it exists, is shown to have the second smallest objective function value among all KKT points. This new property is extended to $p$-regularized subproblem. As a corollary, we show for the first time that finding the local nonglobal minimizer of Nesterov-Polyak subproblem corresponds to a generalized eigenvalue problem.