Local Nonglobal Minima for Solving Large Scale Extended Trust Region Subproblems

TL;DR

This paper extends the understanding of trust region subproblems by characterizing local non-global minima in large-scale extended cases and proposes an efficient algorithm to find the global minimum using eigenvalue problems.

Contribution

It introduces a novel characterization of local non-global minima for large-scale extended trust region subproblems and develops an efficient eigenvalue-based algorithm for solving them.

Findings

The algorithm solves the eTRS by addressing at most three generalized eigenvalue problems.

The paper extends the eigenvalue characterization from TRS to eTRS, including the hard case.

Efficiently finds the global minimum in large-scale eTRS problems.

Abstract

We study large scale extended trust region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving large scale TRS problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS by solving at most three generalized eigenvalue problems.