# The Generalized Trust Region Subproblem: solution complexity and convex hull results

## TL;DR

This paper provides a convex hull characterization and a linear-time algorithm for solving the Generalized Trust Region Subproblem (GTRS) with nonconvex quadratic constraints, improving computational efficiency and extending to variants.

## Contribution

It offers an explicit convex hull description of the GTRS set and develops a linear-time algorithm with error guarantees, applicable to various GTRS variants.

## Key findings

- Convex hull characterized via generalized eigenvalues.
- Linear-time algorithm for epsilon-approximate solutions.
- Applicability to multiple GTRS variants with improved efficiency.

## Abstract

We consider the Generalized Trust Region Subproblem (GTRS) of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint. A lifting of this problem recasts the GTRS as minimizing a linear objective subject to two nonconvex quadratic constraints. Our first main contribution is structural: we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil. This result may be of interest in building relaxations for nonconvex quadratic programs. Moreover, this result allows us to reformulate the GTRS as the minimization of two convex quadratic functions in the original space. Our next set of contributions is algorithmic: we present an algorithm for solving the GTRS up to an epsilon additive error based on this reformulation. We carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms. Notably, our algorithms run in linear (in the size of the input) time. Furthermore, our algorithm for computing an epsilon-optimal solution has a slightly-improved running time dependence on epsilon over the state-of-the-art algorithm. Our analysis shows that the dominant cost in solving the GTRS lies in solving a generalized eigenvalue problem -- establishing a natural connection between these problems. Finally, generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality-, interval-, and hollow- constrained variants of the GTRS. This gives the first linear-time algorithm in the literature for these variants of the GTRS.