# Extended Trust-Region Problems with One or Two Balls: Exact Copositive and Lagrangian Relaxations

## TL;DR

This paper introduces geometric conditions under which copositive relaxations provide exact solutions for certain nonconvex quadratic problems, surpassing traditional Lagrangian relaxations, with practical verifiability via linear optimization.

## Contribution

It establishes new conditions for exact copositive relaxation in extended trust-region problems, improving solution accuracy over standard methods.

## Key findings

- Copositive relaxation is tighter than Lagrangian relaxation.
- Certain quadratic problems have exact copositive relaxation with infinite Lagrangian duality gap.
- Verifiable conditions for exact relaxations via linear optimization.

## Abstract

We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush-Kuhn-Tucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are exact for an extended CDT (two-ball trust-region) problem. Importantly, the sufficient conditions can be verified by solving linear optimization problems.