# An Optimality Gap Test for a Semidefinite Relaxation of a Quadratic Program with Two Quadratic Constraints

## TL;DR

This paper introduces a necessary and sufficient test to determine when solutions to a quadratic program with two quadratic constraints can be obtained from its semidefinite relaxation, even in nonconvex cases, and when strong duality holds.

## Contribution

It generalizes existing methods by providing a test applicable to nonconvex quadratic programs with two constraints, without requiring strict convexity.

## Key findings

- Test confirms no optimality gap in one example
- Test identifies an optimality gap in another example
- Method extends applicability beyond prior assumptions

## Abstract

We propose a necessary and sufficient test to determine whether a solution for a general quadratic program with two quadratic constraints (QC2QP) can be computed from that of a specific convex semidefinite relaxation, in which case we say that there is no optimality gap. Originally intended to solve a nonconvex optimal control problem, we consider the case in which the cost and both constraints of the QC2QP may be nonconvex. We obtained our test, which also ascertains when strong duality holds, by generalizing a closely-related method by Ai and Zhang. An extension was necessary because, while the method proposed by Ai and Zhang also allows for two quadratic constraints, it requires that at least one is strictly convex. In order to illustrate the usefulness of our test, we applied it to two examples that do not satisfy the assumptions required by prior methods. Our test guarantees that there is no optimality gap for the first example -- a solution is also computed from the relaxation -- and we used it to establish that an optimality gap exists in the second.