Exact SDP relaxations for a class of quadratic programs with finite and infinite quadratic constraints

TL;DR

This paper develops new sufficient conditions for the exactness of SDP relaxations in quadratic programs with both finite and semi-infinite quadratic constraints, extending and generalizing previous results.

Contribution

The paper introduces three novel sufficient conditions for SDP relaxation exactness in semi-infinite QCQPs, unifying and strengthening existing theoretical frameworks.

Findings

Proposed conditions are weaker than existing ones, broadening applicability.

Established relationships among different conditions.

Examples demonstrate the effectiveness of the new conditions.

Abstract

We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality constraints (semi-infinite QCQPs). Sufficient conditions for the exactness of SDP relaxations for QCQPs with finitely many constraints have been extensively studied, notably by Argue et al. (MOR, 48:100-126, 2023), Arima et al. (SIOPT, 34:3194-3211, 2024), and Joyse and Yang (MP, 205:539-558, 2024). In this work, we present three new sufficient conditions that generalize the existing conditions in these works for both finite and semi-infinite QCQPs. Specifically, we establish relationships among the proposed and existing conditions, and prove that one of the proposed conditions is the weakest among them. Illustrative examples are also provided to demonstrate the effectiveness of the proposed conditions in comparison to the existing ones.