Constructing QCQP Instances Equivalent to Their SDP Relaxations

TL;DR

This paper explores methods to construct specific QCQP instances that are equivalent to their SDP relaxations, extending previous results and providing ways to derive optimal solutions from SDP relaxations.

Contribution

It introduces constructions of QCQP instances equivalent to their SDP relaxations, including higher-dimensional cases, and discusses solution extraction from SDP.

Findings

Constructed QCQP instances with two variables and extended to higher dimensions.

Demonstrated how to compute optimal QCQP solutions from SDP relaxations.

Extended the class of known equivalent QCQP instances.

Abstract

General quadratically constrained quadratic programs (QCQPs) are challenging to solve as they are known to be NP-hard. A popular approach to approximating QCQP solutions is to use semidefinite programming (SDP) relaxations. It is well-known that the optimal value $\eta$ of the SDP relaxation problem bounds the optimal value $\zeta$ of the QCQP from below, i.e., $\eta \leq \zeta$. The two problems are considered equivalent if $\eta = \zeta$. In the recent paper by Arima, Kim and Kojima [arXiv:2409.07213], a class of QCQPs that are equivalent to their SDP relaxations are proposed with no condition imposed on the quadratic objective function, which can be chosen arbitrarily. In this work, we explore the construction of various QCQP instances within this class to complement the results in [arXiv:2409.07213]. Specifically, we first construct QCQP instances with two variables and then extend them to higher dimensions. We also discuss how to compute an optimal QCQP solution from the SDP relaxation.