Separable QCQPs and Their Exact SDP Relaxations

TL;DR

This paper develops a framework for constructing larger separable QCQPs from smaller ones while preserving the exactness of their SDP relaxations, with applications to various classes of QCQPs.

Contribution

It introduces a sufficient condition for the exactness of SDP relaxations when combining separable QCQPs and identifies classes where this applies.

Findings

Exactness is preserved under separable horizontal connection of QCQPs.

The framework applies to convex, sign-pattern, graph-structural, and homogeneous QCQPs.

Constructive examples demonstrate how to generate new QCQPs with exact SDP relaxations.

Abstract

This paper studies exact semidefinite programming relaxations (SDPRs) for separable quadratically constrained quadratic programs (QCQPs). We consider the construction of a larger separable QCQP from multiple QCQPs with exact SDPRs. We show that exactness is preserved when such QCQPs are combined through a separable horizontal connection, where the coupling is induced through the right-hand-side parameters of the constraints. The proposed framework provides a simple sufficient condition for exactness of the resulting SDPR. We then identify notable classes of QCQPs for which this condition holds, including convex QCQPs, QCQPs defined by sign-pattern and graph-structural conditions, and separable homogeneous QCQPs with a limited number of constraints. Two examples illustrate the constructive nature of the proposed framework, showing how heterogeneous QCQPs can be combined to yield new instances with exact SDP relaxations.