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October 16, 2025Open Access

Extending Exact SDP Relaxations of Quadratically Constrained Quadratic Programs

Key Points

This research shows that additional quadratic inequality constraints can maintain exact SDP relaxation in QCQPs.
By analyzing three classes of QCQPs, the study reveals that rank-one generated cones are pivotal for exact solutions.
Incorporating proposed conditions, the work extends the application of exact SDP relaxations to a broader range of QCQPs.
Illustrative examples demonstrate how these conditions influence the structure of QCQPs and their SDP relaxations.

Abstract

The semidefinite (SDP) relaxation of a quadratically constrained quadratic program (QCQP) is called exact if it has a rank-1 optimal solution corresponding to a QCQP optimal solution. Given an arbitrary QCQP whose SDP relaxation is exact, this paper investigates incorporating additional quadratic inequality constraints while maintaining the exactness of the SDP relaxation of the resulting QCQP. Three important classes of QCQPs with exact SDP relaxations include (a) those characterized by rank-one generated cones, (b) those by convexity, and (c) those by the sign pattern of the data coefficient matrices. These classes have been studied independently until now. By adding quadratic inequality constraints satisfying the proposed conditions to QCQPs in these classes, we extend the exact SDP relaxation to broader classes of QCQPs. Illustrative QCQP instances are provided.

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Cite This Study

Kojima et al. (Fri,) studied this question.

synapsesocial.com/papers/68f147cc724575985c3fcfc9 https://doi.org/https://doi.org/10.48550/arxiv.2504.03204

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