[Paper Review] A Geometric View of SDP Exactness in QCQPs and its Applications

Alex L. Wang, Fatma Kılınç-Karzan|arXiv (Cornell University)|Nov 13, 2020

Advanced Optimization Algorithms Research44 references4 citations

TL;DR

This paper presents a geometric framework for analyzing exactness in semidefinite programming (SDP) relaxations of quadratically constrained quadratic programs (QCQPs). By analyzing the cone of convex Lagrange multipliers Γ and its relatives ΓP and Γ°, the authors derive sufficient (and in some cases necessary) conditions for objective value exactness and convex hull exactness, enabling systematic treatment of exactness when these geometric structures are well-understood.

ABSTRACT

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. Towards understanding when this relaxation is exact, we study general QCQPs and their (projected) SDP relaxations. We present sufficient (and in some cases, also necessary) conditions for objective value exactness (the condition that the objective values of the QCQP and its SDP relaxation coincide) and convex hull exactness (the condition that the convex hull of the QCQP epigraph coincides with the epigraph of its SDP relaxation). Our conditions for exactness are based on geometric properties of $Γ$, the cone of convex Lagrange multipliers, and its relatives $Γ_P$ and $Γ^\circ$. These tools form the basis of our main message: questions of exactness can be treated systematically whenever $Γ$, $Γ_P$, or $Γ^\circ$ is well-understood. As further evidence of this message, we apply our tools to address questions of exactness for a prototypical QCQP involving a binary on-off constraint, quadratic matrix programs, the QCQP formulation of the partition problem, and random and semi-random QCQPs.

Motivation & Objective

• To develop a geometric framework for understanding when SDP relaxations of QCQPs yield exact solutions.
• To address the limitations of prior work that required polyhedral sets of projective convex Lagrange multipliers.
• To unify and extend existing results on objective value and convex hull exactness in QCQPs.
• To apply the framework to concrete problems such as binary on-off constraints, matrix programs, and random QCQPs.
• To establish conditions under which the convex hull of the QCQP epigraph coincides with the epigraph of its SDP relaxation.

Proposed method

• Introduces the cone of convex Lagrange multipliers Γ and its projective variant ΓP to characterize exactness in QCQPs.
• Defines the polar cone Γ° to analyze boundary and extremal behavior of Lagrange multiplier structures.
• Uses geometric duality and semidefinite programming duality to relate QCQP solutions to SDP relaxations.
• Applies dimension-counting arguments and subspace analysis to construct perturbations that test exactness.
• Employs vector and matrix perturbation techniques to bound deviations from exact solutions using ε-approximate feasibility.
• Leverages the S-lemma and rank-one-generated (ROG) cone properties as special cases within the geometric framework.

Experimental results

Research questions

• RQ1 Under what geometric conditions on the Lagrange multiplier cone Γ is the SDP relaxation of a QCQP exactly equal to the original problem’s optimal value?
• RQ2 When does the convex hull of the QCQP epigraph coincide with the epigraph of its SDP relaxation?
• RQ3 How can the framework be applied to QCQPs with non-polyhedral multiplier sets, such as those arising from binary constraints or matrix programs?
• RQ4 What role does the structure of ΓP and Γ° play in certifying exactness for general QCQPs beyond diagonal or single-constraint cases?
• RQ5 Can the geometric approach be used to derive quantitative bounds on the gap between QCQP and SDP solutions in semi-random or structured QCQPs?

Key findings

• The paper establishes sufficient and in some cases necessary conditions for objective value exactness based on the geometric structure of the Lagrange multiplier cone Γ.
• It proves that convex hull exactness holds when the epigraph of the QCQP is equal to the epigraph of its SDP relaxation, which is guaranteed under specific geometric conditions on ΓP and Γ°.
• The framework generalizes the S-lemma and previous results on diagonal QCQPs by unifying them under a common geometric lens.
• For the on-off constraint QCQP, the authors derive exactness conditions using the geometry of Γ and its polar Γ°, showing that exactness holds when the multiplier cone is well-behaved.
• In semi-random QCQPs, the method enables quantitative bounds on the optimality gap via ε-approximate feasibility and perturbation analysis.
• The analysis reveals that when Γ° is well-understood, exactness can be systematically certified, even when ΓP is not polyhedral.