Separable QCQPs and Their 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.

• The paper presents a framework that couples independently exact QCQPs to preserve SDP relaxation exactness through a horizontal connection approach.
• It establishes sufficient conditions for exact SDP relaxations by integrating convex, sign-pattern, and homogeneous QCQP classes.
• The analysis provides rank-based results and practical examples that guide scalable solver development for nonconvex optimization.

Separable QCQPs and Exact Semidefinite Relaxations: A Technical Perspective

Introduction

The paper "Separable QCQPs and Their Exact SDP Relaxations" (2604.02968) investigates the propagation of exactness in semidefinite programming relaxations (SDPRs) for quadratically constrained quadratic programs (QCQPs) with separable structures. QCQPs are central to nonconvex optimization, with applications spanning signal processing, control, and finance. While exact SDPRs are established for certain specialized QCQP classes (e.g., convex problems, sign-pattern-structured cases, and homogeneous separable QCQPs with bounded constraint counts), there has been a lack of systematic understanding regarding the preservation of exactness when these classes are combined through separable (block-wise) constructions. This work addresses that theoretical gap, focusing on separable horizontal connections of independently exact QCQPs and providing general sufficient conditions for the exactness of the aggregate SDPR.

Problem Formulation and Horizontal Connection Framework

The primary construct is the horizontal connection of $\hat{p}$ sub-QCQPs, where each subproblem possesses exact SDPR. The aggregate QCQP formed by coupling the subproblems via shared right-hand side constraint parameters is given by: $$\zeta(\gamma) = \inf \left\{ \sum_{p=1}^{\hat{p}} f^p_0(x^p) : \sum_{p=1}^{\hat{p}} f^p_k(x^p) \trianglelefteq_k \gamma_k, \, 1 \leq k \leq m \right\}$$ The subproblems may be heterogeneous, potentially belonging to distinct structural classes.

The paper establishes that if each sub-QCQP’s SDPR is exact whenever it is bounded (Assumption I) and the aggregate SDPR admits an optimal solution (Assumption II), then the aggregate SDPR is also exact. This result is encapsulated in Theorem 1, which is the paper’s central technical contribution.

Preservation of Exactness and Sufficient Conditions

The exactness preservation result leverages the separable structure of both the primal and dual (SDP) formulations. By exploiting block-diagonal sparsity and chordal decomposition, the proof uses a minimality argument: any improvement to a subproblem’s SDPR would propagate to the full problem, contradicting optimality. Critical to this is the coupling through right-hand sides, ensuring the composite problem remains amenable to blockwise analysis.

Eligible classes of sub-QCQPs for the framework include:

• Convex QCQPs: If all quadratic forms are convex, SDPR is exact for all right-hand sides, independent of coupling.
• Sign-Pattern and Graph-Structural QCQPs: If coefficient matrices satisfy sign-definiteness or have off-diagonal nonpositivity and the aggregate sparsity graph meets certain properties, exactness holds (e.g., as in [Sojoudi and Lavaei, SIAM J. Optim. 24]).
• Separable Homogeneous QCQPs: When all functions are homogeneous quadratics, exactness extends when the number of constraints is at most one plus the number of separable blocks, modulo further rank-based refinements.

This analysis unifies several disparate literatures, demonstrating that diverse exactness guarantees interoperate under the horizontal connection construction.

Rank-Based Exactness for Separable Homogeneous QCQPs

The paper extends classical results by relaxing the classical $m \leq \hat{q}+1$ bound for exactness in homogeneous separable QCQPs via a refined rank-based sufficient condition. The derived result stipulates that, provided at least $m-1$ matrix variables or residuals are nonzero in any optimal SDPR solution, then rank-one block optimality holds, yielding exactness. Explicit discussion illustrates that the sufficiency is not always necessary: certain pathologies or structural degeneracies can induce loss of exactness despite admissible bounds, as shown in the paper’s parametric examples.

Examples and Constructive Applications

Two illustrative examples are presented:

1. Separable Homogeneous QCQP: The dependence of exactness on constraint parameters (and not just problem size) is made explicit, showing rank violation (and hence gap) can occur for measure-zero parameter sets.
2. Heterogeneous Combinations: A larger separable QCQP is synthesized from three distinct subproblems — one convex, one sign-pattern-driven, and one homogeneous separable. The framework’s constructive nature is demonstrated as the exactness of the composite SDPR is certified by verifying the subproblem conditions.

These examples underscore the framework’s utility as a tool for generating new large-scale QCQPs where global optimality can be certified via SDPR.

Implications and Future Directions

This work’s framework augments the toolset for certifying global optimality in structured nonconvex QCQPs, particularly in systems engineering scenarios where independent subsystems, each with tractable SDPRs, are coupled via resource or regularization constraints. Practically, this can accelerate solver development for decomposable large-scale problems and suggests systematic approaches for constructing problem instances where SDPR lower bounds are tight.

Theoretically, the horizontal connection paradigm complements existing “vertical extension” frameworks, where problems are extended by constraint augmentation rather than coupling. Combining both directions offers a roadmap for exploring the full closure under SDPR exactness-preserving operations.

Potential avenues for future development include:

• Extending the horizontal connection results to problems lacking full separability or with overlapping variable blocks,
• Elucidating necessary as well as sufficient conditions for exactness under coupling,
• Exploring algorithmic impacts (e.g., exploiting exactness-preserving block decompositions for scalable SDP solvers).

Conclusion

This paper delivers a principled and general mechanism for constructing separable QCQPs with guaranteed exact SDPRs by horizontally coupling exact subproblems. The unification of prior exactness classes within this framework, combined with the rank-based extension for homogeneous settings, advances both the theoretical and practical frontiers of nonconvex quadratic programming relaxations. Such systematic frameworks are essential for scalable global optimization and structured problem certification in mathematical programming.