Progressive Bound Strengthening via Doubly Nonnegative Cutting Planes for Nonconvex Quadratic Programs

2025-10-03

AI Summary

This paper addresses nonconvex quadratically constrained quadratic programming (QCQP) problems by proposing an iterative tightening framework based on doubly nonnegative (DNN) relaxation and linear semidefinite programming (LSDP) cutting planes. Methodologically, it integrates DNN relaxation, LSDP cut separation, and relaxation strengthening to progressively tighten bound constraints. The key contribution is the first theoretical characterization linking Karush–Kuhn–Tucker (KKT) points to LSDP cutting planes, enabling a finite-step-convergent local search strategy that improves lower bounds efficiently—without branch-and-bound. On a benchmark set of 140 instances with 100 variables, the method reduces the relative optimality gap to within 0.01% for 138 problems within one hour, significantly outperforming state-of-the-art commercial and academic solvers in both efficiency and robustness.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

Security and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: LLM based quality controls for crowd workGraph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We introduce a cutting-plane framework for nonconvex quadratic programs (QPs) that progressively tightens convex relaxations. Our approach leverages the doubly nonnegative (DNN) relaxation to compute strong lower bounds and generate separating cuts, which are iteratively added to improve the relaxation. We establish that, at any Karush-Kuhn-Tucker (KKT) point satisfying a second-order sufficient condition, a valid cut can be obtained by solving a linear semidefinite program (SDP), and we devise a finite-termination local search procedure to identify such points. Extensive computational experiments on both benchmark and synthetic instances demonstrate that our approach yields tighter bounds and consistently outperforms leading commercial and academic solvers in terms of efficiency, robustness, and scalability. Notably, on a standard desktop, our algorithm reduces the relative optimality gap to 0.01% on 138 out of 140 instances of dimension 100 within one hour, without resorting to branch-and-bound.

Problem

• Strengthening convex relaxations for nonconvex quadratic programs via cutting planes
• Computing strong bounds using doubly nonnegative relaxation and linear SDPs
• Achieving tight optimality gaps efficiently without branch-and-bound methods

Innovation

• Cutting-plane framework tightens convex relaxations for QPs
• DNN relaxation generates separating cuts via linear SDP
• Finite-termination local search identifies KKT points efficiently