Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zheng Qu, Defeng Sun, Jintao Xu

TL;DR

This work develops a cutting-plane framework for nonconvex quadratic programs by leveraging a doubly nonnegative relaxation to obtain strong lower bounds and SDP-based cuts. A key theoretical result shows that, at a KKT point satisfying a second-order condition, a valid cut can be obtained via a linear SDP, and a finite-termination local search (generalized mountain climbing) can reach such points. The DCQP algorithm integrates DNN lower bounds and SDP-based cut generation, delivering a global solver that does not rely on branch-and-bound. Empirical results on benchmark and synthetic data demonstrate tight bounds and superior robustness and scalability, particularly in problems with equalities or denser data, achieving near-optimality within one hour on a standard desktop.

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.

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

TL;DR

This work develops a cutting-plane framework for nonconvex quadratic programs by leveraging a doubly nonnegative relaxation to obtain strong lower bounds and SDP-based cuts. A key theoretical result shows that, at a KKT point satisfying a second-order condition, a valid cut can be obtained via a linear SDP, and a finite-termination local search (generalized mountain climbing) can reach such points. The DCQP algorithm integrates DNN lower bounds and SDP-based cut generation, delivering a global solver that does not rely on branch-and-bound. Empirical results on benchmark and synthetic data demonstrate tight bounds and superior robustness and scalability, particularly in problems with equalities or denser data, achieving near-optimality within one hour on a standard desktop.

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.

Paper Structure

This paper contains 41 sections, 18 theorems, 95 equations, 5 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. A Generic Cutting Plane Framework
  3. Main Contributions
  4. Related Work
  5. Organization
  6. Preliminaries
  7. Notations
  8. Nondegenerate KKT Point
  9. Second Order Optimality Conditions
  10. Construction of Valid Cut
  11. Sufficient Conditions
  12. Computationally Tractable Construction of Valid Cuts
  13. Feasibility of the Dual Program
  14. A Local Search Method with Finite Convergence
  15. Generalized Mountain Climbing Method
  16. ...and 26 more sections

Key Result

Theorem 1

A point $\bar{x} \in \mathbb{R}^n$ is a local (resp. locally unique) solution of eq:QP if and only if:

Figures (5)

  • Figure 1: Comparison of the percentage of instances solved versus computational time across four groups of benchmark instances, each comprising 16 instances.
  • Figure 2: Comparison of the percentage of instances solved versus computational time, under a 3600-second time limit, across six groups of synthetic instances without equality constraints, each comprising 20 instances.
  • Figure 3: Comparison of the relative gap versus computational time for four selected instances under a 48-hour time limit.
  • Figure 4: Comparison of the percentage of instances solved versus computational time, under a 3600-second time limit, for the 20 instances in the group qp_u_25_1.
  • Figure 5: Comparison of the relative gap versus computational time for instance 18 in the group qp_u_25_3.

Theorems & Definitions (45)

  • Definition 1
  • Remark 1
  • Definition 2: Local and locally unique solution Mangasarian1980
  • Theorem 1: Mangasarian1980LuisContesse1980
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 35 more