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Globally Solving Concave Quadratic Programs via Doubly Nonnegative Relaxation

Zheng Qu, Tianyou Zeng, Yuchen Lou

TL;DR

This work tackles global optimization for maximizing a convex quadratic over a bounded polyhedron, an NP-hard problem with large-scale applications in biology. It introduces QuadProgCD, a cutting-plane framework that uses the doubly nonnegative (DNN) relaxation to generate deeper valid cuts via a Shor-relaxation-based equivalence, while ensuring robust bounds even when SDP solvers return inexact solutions. The authors prove strong duality for the relevant SDP and provide an explicit mechanism to derive valid upper bounds from approximate solutions, enabling scalable performance. Computational experiments show the method solving hundreds of thousands of high-dimensional instances far faster than traditional solvers, including large biology datasets and $k$ up to 819, indicating strong practical impact for large-scale global QP problems.

Abstract

We consider the problem of maximizing a convex quadratic function over a bounded polyhedral set. We design a new framework based on SDP relaxations and cutting plane methods for solving the associated reference value problem. The major novelty is a new way to generate valid cuts through the doubly nonnegative (DNN) relaxation. We establish various theoretical properties of the DNN relaxation, including its equivalence with the Shor relaxation of an equivalent quadratically constrained problem, the strong duality, and the generation of valid cuts from an approximate solution of the DNN relaxation returned by an arbitrary SDP solver. Computational results on both real and synthetic data demonstrate the efficiency of the proposed method and its ability to solve high-dimensional problems with dense data. In particular, our new algorithm successfully solves in 3 days the reference value problem arising from computational biology for a dataset containing more than 300,000 instances of dimension 78. In contrast, CPLEX or Gurobi is estimated to require years of computational time for the same dataset on the same computing platform.

Globally Solving Concave Quadratic Programs via Doubly Nonnegative Relaxation

TL;DR

This work tackles global optimization for maximizing a convex quadratic over a bounded polyhedron, an NP-hard problem with large-scale applications in biology. It introduces QuadProgCD, a cutting-plane framework that uses the doubly nonnegative (DNN) relaxation to generate deeper valid cuts via a Shor-relaxation-based equivalence, while ensuring robust bounds even when SDP solvers return inexact solutions. The authors prove strong duality for the relevant SDP and provide an explicit mechanism to derive valid upper bounds from approximate solutions, enabling scalable performance. Computational experiments show the method solving hundreds of thousands of high-dimensional instances far faster than traditional solvers, including large biology datasets and up to 819, indicating strong practical impact for large-scale global QP problems.

Abstract

We consider the problem of maximizing a convex quadratic function over a bounded polyhedral set. We design a new framework based on SDP relaxations and cutting plane methods for solving the associated reference value problem. The major novelty is a new way to generate valid cuts through the doubly nonnegative (DNN) relaxation. We establish various theoretical properties of the DNN relaxation, including its equivalence with the Shor relaxation of an equivalent quadratically constrained problem, the strong duality, and the generation of valid cuts from an approximate solution of the DNN relaxation returned by an arbitrary SDP solver. Computational results on both real and synthetic data demonstrate the efficiency of the proposed method and its ability to solve high-dimensional problems with dense data. In particular, our new algorithm successfully solves in 3 days the reference value problem arising from computational biology for a dataset containing more than 300,000 instances of dimension 78. In contrast, CPLEX or Gurobi is estimated to require years of computational time for the same dataset on the same computing platform.
Paper Structure (9 sections, 2 theorems, 31 equations)

This paper contains 9 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Related work
  4. Review on the doubly nonnegative relaxation
  5. Approach and contribution
  6. Notations
  7. Doubly Nonnegative Relaxation
  8. Equivalence of the DNN relaxation with the Shor relaxation
  9. Strong duality

Key Result

theorem 1

The quadratic program prob:original_max_qps is equivalent to

Theorems & Definitions (5)

  • theorem 1: Burer2012
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • remark thmcounterremark