Globally Solving Concave Quadratic Programs via Doubly Nonnegative Relaxation

TL;DR

This paper introduces a novel SDP relaxation approach using doubly nonnegative relaxation and cutting plane methods to efficiently solve large-scale concave quadratic programs, outperforming traditional solvers.

Contribution

The paper presents a new framework leveraging DNN relaxation and valid cuts for globally solving concave quadratic programs, with proven theoretical properties and high computational efficiency.

Findings

Successfully solved a large-scale biological dataset with over 300,000 instances in 3 days.

Demonstrated the method's superiority over CPLEX and Gurobi in computational time.

Established theoretical equivalence and strong duality of the DNN relaxation.

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.