A new dual for quadratic programming and its applications

TL;DR

This paper introduces a novel dual for quadratic programming with strong duality, develops a semi-definite relaxation bound, and proposes a globally convergent branch and cut algorithm for concave quadratic programs, demonstrated on numerical instances.

Contribution

It presents a new dual formulation for quadratic programs, establishes a semi-definite relaxation bound, and develops a globally convergent branch and cut algorithm for concave quadratic programs.

Findings

New dual with strong duality for quadratic programs

Semi-definite relaxation bound improves existing bounds

Proposed algorithm demonstrates global convergence and effectiveness

Abstract

The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound for quadratic programs. We demonstrate that the dual of the bound is a semi-definite relaxation of quadratic programs. In addition, we probe the relationship between this bound and the well-known bounds. In the second part, thanks to the new bound, we propose a branch and cut algorithm for concave quadratic programs. We establish that the algorithm enjoys global convergence. The effectiveness of the method is illustrated for numerical problem instances.