Exact Solution Methods for the $k$-item Quadratic Knapsack Problem

TL;DR

This paper introduces an exact semidefinite optimization approach for solving the 0-1 $k$-item quadratic knapsack problem, improving solution accuracy and efficiency through relaxation strengthening and comparative analysis.

Contribution

It presents a novel semidefinite relaxation method with rank constraints and polyhedral strengthening for the $k$-item quadratic knapsack problem, along with experimental comparisons.

Findings

The proposed method effectively solves various instances of the problem.

Strengthening relaxations improves solution quality.

Comparative analysis highlights advantages over existing methods.

Abstract

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.