# Parametric Convex Quadratic Relaxation of the Quadratic Knapsac Problem

## TL;DR

This paper introduces a parametric convex quadratic relaxation for the quadratic knapsack problem, optimizing perturbations and valid inequalities to obtain tighter upper bounds, with theoretical analysis and practical algorithms.

## Contribution

It proposes a novel parametric quadratic relaxation method with a primal-dual interior point approach and valid inequalities for improved bounds on QKP.

## Key findings

- The relaxation provides tighter upper bounds than existing methods.
- The perturbation approach cannot improve bounds when applied to SDP relaxations.
- Valid inequalities from cover and knapsack problems enhance the relaxation's effectiveness.

## Abstract

We consider a parametric convex quadratic programming, CQP, relaxation for the quadratic knapsack problem, QKP. This relaxation maintains partial quadratic information from the original QKP by perturbing the objective function to obtain a concave quadratic term. The nonconcave part generated by the perturbation is then linearized by a standard approach that lifts the problem to the matrix space. We present a primal-dual interior point method to optimize the perturbation of the quadratic function, in a search for the tightest upper bound for the \QKPp. We prove that the same perturbation approach, when applied in the context of semidefinite programming, SDP, relaxations of the QKP, cannot improve the upper bound given by the corresponding linear \SDP relaxation. The result also applies to more general integer quadratic problems. Finally, we propose new valid inequalities on the lifted matrix variable, derived from cover and knapsack inequalities for the QKP, and present the separation problems to generate cuts for the current solution of the CQP relaxation. Our best bounds are obtained from alternating between optimizing the parametric quadratic relaxation over the perturbation and adding cutting planes generated by the valid inequalities proposed.