A Semidefinite Programming Method for Integer Convex Quadratic Minimization

TL;DR

This paper introduces a semidefinite programming relaxation approach for the NP-hard problem of minimizing convex quadratic functions over integers, providing bounds and a randomized algorithm for solutions.

Contribution

It presents a novel SDP relaxation and a probabilistic method to find suboptimal solutions for integer convex quadratic minimization.

Findings

Effective bounds on the optimal value demonstrated

Randomized algorithm produces good suboptimal solutions

Method tested successfully on various problem sizes

Abstract

We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice ${\bf Z}^n$. We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.