# A proximal DC approach for quadratic assignment problem

## TL;DR

This paper introduces a novel DC-based semi-proximal algorithm for solving the quadratic assignment problem by reformulating it as a rank-constrained DNN problem, demonstrating efficiency and effectiveness in finding optimal or near-optimal solutions.

## Contribution

It proposes a new DC framework and semi-proximal algorithm for the QAP, enabling efficient solution of the rank-constrained DNN reformulation.

## Key findings

- The algorithm converges to a stationary point feasible for the original problem.
- It efficiently finds global optima for most QAP instances.
- It provides good feasible solutions within reasonable computational time.

## Abstract

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm (DCA) is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method (sPALM). We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.