A Discrete Differential Evolution Approach with Local Search for Traveling Salesman Problems

Abstract

Combinatorial optimization problems are very commonly seen in scientific research and practical applications. Traveling Salesman Problem (TSP) is one nonpolynomial-hard combinatorial optimization problem. It can be describe as follows: a salesman, who has to visit clients in different cities, wants to find the shortest path starting from his home city, visiting every city exactly once and ending back at the starting point. There are exact algorithms, such as cutting-plane or facet-finding, are very complex and demanding of computing power to solve TSPs. There here, however, metaheuristics based on evolutionary algorithms that are useful to finding solutions for a much wider range of optimization problems including the TSP. Differential Evolution (DE) is a relatively simple evolutionary algorithm, which is an effective adaptive approach to global optimization over continuous search spaces. Furthermore, DE has emerged as one of the fast, robust, and efficient global search heuristics of current interest. DE shares similarities with other evolutionary algorithms, it differs significantly in the sense that distance and direction information from the current population is used to guide the search process. Since its invention, DE has been applied with success on many numerical optimization problems outperforming other more popular metaheuristics such as the genetic algorithms. Recently, some researchers extended with success the application of DE to combinatorial optimization problems with discrete decision variables. In this paper, the following discrete DE approaches for the TSP are proposed and evaluated: i) DE approach without local search, ii) DE with local search based on Lin-Kernighan-Heulsgaun (LKH) method, and iii) DE with local search based on Variable Neighborhood Search (VNS) and together with LKH method. Numerical study is carried out using the TSPLIB of test TSP problems. In this context, the computational results are compared with the other results in the recent TSP literature. The obtained results show that LKH method is the best method to reach optimal results for TSPLIB benchmarks, but for largest problems, the DE+VNS improve the quality of obtained results.