TSPLIB Benchmark Research Articles

Nonoblivious 2‐Opt heuristics for the traveling salesman problem

The k‐opt heuristics are among the most common techniques for approaching the traveling salesman problem (TSP). They are used either directly or as subroutines in more sophisticated heuristics, such as the celebrated Lin–Kernighan heuristic. The value of k is typically 2 or 3. In this article, we modify the 2‐opt heuristic to be based on a function f of the distances rather than the distances solely. This may be viewed as modifying the local search with the 2‐change neighborhood to be nonoblivious. We denote the corresponding heuristic by (2, f)‐opt. We provide theoretical performance guarantees for it: both lower and upper bounds based on the ones given by Chandra et al. [SIAM J Comput 28 (1999), 1998–2029], obtained originally for the standard 2‐opt heuristic. By a tighter analysis of the neighborhood size, we improve their upper bound for the standard 2‐opt by a factor of , and we show that these bounds hold for (2, f)‐opt for any nonnegative, increasing function f. We then provide experimental evidence based on TSPLIB benchmark problems, showing that (2, f)‐opt with for various values of significantly outperforms 2‐opt. These values of r also depend on the method chosen for constructing the initial tours. Specifically, when the initial tours are random permutations, the improvement over 2‐opt is more than 35% for ; when they are generated by the Nearest Neighbor heuristic, it is about 10% for r = 0.5, 0.55, 0.6. We also see that the average length of the tour generated by (2, f)‐opt is relatively close to the optimum or the known bound. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(3), 201–219 2013

Coordination of Cluster Ensembles via Exact Methods

We present a novel optimization-based method for the combination of cluster ensembles for the class of problems with intracluster criteria, such as Minimum-Sum-of-Squares-Clustering (MSSC). We propose a simple and efficient algorithm-called EXAMCE-for this class of problems that is inspired from a Set-Partitioning formulation of the original clustering problem. We prove some theoretical properties of the solutions produced by our algorithm, and in particular that, under general assumptions, though the algorithm recombines solution fragments so as to find the solution of a Set-Covering relaxation of the original formulation, it is guaranteed to find better solutions than the ones in the ensemble. For the MSSC problem in particular, a prototype implementation of our algorithm found a new better solution than the previously best known for 21 of the test instances of the 40-instance TSPLIB benchmark data sets used in [1], [2], and [3], and found a worse-quality solution than the best known only five times. For other published benchmark data sets where the optimal MSSC solution is known, we match them. The algorithm is particularly effective when the number of clusters is large, in which case it is able to escape the local minima found by K-means type algorithms by recombining the solutions in a Set-Covering context. We also establish the stability of the algorithm with extensive computational experiments, by showing that multiple runs of EXAMCE for the same clustering problem instance produce high-quality solutions whose Adjusted Rand Index is consistently above 0.95. Finally, in experiments utilizing external criteria to compute the validity of clustering, EXAMCE is capable of producing high-quality results that are comparable in quality to those of the best known clustering algorithms.

Objective Function Adjustment Algorithm for Combinatorial Optimization Problems

An improved algorithm of Guided Local Search called objective function adjustment algorithm is proposed for combinatorial optimization problems. The performance of Guided Local Search is improved by objective function adjustment algorithm using multipliers which can be adjusted during the search process. Moreover, the idea of Tabu Search is introduced into the objective function adjustment algorithm to further improve the performance. The simulation results based on some TSPLIB benchmark problems showed that the objective function adjustment algorithm could find better solutions than Local Search, Guided Local Search and Tabu Search.

Lagrangian object relaxation neural network for combinatorial optimization problems

We propose a Lagrangian object relaxation technique that can obtain a more near-optimal solution for the traveling salesman problem (TSP). It consists of two stages. First, a feasible solution is calculated and second, a more near-optimal solution is calculated by a Hopfield neural network (HNN). The Lagrangian object relaxation technique can help the HNN escape from the local minimum by correcting Lagrangian multipliers. The Lagrangian object relaxation neural network is analyzed theoretically and evaluated experimentally through simulating the TSP. The simulation results based on some TSPLIB benchmark problems show that the proposed method can find 100% valid solutions which are near-optimal solutions.

Learning Method of Hopfield Neural Network and Its Application to Traveling Salesman Problem.

A learning method of the Hopfield neural network is presented for efficiently solving combinatorial optimization problems. The learning method adjusts the balance between the constraint term and the cost term of the energy function so as to keep the Hopfield network updating in a gradient descent direction of energy.This paper describes and analyzes the learning method and shows its application to the traveling salesman problem (TSP). The performance is evaluated through simulating 100 randomly generated instances of the 10-city traveling salesman problem and some TSPLIB benchmark problems. The simulation results show that the performance of the proposed learning method on these test problems is very satisfactory in terms of both solution quality and running time. The proposed learning method finds the optimal solutions on the test TSPLIB benchmark problems in very short computation time.