Revisiting the big valley search space structure in the TSP

The solution space of the travelling salesman problem under 2-opt moves has been characterized as having a big-valley structure, in which the evaluation of a tour is positively correlated to the distance of the tour from the global optimum. We examine the big-valley hypothesis more closely and show that while the big-valley structure does appear in much of the solution space, it breaks down around local optima that have solutions whose evaluation is very close to that of the global optimum; multiple funnels appear around local optima with evaluations close to the global optimum. The appearance of multiple funnels explains why certain iterated local search heuristics can quickly find high-quality solutions, but fail to consistently find the global optimum. We then investigate a novel search operator, which is demonstrated to have the ability to escape funnels at evaluations close to the global optimum. Journal of the Operational Research Society (2011) 62, 305–312. doi:10.1057/jors.2010.116 Published online 1 September 2010