Hybridizing Evolutionary Algorithms with Variable-Depth Search to Overcome Local Optima

Abstract

Hybridizing evolutionary algorithms with local search has become a popular trend in recent years. There is empirical evidence for various combinatorial problems where hybrid evolutionary algorithms perform better than plain evolutionary algorithms. Due to the rapid development of a highly active field of research, theory lags far behind and a solid theoretical foundation of hybrid metaheuristics is sorely needed. We are aiming at a theoretical understanding of why and when hybrid evolutionary algorithms are successful in combinatorial optimization. To this end, we consider a hybrid of a simple evolutionary algorithm, the (1+1) EA, with a powerful local search operator known as variable-depth search (VDS) or Kernighan-Lin. Three combinatorial problems are investigated: Mincut, Knapsack, and Maxsat. More precisely, we focus on simply structured problem instances that contain local optima which are very hard to overcome for many common metaheuristics. The plain (1+1) EA, iterated local search, and simulated annealing need exponential time for optimization, with high probability. In sharp contrast, the hybrid algorithm using VDS finds a global optimum in expected polynomial time. These results demonstrate the usefulness of hybrid evolutionary algorithms with VDS from a rigorous theoretical perspective.