A Hyperheuristic Based on Dynamic Iterated Local Search

Two hyperheuristics are proposed as entries for the 2011 Cross-domain Heuristic Search Challenge (CHeSC), namely DynamicILS and BiasILS; both are simple adaptations of Iterated Local Search (ILS). 1 Iterated Local Search Iterated Local Search (ILS) [3] is a metaheuristic search technique which aims to find good solutions to difficult combinatorial optimisation problems in a reasonable amount of computational time. It is an iterative procedure which performs a biased sampling of local optima (hilltops) in the search space by randomly changing the current solution at each iteration (performing a “kick”) and then using a local search heuristic to find a new local optima. Thus ILS aims to avoid getting stuck in local optima (the main disadvantage of heuristics). ILS can operate with “black-box” components; it does not need to know the local search heuristic or how the kick is performed. Algorithm 1 Iterated Local Search [3] s0 = GenerateInitialSolution s∗ = LocalSearch(s0) repeat s′ = PerformKick(s∗, history) s′∗ = LocalSearch(s′) s∗ = AcceptanceCriterion(s∗, s′∗, history) until TerminationCriterion ILS begins (see Algorithm 1) with a local optima from using a local search heuristic on an initial solution (generated by a construction heuristic). It then iteratively performs a kick on the current solution (by randomly changing some elements of the solution), obtains a new local optima (using local search) and then determines if this new solution should be used as the current solution for the next iteration by testing it against some acceptance criterion, e.g., only accepting a better solution or probabilistically accepting a worse solution. The search is terminated when a termination criterion is reached, e.g., the computation time limit expires. Within the HyFlex package, a kick (PerformKick) corresponds to a predefined mutation or a ruin-and-recreate low-level heuristic. The current solution at each iteration in ILS has a neighbourhood, which is the set of solutions that all possible locally optimised kicks can reach (the set of all neighbouring hilltops). Standard implementations of ILS perform a single kick blindly without searching for the “best” kick to perform (to reach the best neighbouring hilltop). In addition, the strength of the kick (how much change is made to the current solution) determines the solutions that are contained within the neighbourhood (hilltops closer or further away). 2 Dynamic Iterated Local Search A key decision when using ILS is the determination of the kick’s nature and strength. A new method of dynamically adjusting the kick strength during the search was proposed by Liddle [2] and applied to the Travelling Salesman Problem (TSP). Here we extend this idea within the HyFlex package for the CHeSC competition. We propose a variation of ILS, called Dynamic Iterated Local Search (DynamicILS, see Algorithm 2). DynamicILS dynamically samples from a set of available kick strengths K, choosing more often those kick strengths in K that seem to be performing well so far. For ILS, |K| experiments (one for each kick strength) would be needed to establish the most effective k ∈ K, but DynamicILS has the potential to achieve the same or similar results using only a single experimental run. Algorithm 2 Dynamic Iterated Local Search s0 = GenerateInitialSolution s∗ = LocalSearch(s0) w = InitialiseWeights(s∗) repeat s′ = PerformDynamicKick(s∗,w, history) s′∗ = LocalSearch(s′) w = UpdateWeights(s∗, s′,w) s∗ = AcceptanceCriterion(s∗, s′∗, history) until TerminationCriterion Weights for Kick Strengths: We use weights wk for each k ∈ K in a probabilistic selection method. These wk are updated during the search. This concept is similar to the use of pheromones in Ant Colony Optimization [1]. K is specified by lower and upper kick strength limits kl and ku, thus K = {kl, kl+1, . . . , ku− 1, ku}, with m = ku − kl + 1 possible kick strengths to sample from. Within the HyFlex package, these are subsequently scaled to (0, 1). Memory: A simple implementation of dynamic sampling is to use a single weights vector w, where the next kick strength is chosen according to w only, that is, no memory is retained. Alternatively, one level of memory could be incorporated into the kick strength choice, in order to evolve a balance between intensification and diversification. In this case a weights matrix W is kept, where each row j of W (j = 1, . . . ,m) corresponds to the weights vector wj : the weights for choosing the next kick strength given that the previous kick strength was j. This is not included in the HyFlex implementation as it was found not to be as effective as the simpler version over the given test domains and the TSP. Probabilistic Selection Method: The proposed method for probabilistically selecting a kick strength j from the set of possible kick strengths K, when performing a dynamically sampled kick in D-ILS is as follows. First determine the row vector of weights w. If not using memory, this is just w, the vector of weights. If using memory assign w = wi, row i of the weights matrix W , which corresponds to the previous kick strength i ∈ {1, . . . ,m}. Then calculate the probability vector p from w as follows. Normalise the weights to be non-negative: