Optimising real parameters using the information of a mesh of solutions: VMO algorithm Amilkar Puris and Rafael Bello Department of Computer Science University of Las Villas, Cuba Email: {ayudier, rbellop}@uclv.edu.cu Daniel Molina Department of Computer Science and Engineering University of Cadiz, Spain Email: daniel.molina@uca.es Francisco Herrera Department of Comp. Sci. and A.I. University of Granada, Spain Info Contact: http://decsai.ugr.es/ herrera/ Email: herrera@decsai.ugr.es Abstract—Population-based Meta-heuristics are algorithms that can obtain very good results for complex continuous op- timisation problems, using the information of a population of solutions. In these algorithms the distribution of solutions is crucial because it has a strong inﬂuence of the exploration new regions. In this work, we present a population algorithm, Variable Mesh Optimisation (VMO), in which a set of nodes (potential solutions) is distributed as a mesh. This mesh is initially homogeneously distributed, and then the mesh evolves to a heterogeneous structure resampling the space toward the best neighbours, maintaining at the same time a controlled diversity (avoiding solutions too close to each other). We use a benchmark of multimodal continuous functions to study the inﬂuence of the different components of the proposal, and to compare the proposed algorithm with other basic population-based meta- heuristics in the literature. The results show that VMO is a very competitive algorithm. Index Terms—continuous optimisation, meta-heuristics, popu- lation meta-heuristics, variable mesh optimisation I. I NTRODUCTION Population based meta-heuristics (PMHs) are meta- heuristics that use a solution sets (called population) that evolves during the iterations of the algorithms, using the information of these solutions to make a heuristic sampling of the domain search. Real coded Genetic Algorithms (GAs) [1], Particle Swarm optimisation (PSO) [2], Estimation of Distribution Algorithms (EDAs), Scatter Search (SS) [3], Difference Evolution (DE) [4] are, among others, examples of PMHs. PMHs introduce different ways of exploring the search space. They present powerful communication or cooperation mechanisms (depending on the context) in order to converge the population toward promising regions of the search domain. In these mechanisms, the best solutions usually have a strong inﬂuence over the remaining ones of the population. For instance, they could have a greater probability of survival into the population, like in genetic algorithms [5]. In other algorithms, remaining solutions are oriented to the best ones directly, as in PSO [2], or more indirectly, as in DE [4], [6]. This type of meta-heuristics, also implements several mech- anisms to introduce or maintain diversity into the population. This combination of a convergence mechanism with strategies to introduce the diversity allow PMHs to obtain very good results in continuous optimisation problems. With these two facts in mind, we present a new PMH called Variable Mesh optimisation, VMO, for real parameter optimi- sation. In this algorithm, the population is represented by a set of nodes (potential solutions) that are initially distributed as a mesh, using a uniform distribution. Then, VMO evolves this mesh creating more solutions around the most promising regions of the mesh, creating a heterogeneous structure of the mesh. This heterogeneous structure can explore with a better performance, as it is said in [7]. VMO differs from other similar PMHs in the fact that in each iteration it creates new solutions around the current solutions of the mesh, and not only around the best ones, dividing the mesh in more solutions in the most promising regions. Also, to avoid the risk of premature convergence that could be associated with this type of structures, VMO maintain the population diversity by selecting the best representatives of the mesh for the next iteration. The search process developed by VMO can be described by the following two operations: • Expansion: this mechanism explores around the best solutions found, by creating new nodes between each node of the mesh and its best neighbour, and around the external borders of the mesh. • Contraction: a clearing process removes all nodes that are too close to others with best ﬁtness. The aim is to maintain the population size and to foment mesh diversity. We study, based on experimental work, the inﬂuence of its different components, showing that the ideas underlying this technique can lead to successful results. Then, we compare the performance of the VMO with other basic PMHs with multimodal functions on continuous domains, showing that VMO is a competitive model. This paper is organised as follows: In Section II, a detailed description of the VMO is given, emphasizing the expansion and contraction processes of the mesh. In Section III, the experimental framework and the statistical tests used to vali- date the experimental results are presented. In Section IV, we analyse the behaviour of several VMO components. In Section V, the proposed model is compared with others, to probe if its results improve the obtained by other PMHs of the literature. Finally, in Section VI, we present the main conclusions and U.S. Government work not protected by U.S. copyright WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IEEE CEC 2553