Multimodal Optimization Using Niching Differential Evolution with Index-based Neighborhoods Michael G. Epitropakis * , Vassilis P. Plagianakos † , and Michael N. Vrahatis * * Computational Intelligence Laboratory, Department of Mathematics, University of Patras, GR-26110 Patras, Greece, Email: {mikeagn,vrahatis}@math.upatras.gr † Department of Computer Science and Biomedical Informatics, University of Central Greece, GR-35100 Lamia, Greece, Email: vpp@ucg.gr Abstract—A new family of Differential Evolution mutation strategies (DE/nrand) that are able to handle multimodal func- tions, have been recently proposed. The DE/nrand family in- corporates information regarding the real nearest neighborhood of each potential solution, which aids them to accurately locate and maintain many global optimizers simultaneously, without the need of additional parameters. However, these strategies have increased computational cost. To alleviate this problem, instead of computing the real nearest neighbor, we incorporate an index- based neighborhood into the mutation strategies. The new muta- tion strategies are evaluated on eight well-known and widely used multimodal problems and their performance is compared against five state-of-the-art algorithms. Simulation results suggest that the proposed strategies are promising and exhibit competitive behavior, since with a substantial lower computational cost they are able to locate and maintain many global optima throughout the evolution process. I. I NTRODUCTION Handling multimodal functions is a very important and chal- lenging task in evolutionary computation community, since most hard real-world problems exhibit highly multimodal landscapes. They are likely to have several global and/or local minima, and in many cases it is desirable to accurately locate as many as possible. To this end, several Evolutionary Algorithms (EAs) have been recently extended to handle such landscapes through the concept of the niche formation. Niche formation is a common biological phenomenon [1]. A niche can be defined as a subspace in the environment that can support different types of life. In general, niches indirectly impose reproduction restrictions to aid the differentiation of the species and thus maintain their diversity. Many natural environments can lead to niche formation, such as remote islands, high mountains and isolated valleys. Many well- known EAs have been developed, to mimic the biological niche formation and take advantage of its characteristics. The- ses methodologies are characterized as Niching methods [1]. Niching methods tend to maintain the diversity within their population and allow a parallel convergence into multiple solutions. Various niching techniques have been proposed and successfully applied to different EAs, such as, crowding [2], [3], fitness sharing [3], [4], clearing [5], specialized evolution operators [6], clustering [7], stretching and deflation [8], [9], parallelization [10], restricted tournament selection [11], [12], and speciation [13]. In the paper at hand, we consider the Differential Evolu- tion (DE) algorithm which has been proposed by Storn and Price [14]. DE has been successfully applied in a plethora of optimization problems [14]–[17]. In this work, the objective is to locate as many global optimizers of a multimodal function as possible. The DE literature includes several dif- ferent variants that incorporate the aforementioned niching techniques and attempt to handle multimodal landscapes. In particular, Thomsen extends DE with both a crowding and a fitness sharing technique, namely Crowding DE (CDE) and Sharing DE [18], and shows that the CDE variant is a more promising approach, since outperforms the Sharing DE in all tested problems [18]. In turn, Species-based DE (SDE) [19], [20] incorporates the speciation concept to handle multimodal functions. SDE locates multiple global optima simultaneously through the adaptive formation of multiple species, which are evolved through DE. Although SDE is computationally more efficient than the Crowding DE, it incorporates a user-specified and problem dependent parameter called species radius, which should be properly chosen. Additionally, DE using local selection (DELS) [21] employs a new mutation strategy that divides the mutation operation into the local and the global mutation stages. It selects a different mutation strategy, with a pre-specified probability, to perform either a global or a local mutation. The global mutation enhance the exploratory ability of the algorithm, while the local mutation its exploitative behavior. DELS has been further hybridized with a multi-start gradient-based local search, as well as with the crowding technique [22]. In turn, Zaharie proposed a parallel approach that utilizes an “island model” approach to locate in parallel many global optima [10], while in [12] a DE extension with an ensemble of the restricted tournament selection (ERTS-DE) has been proposed. Finally, several other EAs have been proposed, which attempt to handle multimodal landscapes [9], [23]–[26]. Recently, we have introduced a new family of mutation strategies that are able to efficiently handle multimodal func- tions, namely the DE/nrand family [6]. The DE/nrand family incorporates information regarding the real nearest neighbor- hood of each potential solution, which aids them to accurately locate and maintain many global optimizers, without the need of additional parameters. Nevertheless, the calculation of the distances between the individuals demands a high