Most real-world optimization problems often come with multiple global optima or local optima. Therefore, increasing niching metaheuristic algorithms, which devote to finding multiple optima in a single run, are developed to solve these multimodal optimization problems. However, there are two difficulties urgently to be solved for most existing niching metaheuristic algorithms: how to set the niching parameter values for different optimization problems and how to jump out of the local optima efficiently. These two difficulties limit their practicality largely. Based on Whale Swarm Algorithm (WSA) we proposed previously, this paper presents a new multimodal optimizer named WSA with Iterative Counter (WSA-IC) to address these two difficulties. On the one hand, WSA-IC improves the iteration rule of the original WSA for multimodal optimization, which removes the need of specifying different values of attenuation coefficient for different problems to form multiple subpopulations, without introducing any niching parameter. On the other hand, WSA-IC enables the identification of extreme points during the iterations relying on two new parameters (i.e., stability threshold (T_ s ) and fitness threshold (T_ f )), to jump out of the located extreme points. Moreover, the convergence of WSA-IC is proved. Finally, the proposed WSA-IC is compared with several niching metaheuristic algorithms on CEC2015 niching benchmark test functions and on five additional high-dimensional multimodal functions. The experimental results demonstrate that WSA-IC statistically outperforms other niching metaheuristic algorithms on most test functions.
Since increasing niching methods were put forward, a large number of multimodal optimization algorithms combining the metaheuristic algorithms with these niching methods have been proposed. In this section, a brief overview of multimodal optimization algorithms is presented. According to whether the prior knowledge of the fitness landscape is needed, these multimodal optimization algorithms are classified into prior knowledge based methods and non-prior knowledge based methods. More references and discussions about multimodal optimization algorithms can be viewed in references @cite_19 @cite_27 .
@cite_19 @cite_27
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A technique is described that allows unimodal function optimization methods to be extended to locate all optima of multimodal problems efficiently. We describe an algorithm based on a traditional genetic algorithm (GA). This technique involves iterating the GA but uses knowledge gained during one iteration to avoid re-searching, on subsequent iterations, regions of problem space where solutions have already been found. This gain is achieved by applying a fitness derating function to the raw fitness function, so that fitness values are depressed in the regions of the problem space where solutions have already been found. Consequently, the likelihood of discovering a new solution on each iteration is dramatically increased. The technique may be used with various styles of GAs or with other optimization methods, such as simulated annealing. The effectiveness of the algorithm is demonstrated on a number of multimodal test functions. The technique is at least as fast as fitness sharing methods. It provides an acceleration of between 1 and l0p on a problem with p optima, depending on the value of p and the convergence time complexity. Multimodal optimization (MMO) aiming to locate multiple optimal (or near-optimal) solutions in a single simulation run has practical relevance to problem solving across many fields. Population-based meta-heuristics have been shown particularly effective in solving MMO problems, if equipped with specifically-designed diversity-preserving mechanisms, commonly known as niching methods. This paper provides an updated survey on niching methods. This paper first revisits the fundamental concepts about niching and its most representative schemes, then reviews the most recent development of niching methods, including novel and hybrid methods, performance measures, and benchmarks for their assessment. Furthermore, this paper surveys previous attempts at leveraging the capabilities of niching to facilitate various optimization tasks (e.g., multiobjective and dynamic optimization) and machine learning tasks (e.g., clustering, feature selection, and learning ensembles). A list of successful applications of niching methods to real-world problems is presented to demonstrate the capabilities of niching methods in providing solutions that are difficult for other optimization methods to offer. The significant practical value of niching methods is clearly exemplified through these applications. Finally, this paper poses challenges and research questions on niching that are yet to be appropriately addressed. Providing answers to these questions is crucial before we can bring more fruitful benefits of niching to real-world problem solving. Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and when needed, the current solution may be switched to a more suitable one while still maintaining the optimal system performance. Niching particle swarm optimizers (PSOs) have been widely used by the evolutionary computation community for solving real-parameter multimodal optimization problems. However, most of the existing PSO-based niching algorithms are difficult to use in practice because of their poor local search ability and requirement of prior knowledge to specify certain niching parameters. This paper has addressed these issues by proposing a distance-based locally informed particle swarm (LIPS) optimizer, which eliminates the need to specify any niching parameter and enhance the fine search ability of PSO. Instead of using the global best particle, LIPS uses several local bests to guide the search of each particle. LIPS can operate as a stable niching algorithm by using the information provided by its neighborhoods. The neighborhoods are estimated in terms of Euclidean distance. The algorithm is compared with a number of state-of-the-art evolutionary multimodal optimizers on 30 commonly used multimodal benchmark functions. The experimental results suggest that the proposed technique is able to provide statistically superior and more consistent performance over the existing niching algorithms on the test functions, without incurring any severe computational burdens. One of the most critical issues that remains to be fully addressed in existing multimodal evolutionary algorithms is the difficulty in pre-specifying parameters used for estimating how far apart optima are. These parameters are typically represented as some sorts of niching parameters in existing EAs. Without prior knowledge of a problem, it is almost impossible to determine appropriate values for such niching parameters. This paper proposes a PSO for multimodal optimization that removes the need of these niching parameters. Our results show that the proposed algorithm, Fitness Euclidean-distance Ratio based PSO (FER-PSO) is able to reliably locate multiple global optima on the search landscape over some widely used multimodal optimization test functions, given that the population size is sufficiently large.