Survey PaperAn alternative way of evolutionary multimodal optimization: density-based population initialization strategy☆
Introduction
For the complex optimization problems in the real world, there are often multiple global optima or acceptable local optima. These problems are known as multimodal optimization problems (MMOPs). Finding all the global optima (and acceptable local optima on request) of MMOPs as much as possible could provide more information about the problem and more choices for decision makers, and decision makers could quickly obtain suitable candidate solutions according to their personal preferences [1], [2].
Evolutionary algorithms (EAs) [3] and swarm intelligence (SI) [4] are population-based algorithms, which have been widely used to deal with MMOPs owing to their favorable performances [5]. The general idea of EAs and SI is to start with a set of random candidate solutions, use the search strategies to iterate the population to find an optimal solution. However, the basic EAs and SI that often find one optimal solution cannot be directly used to solve MMOPs. Therefore, how to enable EAs and SI to simultaneously find the optimal solutions from multiple potential regions and how to increase the diversity of the population have become the main challenges in the evolutionary multimodal optimization community.
There has been much work for MMOPs to date, among which niching methods [2] and multiobjectivization methods [6] are two popular types of multimodal optimization algorithms. Generally, the niching methods divide the population into multiple subpopulations to enhance the ability of EAs and SI to search for multiple global peaks, among which typical methods include nearest-better clustering (NBC) [7], [8], speciation techniques [9], a niching strategy based on local binary pattern [10], and neighborhood mutation [11]. Moreover, Biswas et al. [12] combined crowding strategy with normalized search neighborhood for effectively locating multiple solutions. In [13], the authors proposed the locally informative niching differential evolution, which used local information to guide population evolution. Apart from niching methods, some work has been done to transform an MMOP into a multiobjective optimization problem [14], [15], [16], [17], which design different objectives to ensure that all the optimal solutions are non-dominated.
In Ref[17]., the authors employed Latin hypercube sampling to generate an initial population for solving MMOPs, which is more uniform than the pseudo-random initial population. However, most evolutionary multimodal optimization algorithms have not paid much attention to the distribution of the initial population, and the pseudo-random numbers generator (PRNG) is a mainstream method for generating a random initial population because of its simplicity, convenience, and powerful adaptability [18]. Fortunately, much work has been done to study the effects of initial population on the performance of EAs and SI [18], [19], and some literature indicated that different population initialization strategies could influence the convergence speeds of algorithms [20], the quality of the final solutions [21], and the stability of the results [22]. Among a considerable number of population initialization strategies, many strategies try to generate a more uniform initial population, such as centroidal Voronoi tessellations (CVT) [23], [24], quasi-random numbers [25], [26], uniform design (UD) [27], and orthogonal design (OD) [28], [29]. Additionally, opposition based learning (OBL) is adopted in [30] as the population initialization strategy, where the worse half of the population is replaced by their oppositions.
This paper is an extended version of our previous conference paper [31]. Our previous work [31] proposed a density-based population initialization strategy (DPIS), which tends to generate a uniform initial population with a certain degree of randomness. Compared with typical initialization strategies, experimental results indicate that the proposed population initialization strategy is beneficial to solve MMOPs. However, the boundary of the search space is prone to be estimated low-density in DPIS, which reduces the uniformity of the population distribution. In this paper, we further study the density-based population initialization strategy for evolutionary multimodal optimization. In summary, our contributions are as follows:
- 1.
We propose an improved density-based population initialization strategy (IDPIS). Furthermore, we propose a variant of DPIS, i.e., fitness-weighted DPIS (FDPIS), which encourages individuals to appear in the areas with higher fitness by evaluating the fitness of the current population. FDPIS could help the algorithm catch some potential optimal regions faster when the algorithm has the raw landscape knowledge.
- 2.
We design a multi-species cooperative coevolution (MSCC) algorithm to solve MMOPs, which employs IDPIS and FDPIS to construct a large initial population. MSCC dynamically allocates computing resources based on the possibility of searching for optima for each species, and gives the priority to the best species as well as the species most likely to find a new optimal solution. MSCC adopts covariance matrix adaptation evolution strategy (CMA-ES) [32] as the basic optimizer and nearest-better clustering (NBC) [7] as the niching strategy, and is tested on the CEC-2013 multimodal optimization benchmark. Experimental results demonstrate that the performance of the proposed algorithm is very competitive.
The rest of this paper is organized as follows. Section 2 introduces the related work. Section 3 describes the details of the proposed algorithms, including IDPIS, FDPIS, and MSCC. Section 4 presents the experimental design and results, and the discussion about the uniformity and the quality of the population is shown in Section 5. Finally, Section 6 concludes this paper.
Section snippets
Multimodal optimization problems
Multimodal optimization problems have multiple global optimal solutions or acceptable local optimal solutions. In this paper, without loss of generality, we assume that the objective of MMOP is of a maximization form, and only the global optimal solutions are considered. Therefore, an MMOP is formalized as follows.where is an objective function, is the search apace, is a set of global optimal solutions of the function , and holds, i.e., the number of optimal
Proposed algorithm
Although improving distribution and increasing the size of the initial population could learn more information about the problem landscape, there are two obstacles to employing such a population as an initial population for evolutionary multimodal optimization. First, how to construct a large population for EAs. Second, how to balance the exploration and exploitation of such a population. In our previous work [31], we proposed a population initialization strategy, density-based population
Experiments
In this section, the experimental setting and results on the CEC-2013 multimodal optimization benchmark problems are provided.
Discussion
In this section, the uniformity of the population formed by IDPIS, and the quality of the population formed by IDPIS and FDPIS are discussed and analyzed. The parameter settings of IDPIS and FDPIS in this section are the same as those in Section 4.
Conclusion
In this paper, we make an attempt to solve MMOPs by employing a large population. First, we improve DPIS to generate a uniform initial population with a certain degree of randomness. Then, we further propose the FDPIS to improve the quality of the initial population. IDPIS and FDPIS are used to construct a large population as an initial population for the multimodal optimization algorithm, and we propose MSCC to balance the exploration and the exploitation of such a population in the
Declaration of Competing Interest
there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
We confirm that we have given due
CRediT authorship contribution statement
Peilan Xu: Methodology, Writing – original draft, Validation. Wenjian Luo: Methodology, Validation, Writing – original draft, Funding acquisition, Project administration. Jiafei Xu: Methodology, Software. Yingying Qiao: Software, Validation. Jiajia Zhang: Software, Validation, Writing – review & editing. Naijie Gu: Validation, Writing – review & editing, Project administration.
Declaration of Competing Interest
The authors declare no conflict of interest.
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2022, Information SciencesCitation Excerpt :For a particular EA, the reproducibility of its optimal solution is achieved in the statistical sense. Some studies have used different methods to solve the reproducibility problems [26,27]; the recently developed niching algorithms [28,29,22,23,30] with various basic optimizers have achieved satisfactory performance for MMOPs. These methods aim to balance the exploration ability of the global space and the exploitation ability of multiple optimal regions with some improved strategies; however, the theoretical basis of EAs remains a challenging topic.
Multimodal optimization via dynamically hybrid niching differential evolution
2022, Knowledge-Based SystemsCitation Excerpt :In NSAMA [37], in order to balance diversity and convergence, the exploited population and the explored population were divided according to the potential of the niche, and an adaptive Cauchy-based local search scheme was proposed to implement local search in the possible location of the optimal solution. Xu et al. used a large initial population to solve the MMOP problem and proposed an improved density-based population initialization strategy (IDPIS) in a DP-MACC-ES method [38]. In a local binary pattern-based adaptive differential evolution (LBPADE) [39], the local binary pattern uses neighbors’ information to extract relevant pattern information to form multiple niches.
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This work is supported by National Natural Science Foundation of China (No. 61573327).