Regular Paper
Novel benchmark functions for continuous multimodal optimization with comparative results

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Abstract

Multi-modal optimization is concerned with locating multiple optima in one single run. Finding multiple solutions to a multi-modal optimization problem is especially useful in engineering, as the best solution may not always be the best realizable due to various practical constraints. To compare the performances of multi-modal optimization algorithms, multi-modal benchmark problems are always required. In this paper, 15 novel scalable multi-modal and real parameter benchmark problems are proposed. Among these 15 problems, 8 are extended simple functions while the rest are composition functions. These functions coordinate rotation and shift operations to create linkage among different dimensions and to place the optima at different locations, respectively. Four typical niching algorithms are used to solve the proposed problems. As shown by the experimental results, the proposed problems are challenging to these four recent algorithms.

Introduction

In real world optimization, many problems often contain multiple global/local optima. Multimodal optimization aims to find a number of global or good local optima so that the user is able to select the most appropriate one among these potential solutions. Multiple solutions could also be analyzed to discover hidden properties or relationships of the concerned functional landscape [1], [23]. In recent decades, Evolutionary Algorithms (EAs) have become a common choice for solving multimodal problems due to their use of a population, which allows multiple solutions to be searched simultaneously in one execution [2], [3].

Detection and maintenance of multiple solutions are two challenging tasks for the EAs while solving multimodal optimization problems as EAs are originally designed to solve single-peak global optimization problems. Various techniques that commonly referred to as niching methods are incorporated in the original EAs to make them suitable for solving multimodal optimization problems [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Niching is a technique of finding and preserving multiple stable sub-populations or niches and preventing the population from converging to a single point. The primary objective of any niching technique is to preserve the diversity of the population. The techniques for multimodal optimization are usually based on diversity maintenance techniques borrowed from other domains [16]. Many niching methods have been developed in the past few years, including crowding [8], fitness sharing [10], restricted tournament selection [9], speciation [11], and geographical neighborhood-based mutation [6]. Crowding [8] technique is inspired by the competition among similar individuals in a natural population. In order to maintain the diversity of the population, it compares the offspring with several randomly sampled individuals from the current population. Restricted Tournament Selection [10] is similar to crowding. It selects a random sample of ‘w’ (window size) individuals from the population and compares the offspring with the closest one within the sampled individuals. In fitness sharing [9], the population is partitioned into different subgroups according to the similarity of the individuals and individuals share their information within the same subgroup. Speciation [11] divides the population into different species based on their similarity and the size of the specie is determined by a user specified value. Geographical neighborhood-based mutation [6], [22] restricts the production of offspring as well as the competition within a local area (most often a compact Euclidean ball surrounding the current parent) which can maintain the diversity of the population and improve independent multiple local convergences.

Multimodal benchmark test functions are needed to comparatively assess the effectiveness of a newly proposed niching technique. It is noticeable that most of the commonly used test functions are relatively too simple to meaningfully compare high performing algorithms. The dimensions of these problems are small and non-scalable as well as the positions of the optima are also mostly periodic without linkages among the decision variables. Although some scalable test functions were proposed in literature [17], they do not contain different rotations for different subset of decision variables. Without these operations, the test functions can be biased to certain algorithms. For these reasons, this paper constructs 15 novel scalable multi-modal benchmark functions. The characteristics of these functions can be changed independently by using rotation and shift operations. The linkages among different dimensions are also created by these operations. The guidelines for designing these functions are listed as follows:

  • 1.

    The functions should be constructed or obtained by using commonly used simple functions.

  • 2.

    The functions should be scalable. (As the dimensions increase, the number of optima also increases which will pose extra difficulties).

  • 3.

    Rotation and shift operations should be included in the problems.

  • 4.

    The positions of optima should be changeable.

As an illustration, four recent niching algorithms are selected to demonstrate the levels of difficulties of these benchmark functions. The remainder of this paper is organized as follows. Section 2 presents the proposed benchmark functions. Section 3 introduces the optimization methods used in the experimental part. Experimental setup and experimental results are presented in 5 Experimental results, 6 Conclusions, respectively. Section 6 concludes the paper.

Section snippets

Novel test functions

This section introduces the novel multimodal benchmark test functions proposed in this paper. (Note that the sources codes of these functions can be downloaded from http://www3.ntu.edu.sg/home/epnsugan/) The Contour maps of these novel test functions are presented in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15. These functions are divided into two categories as extended simple functions and composition functions.

Optimization algorithms

This section introduces the four optimization algorithms used in the experimental section. These algorithms are compared using the 15 new benchmark functions to evaluate their ability to find multiple optima.

Experimental setting

All algorithms are implemented by using Matlab 2013 and executed on a PC with Intel® Core™ i5 CPU and 4 Gb of RAM. The operating system is Microsoft Windows 7. The settings of the 4 algorithms are adopted exactly as in their original works [1], [8], [17], [21].

Performance measure

Various performance measures have been used in the literature such as average number of optima found, success rate and peak ratio [3]. Different measures have different advantages and disadvantages. If we consider the success rate as an

Experimental results

To compare the performance of the 4 algorithms on the proposed test problems, 25 independent runs are executed for each problem. The best run, worst run as well as the mean value (in boldface) of the simple functions are presented in Table 3, Table 4, Table 5, Table 6. In Tables 7 and 5 best optima (the objective values) of median run found by each algorithms are presented. The mean values of the 5 best optima are also calculated and highlighted in boldface in Table 7. Comparing the results in

Conclusions

In this paper, 15 novel multimodal benchmark test functions have been presented. These functions offer an easy way to construct scalable functions and a challenging environment for comparing novel multimodal optimization algorithms with the state-of-the-art. These functions are divided into two categories, namely as the extended simple functions and composition functions, respectively. The extended simple functions were obtained through rotating and shifting some well known multimodal test

Acknowledgments

This research is partially supported by National Natural Science Foundation of China (61305080, 61473266, 61379113) and Postdoctoral Science Foundation of China (2014M552013) and the Scientific and Technological Project of Henan Province (132102210521).

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