An improved firefly algorithm for global continuous optimization problems

https://doi.org/10.1016/j.eswa.2020.113340Get rights and content

Highlights

  • Logarithmic spiral path enhances search agents’ exploitation.

  • The exploration rate is made dynamic to improve the convergence.

  • The new procedure is tested by minimizing the 29 well known functions.

  • The proposed optimizer is validated using the 6 real cases.

Abstract

Global continuous optimization is populated by its implementation in many real-world applications. Such optimization problems are often solved by nature-inspired and meta-heuristic algorithms, including the firefly algorithm (FA), which offers fast exploration and exploitation. To further strengthen FA’s search for global optimum, a Levy-flight FA (LF-FA) has been developed through sampling from a Levy distribution instead of the traditional uniform one. However, due to its poor exploitation in local areas, the LF-FA does not guarantee fast convergence. To address this problem, this paper provides an adaptive logarithmic spiral-Levy FA (AD-IFA) that strengthens the LF-FA’s local exploitation and accelerates its convergence. Our AD-IFA is integrated with logarithmic-spiral guidance to its fireflies’ paths, and adaptive switching between exploration and exploitation modes during the search process. Experimental results show that the AD-IFA presented in this paper consistently outperforms the standard FA and LF-FA for 29 test functions and 6 real cases of global optimization problems in terms of both computation speed and derived optimum.

Introduction

Many engineering and business optimization problems are regarded as global optimization problems with many local optima. For example, the design of electronic systems, vehicle routine planning in large-scale traffic networks, the inverse problem of chemical kinetics and gene recognition in bioinformatics. In order to solve these optimization problems, two classes of methods have been developed: cooperative co-evolution (CC) methods (Potter, 1997) and non-decomposition methods (Hoang, 2008). The CC methods decompose the global continuous optimization to multiple low-dimensional sub-components. They can be further divided into two categories: static grouping based CC methods and dynamic grouping based CC methods (Chen, Johansson, Hsu, Liao, & Lin, Potter, De Jong, 1994). However, CC methods demand high computation costs, especially in non-linear continuous functions (Maeda, Fukano, Yamamichi, Nitta, & Kurata, 2011). Therefore, non-decompositional methods without a divide-and-conquer strategy is developed for improved computational efficiency. Among various non-decompositional methods, meta-heuristics algorithms become dominant with their strong exploration ability in global space. According to the nature-inspired origins, these meta-heuristic algorithms use two main search methods: evolutionary computation and swarm intelligence. The evolutionary computation uses reputation, mutation, recombination and selection strategies to obtain the best offspring as the best optimum (Ashlock, 2006), as in genetic algorithm and differential evolution. The swarm intelligence imitates animals’ behaviors for searching preys or other mates to update their positions (Banks, Vincent, & Anyakoha, 2007), as in particle swarm optimization (PSO) and ant colony optimization. Compared with evolutionary computation, swarm intelligence searches space faster (Nazir, Majid-Mirza, & Ali-Khan, 2014). To accelerate the speed for exploring and exploiting, inspired by the flashing patterns and behaviors of fireflies, Yang (2008) has firstly proposed the firefly algorithm (FA). According to Łukasik and Żak (2009), the FA performs better than PSO on continuous constrained optimization problems.

Due to its advantages, the FA has been widely used in many engineering applications. Senthilnath, Omkar, and Mani (2011) applied the FA for searching for the centers of the clusters by minimizing the distance sum of the patterns to their centers. Yang, Hosseini, and Gandomi (2012) proposed a FA method to address the economic dispatch problem for practical power systems management. Kavousi-Fard, Samet, and Marzbani (2014) used the FA in searching the best parameter in support vector regression for accurate short load forecasting. Wang et al. (2012) introduced the FA to path planning for uninhabited combat air vehicles. In addition, according to Sayadi, Ramezanian, and Ghaffari-Nasab (2010), FA was also popularly used to solve NP-hard scheduling problems, traveling salesman problems (Jati et al., 2011) and digital image compression (Horng, 2012). For parameter estimation in expert systems, Sánchez, Melin, and Castillo (2017) introduced the FA to the modular granular neural networks for ear recognition and face recognition. Langari, Sardar, Mousavi, and Radfar (2020) incorporated FA to fuzzy clustering to protect the anonymized database and minimize the information loss. For online social networks, Jain and Katarya (2019) used the FA to discover the opinion lead in local communities. Moreover, Yang & He, 2013 discussed two main reasons for the effectiveness of FA: the intelligent subdivision and the ability of dealing with multi-modality. From the inspiration of the dependence of FA on the variation of light intensity and their attractiveness, these two factors actually determine the searching performance (Fister, Fister Jr, Yang, & Brest, 2013).

Furthermore, to improve the performance of FA, some parameter adjustment mechanisms, e.g., fuzzy controllers, are incorporated, which can update the systemic parameters in FA based on the search process (Castillo, Soto, Valdez, 2018, Lagunes, Castillo, Valdez, Soria, Melin, 2018). The search path of the firefly is investigated for enhancing the exploration and exploitation of the fireflies. In 2010, Yang (2010a) introduced a Lévy flight move strategy to the FA, called the Lévy-flight FA (LF-FA). While LF-FA has a high probability of jumping out of a local minima, exploitation in local space is not promoted. Therefore, there are three primary motivations of our FA design in this paper:

  • a.

    A new position update path can be developed to enhance the exploitation in local space;

  • b.

    An intelligent controller based on the search process is required to maintain the balance between the exploration and the exploitation; and

  • c.

    For the global continuous optimization problems, a more efficient firefly algorithm is demanded with less computational cost and a better optimum.

In this paper, two novel modifications to FA are developed for continuous global optimization problems. The first one is that the logarithmic spiral path is proposed to improve the exploitation of fireflies in local space. The main idea is to combine this new path with light intensity and attractiveness to strengthen the exploitation based on the nature of the logarithmic spiral (Tamura & Yasuda, 2011). Then, an adaptive switch (ratio) is presented based on the searching dynamic process to determine the task mode: exploration in global space or exploitation in local space. As the switch is more sensitive to the local optimum, the ratio will dramatically auto-decrease to trigger the exploration mode during any potential trap. When the ratio is large, the exploitation mode will be tested. Combining these two designs gives an improved FA, which we referred to as AD-IFA.

This paper makes three main contributions as follows:

  • 1)

    A logarithmic-spiral path is designed to improve the exploitation of search agents for local searching, leading to a speed-up of the FA convergence;

  • 2)

    An adaptive switch (ratio) is proposed to balance the exploitation and the exploration during the search process. According to the variation of current fitness function value, the search agent can adapt its strategy to update the next position;

  • 3)

    A novel adaptive logarithmic spiral-Lévy FA is developed for a more stable and high accurate optimal result;

In order to demonstrate our AD-IFA, 29 benchmark functions will be utilized (Yelghi, Köse, 2018, Surjanovic & Bingham, 2017). Furthermore, our AD-IFA is also tested in 6 real engineering applications to show its superiority to existing FA methods. The source code of our AD-IFA is available at: https://github.com/wujrtudou/AdaptiveFireflyAlgorithm.git

This paper is organized as follows: Firstly, related work on FA and the Lévy-flight FA are reviewed in Section 2. Section 3 outlines the logarithmic spiral path, the adaptive switch (ratio) and our AD-IFA. Then, experimental settings, benchmark functions and simulation results are given in Section 4. Section 5 shows the efficiency of our AD-IFA through six engineering applications. Finally, Section 6 concludes the paper.

Section snippets

Related work

In this section, the FA and LF-FA will be outlined. In addition, the capacity of these two meta-heuristics optimization algorithms will also be explored.

An improved firefly algorithm

This section starts with an exploration of the inspiration for logarithmic spiral paths. Then, an adaptive switch (ratio) is designed for the selection of searching methods to balance the exploration and exploitation based on the change in global fitness function values. After that, to combine with the new adaptive switch (ratio), an improved FA is presented, which we refer to as AD-IFA.

Numerical simulations

In this section, the improved firefly algorithm is tested on 29 test functions. More specifically, the logarithmic spiral and the adaptive switch will be tested, respectively. Nine benchmark functions are selected for analysis of our new optimizer. Twenty additional functions and their simulation results are recorded in Appendix A.

Case studies of real-world applications

In this section, the performance of the four mentioned firefly algorithms, FA, LF-FA, LS-LF-FA, and our AD-IFA, is evaluated in six engineering real applications (Gandomi, Yang, & Alavi, 2013): cantilever beam (Fleury & Braibant, 1986), corrugated bulkhead design (Moses & Onoda, 1969), pressure vessel design (Gandomi et al., 2013), a three-bar truss design (Nowacki, 1973), tubular column design (Rao, 2019), and welded beam design (Coello, 2000). Their mathematical formulations are presented in

Conclusions

The original FA suffers from low exploration in global space and low exploitation in local space. The modified version, LF-FA, focuses on the exploration but results in slowed convergence because of poor exploitation in local space. In this research, the logarithmic-spiral path has been applied in the FA optimizer to strengthen the exploitation in local space and accelerate the convergence of the search process. Then, an adaptive switch has been created based on the current fitness function

CRediT authorship contribution statement

Jinran Wu: Conceptualization, Investigation, Methodology, Validation, Software, Writing - original draft. You-Gan Wang: Supervision, Funding acquisition, Project administration, Writing - original draft, Writing - review & editing. Kevin Burrage: Supervision, Methodology, Writing - review & editing. Yu-Chu Tian: Supervision, Writing - review & editing. Brodie Lawson: Formal analysis, Visualization, Writing - original draft, Writing - review & editing. Zhe Ding: Conceptualization, Writing -

Declaration of Competing Interest

All authors declare no conflict interest.

Acknowledgements

Computational (and/or data visualization) resources and services used in this work were provided by the HPC and Research Support Group, Queensland University of Technology, Brisbane, Australia. This work was supported in part by the ARC Center of Excellence for Mathematical and Statistical Frontiers. This work was supported by the Australian Research Council project [grant number DP160104292, 2016].

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