Elsevier

Applied Soft Computing

Volume 66, May 2018, Pages 232-249
Applied Soft Computing

A hybrid firefly and particle swarm optimization algorithm for computationally expensive numerical problems

https://doi.org/10.1016/j.asoc.2018.02.025Get rights and content

Highlights

Abstract

Optimization in computationally expensive numerical problems with limited function evaluations provides computational advantages over constraints based on runtime requirements and hardware resources. Convergence success of a metaheuristic optimization algorithm depends on directing and balancing of its exploration and exploitation abilities. Firefly and particle swarm optimization are successful swarm intelligence algorithms inspired by nature. In this paper, a hybrid algorithm combining firefly and particle swarm optimization (HFPSO) is proposed. The proposed algorithm is able to exploit the strongpoints of both particle swarm and firefly algorithm mechanisms. HFPSO try to determine the start of the local search process properly by checking the previous global best fitness values. In experiments, several dimensional CEC 2015 and CEC 2017 computationally expensive sets of numerical and engineering, mechanical design benchmark problems are used. The proposed HFPSO is compared with standard particle swarm, firefly and other recent hybrid and successful algorithms in limited function evaluations. Runtimes and convergence accuracies are statistically measured and evaluated. The solution results quality of this study show that the proposed HFPSO algorithm provides fast and reliable optimization solutions and outperforms others in unimodal, simple multimodal, hybrid, and composition categories of computationally expensive numerical functions.

Introduction

Finding optima in many real-world optimization problems requires expensive evaluations in terms of computation. Because of some limitations in studies like project time requirements and computation resource constraints, the optimization process should be conducted quickly and it should not be too complex [1]. Many standard optimization algorithms require a large number of function evaluations. These algorithms usually give satisfactory results by use of their special information transfer mechanisms with a number of first candidate solutions in a number of fitness evaluations. Due to the evaluation of each candidate resolution, these processes often require computer-intensive computer resources and runtime. For this reason, the study and development of successful optimization algorithms for evaluating a limited number of functions is emerging and developing study area. In recent years, many new approaches have been introduced and published. Some satisfactory results have been obtained from these methods with limited function evaluations [[1], [2], [3]].

Nature-inspired algorithm is a computational technique inspired by observation in nature [[4], [5], [6]]. Swarm intelligence is a set of social behaviors pre-determined by some rules. Some of the single individuals do not act wisely enough. However, behavior of all individuals in together can be wise enough with the help of the swarm cooperation [7]. One of the most well-known and researched nature-inspired and swarm intelligence methods is particle swarm optimization (PSO) algorithm. PSO inspired by living or hunting life styles of birds and fish. It is widely used in difficult and complex optimization problems. Since the first introduction of PSO [8], various PSO algorithm variants have been developed by participants to find better algorithm in some specific problems. These developments in PSO variants can be classified roughly into four categories: The first category algorithms are based on parameter settings focusing on the optimization of inertia weight and acceleration coefficients parameters. The second variant algorithm category is based on the neighborhood topology, which describes the connection of particles to each other. The algorithms of the third variant category are based on learning strategies consisting of teaching and peer learning of the particle bests and global best positions and the last fourth category is hybrid variants using the mixture of PSO with other suitable optimization algorithms [9]. Primary objectives of hybrid PSO variants are to establish a balance between exploration and exploitation and to avoid premature convergence. Moreover, hybridization can contribute to the use of the PSO's powerful capabilities and to remove the weakness [10]. Known strong PSO abilities are easy implementation, less computing resource requirements, and fast convergence. On the other hand, it also has some weaknesses such as premature convergence or trapping in local optima and slow convergence rate in the exploitation where particles are close to each other or global optimum [4]. Firefly Algorithm (FA) is another nature-inspired and swarm intelligence optimization algorithm that mimics fireflies in the nature. FA has some advantages over PSO [11] algorithm. These advantages are: FA does not have an individual best or a certain global best and this prevents trapping in local minima or premature convergence disadvantages. Also, the fireflies of FA algorithm do not have velocity characteristic. Thus, other problems that are based on fast or slow velocity, can be prevented [12]. Although FA has good characteristic in local search, sometimes it is unable to escape from local search completely and traps in a local minima [13].

Hybrid optimization technique is a successful combination of metaheuristic algorithm with another optimization algorithm that can display a more robust behavior and exhibit greater flexibility against complex and difficult problems [14]. Local search algorithms iteratively scan the search space to find a preferable solution than existing solution using appropriately defined neighborhood mechanism [14]. Metaheuristic is composed of some iterative generation operations that efficiently combines different sub-heuristics to discover a search space. Some learning strategies are used to find global optimum areas [14,15]. Population-based metaheuristics are natural approaches that explore the search field by manipulating the population and final results highly depend on their unique manipulating methods [14]. Population-based metaheuristics methods are better at describing local optima than other trajectory methods which can be easily influenced by local optima. Therefore, metaheuristic hybrids, which can combine the strengths of both population-based and trajectory methods in a proper way, are usually very efficient and successful [14]. For example, Li et al. [16] proposed an approach that combines, global search ability of genetic algorithm (GA) and the fast convergence mechanism of PSO to find global optimum more precisely. A numerous of studies propose different types of hybrid optimization techniques that combine their powerful mechanisms. For this reason, they are often more productive in terms of running time and/or solution results quality [17]. Memetic algorithms are based on exploitation systematic and the combination of population-based and trajectory metaheuristics synergistic. A known technique to create this hybridation is to include local search add-ons in an evolutionary algorithm [18].

In this study, CEC 2015 and CEC 2017 [1,19], realistic engineering design optimization benchmark problems are used in a limited number of function evaluations. Therefore, fast convergence ability of PSO and good local search ability of FA are used in a hybrid way and obtained results are compared with other recent hybrid PSO and FA approaches. Rest of the paper is organized as follows: Literature review is given in Section 2. PSO and FA algorithms are briefly explained in Sections 3 and 4, respectively. Other recent hybrid-related works are analyzed and explained in Section 5. Proposed hybrid algorithm is explained in Section 6. Detailed flowchart and pseudo codes of the proposed algorithm are shown and given. Computationally expensive CEC 2015 and CEC 2017, engineering and mechanical design problem sets are explained and presented in Section 7. Experimental setup, results and discussions are also given by using of figures and tables. Finally, conclusions of the whole paper are detailed in Section 8.

Section snippets

Literature review

Petalas et al. [20] propose a new efficient and robust memetic particle swarm optimization (MPSO) algorithm that improves local search ability of the standard particle swarm algorithm. The proposed algorithm is tested with various unconstrained, constrained, minimax and integer programming problems. The obtained results show, memetic approach of PSO outperforms the local and global versions of the standard PSO.

Wang et al. [21] propose a new firefly algorithm (NFA) with local search for

Particle swarm optimization algorithm

Particle swarm algorithm is a swarm and population-based optimization technique that is inspired by social behaviors of bird and fish swarms. Particle swarm optimization (PSO) has been proposed by Kennedy and Eberhart [8,30]. Over the years, it becomes one of the most important algorithms and attracts increasing attention, since, as a new swarm intelligence-based algorithm, it makes it possible to solve complex optimization problems [8,31]. Special information transfer mechanism among particles

Firefly optimization algorithm

Firefly algorithm was inspired by fireflies, a kind of insect that lives in the nature. Most fireflies generate rhythmic and short flashes. Type of flashes is often unique for a specific kind. Flashing light is generated by a bioluminescence process, and its real function is still a matter of debate among researchers. Fireflies use their chemical light attractiveness for communication, hunting, and warning their enemies [11,41,42]. In Fig. 2, Firefly algorithm flowchart is given.

Inverse square

Hybrid particle swarm optimization and firefly (HPSOFF) algorithm

Arunachalam et al. [43] have proposed a hybrid particle swarm optimization and firefly algorithm (HPSOFF). In their paper, they propose a new approach, which has conflicting economic and emission objectives, to Combined Economic and Emission Dispatch (CEED) problem using a Hybrid Particle Swarm Optimization and Firefly (HPSOFF) algorithm that is shown in Fig. 3. In standard FA algorithm, initial random population is updated in each iteration until the last iteration. Thus, final result of

Proposed hybrid firefly and particle swarm optimization (HFPSO) algorithm

Achieving a reliable success in evaluating a limited number of functions is the main goal of the proposed HFPSO algorithm. Therefore, speed of convergence is important in the early stage of iterations. Particle swarm optimization algorithm has faster convergence ability rather than some other algorithms in some problems [45,46]. In local search stage of PSO, this fast convergence ability decreases and slows down especially when searching in the solution space close to a global optimal solution.

Experimental setup

Computationally expensive numerical CEC 2015, CEC 2017 and realistic engineering and mechanical design problems are used in experiments as benchmark test sets. Basic PSO, FA and other recent hybrid-related FFPSO and HPSOFF algorithms are compared with the proposed HFPSO algorithm. All optimized input parameters are set same for all hybrid algorithms in experiments and they are optimized and obtained from previous original papers [51]. Size of the swarm (Pop) and dimension of a particle (D) are

Conclusions

In this paper, a hybrid algorithm combining PSO and FA is proposed. Advantages and strengths of PSO and FA are combined and disadvantages of these algorithms i.e. premature convergence and local optima are tried to be mitigated. Hybrid use of these algorithms helps to provide a balance between exploration and exploitation processes. The general results show that this hybridization provides good results over standard PSO and FA in the limited function evaluations of computationally expensive

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