Computational Intelligence and Neuroscience

Volume 2018 (2018), Article ID 6094685, 27 pages

https://doi.org/10.1155/2018/6094685

## PS-FW: A Hybrid Algorithm Based on Particle Swarm and Fireworks for Global Optimization

School of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, China

Correspondence should be addressed to Yang Liu

Received 21 September 2017; Accepted 10 January 2018; Published 20 February 2018

Academic Editor: Raşit Köker

Copyright © 2018 Shuangqing Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Particle swarm optimization (PSO) and fireworks algorithm (FWA) are two recently developed optimization methods which have been applied in various areas due to their simplicity and efficiency. However, when being applied to high-dimensional optimization problems, PSO algorithm may be trapped in the local optima owing to the lack of powerful global exploration capability, and fireworks algorithm is difficult to converge in some cases because of its relatively low local exploitation efficiency for noncore fireworks. In this paper, a hybrid algorithm called PS-FW is presented, in which the modified operators of FWA are embedded into the solving process of PSO. In the iteration process, the abandonment and supplement mechanism is adopted to balance the exploration and exploitation ability of PS-FW, and the modified explosion operator and the novel mutation operator are proposed to speed up the global convergence and to avoid prematurity. To verify the performance of the proposed PS-FW algorithm, 22 high-dimensional benchmark functions have been employed, and it is compared with PSO, FWA, stdPSO, CPSO, CLPSO, FIPS, Frankenstein, and ALWPSO algorithms. Results show that the PS-FW algorithm is an efficient, robust, and fast converging optimization method for solving global optimization problems.

#### 1. Introduction

Global optimization problems are common in engineering and other related fields [1–3], and it is usually difficult to solve the global optimization problems due to many local optima and complex search space, especially in high dimensions. For solving optimization problems, many methods have been reported in the past few years. Recently, the stochastic optimization algorithms have attracted increasing attention because they can get better solutions without any properties of the objective functions. Therefore, many effective metaheuristic algorithms have been presented, such as simulated annealing (SA) [4], differential evolution (DE) [5], genetic algorithm (GA) [6], particle swarm optimization (PSO) [7], ant colony optimization (ACO) [8], artificial bee colony (ABC) [9], and fireworks algorithm (FWA) [10].

Among these intelligent algorithms, the PSO and FWA have shown pretty outstanding performance in solving global optimization problems in the last several years. PSO algorithm is a population-based algorithm originally proposed by Kennedy and Eberhart [7], which is inspired by the foraging behavior of birds. Fireworks algorithm is a new swarm intelligence algorithm that is motivated by observing fireworks explosion. Owing to the less decision parameters, simple implementation, and good scalability, PSO and FWA have been widely applied since they were proposed, including shunting schedule optimization of electric multiple units depot [11], optimal operation of trunk natural gas pipelines [12], location optimization of logistics distribution center [13], artificial neural networks design [14], warehouse-scheduling [15], fertilization optimization [16], power system reconfiguration [17], and multimodal function optimization [18].

Although PSO and FWA are highly successful in solving some classes of global optimization problems, there are certain problems that need to be addressed when they are extended to handling complex high-dimensional optimization problems. The PSO algorithm has a significant efficiency in unimodal problems, but it can easily be trapped in local optima for multimodal problems. Moreover, the FWA is difficult to converge for the optimization problems which do not have their optimal solutions at the origin. This is because the two algorithms cannot keep the balance between the exploration and exploitation properly. Due to the optimal particle dominating the solving process, the PSO algorithm has inferior swarm diversity in the later stage of iterations and relatively poor exploration ability [19], while the fireworks and sparks in FWA are not well-informed by the whole swarm [20] and the FWA framework lacks the local search efficiency for noncore fireworks [21]. In order to improve the performance of PSO and FWA, a considerable number of modified algorithms have been proposed. For example, Nickabadi et al. presented AIWPSO algorithm, in which a new adaptive inertia weight approach was adopted [22]. By embedding a reverse predictor and adding a repulsive force into the basic algorithm, the RPPSO was developed [23]. Wang and Liu used three strategies to ameliorate the standard algorithm, including best neighbor replacement, abandoned mechanism, and chaotic searching [24]. Souravlias and Parsopoulos introduced a PSO-based variant, which could dynamically assign different computational budget for each particle based on the quality of its neighbor [25]. Based on self-adaption principle and bimodal Gaussian function, the advanced fireworks algorithm (AFWA) was proposed [26]. Liu et al. presented several methods for computing the explosion amplitude and number of sparks [27]. Pei et al. proposed to use the elite point of approximation landscape in the fireworks swarm and discussed the effectiveness of surrogate-assisted FWA [28]. Zheng et al. improved the new explosion operator, mutation operator, selection strategy, and mapping rules of FWA, which led to the formation of enhanced fireworks algorithm (EFWA) [29, 30] and dynamic search in fireworks algorithm (dynFWA) [31]. Zheng et al. proposed the new cooperative FWA framework (CoFFWA), in which the independent selection method and crowdedness-avoiding cooperative strategy were contained [21]. Li et al. investigated the operators of FWA and introduced a novel guiding spark in FWA [32] and proposed the adaptive fireworks algorithm (AFWA) [33] and bare bones fireworks algorithm (BBFWA) [34].

Hybrid algorithms can utilize various exploration and exploitation strategies for high-dimensional multimodal optimization problems, which have gradually become the new research areas. For example, Valdez et al. combined the advantages of PSO with GA and proposed a modified hybrid method [35]. In the new PS-ABC algorithm introduced by Li et al., the global optimum could be obtained by combining the local search phase in PSO with two global search phases in ABC [19]. Pandit et al. presented the SPSO-DE, in which the domain information of PSO and DE was shared with one another to overcome their respective weaknesses [36]. Through changing the generation and selection strategy of explosive spark, Gao and Diao proposed the CA-FWA [37]. Zhang et al. proposed BBO-FW algorithm which improved the interaction ability between fireworks [38]. By combining the FWA with the operators of DE, a novel hybrid optimization algorithm was proposed [20].

In this paper, by utilizing the exploitation ability of PSO and the exploration ability of FWA, a novel hybrid optimization algorithm called PS-FW is proposed. Based on the solving process of PSO algorithm, the operators of FWA are embedded into the update operation of the particle swarm. In the iteration process, in order to promote the balance of exploitation and exploration ability of PS-FW, we presented three major techniques. Firstly, the abandonment and supplement strategy is used, to abandon a certain number of particles with poor quality and to supplement the particle swarm with new individuals generated by FWA. Meanwhile, considering the information exchanges between the optimal firework and its neighbor in each dimension, the method for obtaining the explosion amplitude is designed as adaptive, and the mode of generating the explosion sparks is modified by combing the greedy algorithm. Furthermore, the conventional Gaussian mutation operator is abandoned, and the novel mutation operator based on the thought of the social cognition and learning is proposed. The performance of PS-FW is compared with several existing optimization algorithms. The experimental results show that the proposed PS-FW is more efficacious in solving the global optimization problems.

The rest of the paper is organized as follows: Section 2 describes the standard PSO and FWA. Section 3 presents the PS-FW algorithm, in which the algorithm details are proposed. Section 4 introduces the simulation results over 22 high-dimensional benchmark functions and the corresponding comparisons between PS-FW and other algorithms are executed. Finally, the conclusion is drawn in Section 5.

#### 2. Related Work

##### 2.1. PSO Algorithm

In PSO algorithm, the particles scatter in search space of the optimization problems and each particle denotes a feasible solution. Each particle contains three aspects of information: the current position , the velocity , and the previous best position . Assume that the optimization problem is -dimensional and represents the size of the swarm population; then the position and velocity of th () particle can be denoted as and , respectively, while the previous best position is represented as . Besides, the best position encountered by the entire particles so far is known as current global best position . In each generation, and are updated by the following equations: where and are two learning factors that indicate the influence of the cognitive and social components, and are the random real numbers in interval , respectively, and is the inertia weight which controls the convergence speed of the algorithm.

##### 2.2. Fireworks Algorithm

In FWA, a firework or a spark denotes a potential solution of optimization problems, while the process of producing sparks from fireworks represents a search in the feasible space. As in other optimization algorithms, the optimal solutions are obtained by successive iterations. In each iteration, the sparks can be produced by two ways: the explosion and the Gaussian mutation. The explosion of fireworks is dominated by the explosion amplitude and the number of explosion sparks. Compared to the fireworks with lower fitness, the fireworks with better fitness will have smaller explosion amplitude and more explosion sparks. Suppose that denotes the number of fireworks; then the th () firework can be denoted as for -dimensional optimization problems. Besides, the explosion amplitude can be obtained by (3) and the sparks number can be calculated by (4): where denotes the objective function value of the th firework, , and are the explosion amplitude and the number of explosion sparks of the th firework, respectively, , , and are two constants that dominate the explosion amplitude and the number of explosion sparks, respectively, and is the machine epsilon.

Moreover, the bounds of are defined as follows: where , are two constants that control the minimum and maximum of population size, respectively.

In order to generate each explosion spark of th firework, an offset is added to according to the following equation: where is the th explosion spark of th firework and , where is a -dimensional vector which has values of 1 and values of 0, where denotes the number of randomly selected dimensions of and , , where and are random numbers in the intervals and , respectively.

Another type of sparks known as the Gaussian sparks is generated based on the Gaussian mutation operator. In each generation, a certain number of Gaussian sparks are generated and each Gaussian spark is transformed from a firework which is selected randomly. For the selected firework , its Gaussian spark is generated based onwhere is the th Gaussian spark, is a -dimensional vector whose values are 1 in each dimension, is a -dimensional vector which has values of 1 and values of 0, represents the number of randomly selected dimensions of and , and represents a random number subordinated to the Gaussian distribution with the mean of 1 and the standard deviation of 1.

For the purpose of passing information to the next generation, new fireworks populations are chosen to continue the iteration. All the fireworks, the explosion sparks, and Gaussian sparks have the chance to be selected for the next iteration. The location with best fitness is kept for the next generation, while the other locations are selected based on the selection operator and the selection operator is denoted as follows: where denotes the set comprised of all the original fireworks and both types of sparks, , , and are th, th, and th location of , respectively, is the distance between th location and the rest of all the locations, and denotes the probability of being selected for the th location.

#### 3. Hybrid Optimization Algorithm Based on PSO and FWA

The exploitation process focuses on utilizing the existing information to look for better solutions, whereas the exploration process attaches importance to seek the optimal solutions in the entire space. For PSO, under the guidance of their historical best solutions and the current global best solution, the particles can quickly find better solutions, and the excellent exploitation efficiency of algorithm is shown. In FWA, the fireworks can find the global optimal solution in the whole search space by performing explosion and mutation operations while the outstanding exploration capability of FWA is demonstrated. To utilize the advantages of the two algorithms, a hybrid optimization method (PS-FW) based on PSO and FWA is proposed.

##### 3.1. Feasibility Analysis

The formation of a hybrid algorithm is mainly due to the effective combination of the operators of its composition algorithms in a certain way. To clarify the performance enhancement caused by combining the PSO algorithm with fireworks algorithm, we draw Figures 1 and 2 to illustrate the optimization mechanism. As shown in Figure 1, for standard PSO algorithm, the th particle moves from point 1 to point 4 under the common influence of velocity inertia, self-cognition, and social information. When the operators of FWA are added, the particle is transformed into firework and performs explosion and mutation operations and eventually reaches the position of firework or sparks, such as point 5 shown in Figure 1. By performing the operators of FWA, the particle can explore better solutions in multiple directions and jump out of the local optima region as depicted in Figure 1. Thus we can argue that the operators of FWA improve the global search ability of PSO algorithm. As we know, the searching region is determined by the explosion amplitude and fireworks with poor quality have bigger amplitude, which may lead to an uncomprehensive search without considering the cooperation with other fireworks. When the firework with poor quality generates the explosion sparks and mutation sparks, the new selected location may skip over the global optima region without the attraction from the rest of fireworks and arrive at point 2. By adding the operators of PSO after the th firework updates its location, the information of its own historical best location and current global best location is taken into account; then the new solution is found in point 5, which is shown in Figure 2. Therefore, the operators of PSO could strengthen the local search efficiency of FWA. Based on the above analysis, it is concluded that the combination of PSO and FWA is an effective way to form a superior optimization algorithm.