Computational Intelligence and Neuroscience

Volume 2019, Article ID 1589303, 24 pages

https://doi.org/10.1155/2019/1589303

## An Enhanced Lightning Attachment Procedure Optimization with Quasi-Opposition-Based Learning and Dimensional Search Strategies

School of Civil Engineering, Guangzhou University, Guangzhou, China

Correspondence should be addressed to Weili Luo; nc.ude.uhzg@oullw

Received 15 February 2019; Revised 15 June 2019; Accepted 17 July 2019; Published 1 August 2019

Academic Editor: Bruce J. MacLennan

Copyright © 2019 Tongyi Zheng and Weili Luo. 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

Lightning attachment procedure optimization (LAPO) is a new global optimization algorithm inspired by the attachment procedure of lightning in nature. However, similar to other metaheuristic algorithms, LAPO also has its own disadvantages. To obtain better global searching ability, an enhanced version of LAPO called ELAPO has been proposed in this paper. A quasi-opposition-based learning strategy is incorporated to improve both exploration and exploitation abilities by considering an estimate and its opposite simultaneously. Moreover, a dimensional search enhancement strategy is proposed to intensify the exploitation ability of the algorithm. 32 benchmark functions including unimodal, multimodal, and CEC 2014 functions are utilized to test the effectiveness of the proposed algorithm. Numerical results indicate that ELAPO can provide better or competitive performance compared with the basic LAPO and other five state-of-the-art optimization algorithms.

#### 1. Introduction

Optimization problems can be found in many engineering application domains and scientific fields which have a complex and nonlinear nature. It is usually difficult to solve these optimization problems using classical mathematical methods since such methods are often inefficient and have a requirement of strong math assumptions. Due to the limitations of classical approaches, many natural-inspired stochastic optimization algorithms have been proposed to conduct global optimization problems in the last two decades. Such optimization algorithms were commonly simple and easy to implement, and these features make the possibility to solve highly complex optimization problems. These metaheuristics can be roughly classified into three categories: evolutionary algorithms, swarm intelligence, and physical-based algorithms.

Evolutionary algorithms are generic population-based metaheuristics, which imitate the evolutionary behavior of biology in nature such as reproduction, mutation, recombination, and selection. The first generation starts with randomly initialized solutions and further evolves over successive generations. The best individual among the whole population in the final evolution is considered to be the optimization solution. Some of the popular evolutionary algorithms are genetic algorithm (GA) [1], genetic programming (GP) [2], evolution strategy (ES) [3], differential evolution (DE) algorithm [4], and biogeography-based optimizer (BBO) [5].

Swarm intelligence algorithms mimic the collective behavior of swarms, herds, schools, or flocks of creatures in nature, which interact with each other and utilize full information about their environment with the progress of algorithm. For example, honey bees are capable of guaranteeing the survival of a colony without any external guidance. In other words, no one tells honey bees how and where to find food sources; instead, they cooperatively seek the food sources even that is located far away from their nests. In this category, particle swarm optimization (PSO) [6], ant colony optimization (ACO) [7], and artificial bee colony algorithm (ABC) [8] can be regarded as representative algorithms. Some other popular swarm intelligence algorithms are firefly mating algorithm (FMA) [9], shuffled frog leaping algorithm (SFLA) [10], bee collecting pollen algorithm (BCPA) [11], cuckoo search (CS) algorithm [12], dolphin partner optimization (DPO) [13], bat-inspired algorithm (BA) [14], firefly algorithm (FA) [15], and hunting search (HUS) algorithm [16]. Some of the recent swarm intelligence algorithms are fruit fly optimization algorithm (FOA) [17], dragonfly algorithm (DA) [18], artificial algae algorithm (AAA) [19], ant lion optimizer (ALO) [20], shark smell optimization algorithm (DSOA) [21], whale optimization algorithm (WOA) [22], crow search algorithm (CSA) [23], grasshopper optimization algorithm (GOA) [24], mouth brooding fish algorithm (MBFA) [25], spotted hyena optimizer (SHO) [26], butterfly-inspired algorithm (BFA) [27], squirrel search algorithm (SSA) [28], Andean condor algorithm (ACA) [29], and pity beetle algorithm (PBA) [30].

The third category is physical-based algorithms which are based on the basic physical laws such as gravitational force, electromagnetic force, and inertia force. Some of the prevailing algorithms of this category are simulated annealing (SA) [31], gravitational search algorithm (GSA) [32], big-bang big-crunch (BBBC) algorithm [33], charged system search (CSS) [34], black hole (BH) algorithm [35], central force optimization (CFO) [36], small-world optimization algorithm (SWOA) [37], artificial chemical reaction optimization algorithm (ACROA) [38], ray optimization (RO) algorithm [39], galaxy-based search algorithm (GbSA) [40], and curved space optimization (CSO) [41], gravitational search algorithm (GSA) [32], and multiverse optimizer (MVO) [42].

Regardless of the difference among the three categories of algorithms, a common point lies in that besides tuning of common control parameters such as population size and number of generations, the metaheuristic algorithms necessitate tuning of algorithm-specific parameters during the course of optimization. For instance, GA requires tuning of cross-over probability, mutation probability, and selection operator [43]; SA requires tuning of initial temperature and cooling rate [31]; PSO requires tuning of inertia weight and learning factors [6]. The improper tuning of these parameters either increases the computational cost or leads to the local optimal solution.

Recently, a new physical-based metaheuristic algorithm named lightning attachment procedure optimization (LAPO) [44] was proposed, which does not require tuning of any algorithm-specific parameters. Instead, an average value of all solutions was employed to adjust the lightning jump behavior of moving towards or away from a jumping point (or position) in a self-adaptive manner. This is an important reason that LAPO is not easily stuck in the local optimal solution and has a good exploration and exploitation abilities. LAPO has already proved its superiority in solving a number of constrained numerical optimization problems [44].

In this paper, an enhanced lightning attachment procedure optimization, namely, ELAPO is developed to increase the convergence speed during the search process of LAPO while maintaining the key feature of the LAPO as free from algorithm-specific parameters tuning. In ELAPO, a concept of opposition-based learning (OBL) is incorporated for enhancing the searching ability of metaheuristic algorithms. The motivation is that the current estimates and their corresponding opposites are considered simultaneously to find the better solutions, thereby enabling the algorithm to explore a large region of the search space in every generation. This concept was found to be effective in improving the performance of well-known optimization algorithms such as genetic algorithms (GA) [45], differential evolution (DE) [46, 47], particle swarm optimization (PSO) [48, 49], biogeography-based optimization (BBO) [50, 51], harmony search (HS) algorithm [52, 53], gravitational search optimization (GSO) [54, 55], group search algorithm (GSA) [56, 57], and artificial bee colony (ABC) [58]. Meanwhile, a dimensional search strategy is proposed to intensively exploit a local search for each variable of the best solution in each iteration, thus resulting in a higher quality of solution at the end of iteration and strengthening the exploitation of the algorithm. To evaluate the effectiveness of the proposed algorithms, ELAPO is applied to 32 benchmark functions and compared with the basic LAPO and six representative swarm intelligence algorithms (SSA [28], Jaya [59], IBB-BC [60], ODE1 [61], and ALO [20]). The effectiveness of the two strategies is also discussed.

The rest of this paper is organized as follows: Section 2 briefly recapitulates the basic LAPO. Next, the proposed ELAPO is presented in a detailed way in Section 3. Numerical comparisons are illustrated in Section 4. Finally, Section 5 gives the concluding remarks.

#### 2. Basic Algorithm

LAPO is a new nature-inspired global optimization, which mimics the lightning attachment procedure including the downward leader movement and the upward leader propagation. The lightning is a sudden electrostatic discharge occurring between electrically charged regions of a cloud, which moves toward or away from the ground in a stepwise movement. After each step, the downward leader stops and then moves to a randomly selected potential point that may have higher value of electrical field. The upward leader starts from sharp points and goes towards the downward leader. The branch fading feature of lightning is taken effect when the charge of a branch is lower than a critical value. In the case where the two leaders join together, a final strike occurs and the charge of the cloud is neutralized.

##### 2.1. Parameters and Initialization of Test Points

Main parameters of the LAPO consist of the maximum number of iterations , the number of test points , the number of decision variables *n*, and the upper and lower bounds for decision variable and . These parameters are given at the beginning of the algorithm. Similar to other nature-inspired optimization algorithms, an initial population is required. Each population is regarded as a test point in the feasible search space, which could be an emitting point of the downward or upward leader. The test points are randomly initialized as follows:where is a uniformly distributed random number in the range [0, 1]. The electric field (i.e., fitness value) of each test point is calculated based on the objective function:

##### 2.2. Downward Leader Movement toward the Ground

In this phase, all the test points are considered as the downward leader and move down towards the ground. The average value of all test points and its corresponding fitness value are calculated as follows:

Given the fact that the lightning has a random behavior, for test point *i*, a random point *k* is selected among the population (*i* ≠ *k*), and the new test point is updated based on the following rules: (i) if the electric field of point *k* is higher than the average electric field, thenand (ii) if the electric field of point *k* is lower than the average electric field, then

If the electric field of the new test point is better than the old one, the branch sustains; otherwise, it fades. This feature is mathematically formulated as

##### 2.3. Upward Leader Movement

In the upward movement phase, all the test points are considered as the upward leader towards the cloud. The new test points are generated as follows:where and are the best and the worst solutions of the population and *S* is an exponent factor that is a function of the number of iterations and the maximum number of iterations :

From a computational point of view, this iteration-dependent exponent factor is important for the balance of exploration and exploitation capabilities of the algorithm. Similar to the downward movement, the branch fading feature also occurs in this phase.

##### 2.4. Enhancement of the Performance

In order to enhance the performance of LAPO, in each iteration, the worst test point is replaced by the average test point if the fitness of the former is worse than the latter:

##### 2.5. Stopping Criterion

The algorithm terminates if the maximum number of iterations is satisfied. Otherwise, the procedures of downward and upward leader movements and of performance enhancement are repeated.

##### 2.6. Procedure of the Basic LAPO

The complete computational procedure of the basic LAPO is provided in Algorithm 1.