Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 128902, 15 pages

http://dx.doi.org/10.1155/2015/128902

## The Cellular Differential Evolution Based on Chaotic Local Search

^{1}Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University, Shanghai 200072, China^{2}School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China

Received 27 January 2015; Revised 30 April 2015; Accepted 4 May 2015

Academic Editor: George S. Dulikravich

Copyright © 2015 Qingfeng Ding and Guoxin Zheng. 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

To avoid immature convergence and tune the selection pressure in the differential evolution (DE) algorithm, a new differential evolution algorithm based on cellular automata and chaotic local search (CLS) or ccDE is proposed. To balance the exploration and exploitation tradeoff of differential evolution, the interaction among individuals is limited in cellular neighbors instead of controlling parameters in the canonical DE. To improve the optimizing performance of DE, the CLS helps by exploring a large region to avoid immature convergence in the early evolutionary stage and exploiting a small region to refine the final solutions in the later evolutionary stage. What is more, to improve the convergence characteristics and maintain the population diversity, the binomial crossover operator in the canonical DE may be instead by the orthogonal crossover operator without crossover rate. The performance of ccDE is widely evaluated on a set of 14 bound constrained numerical optimization problems compared with the canonical DE and several DE variants. The simulation results show that ccDE has better performances in terms of convergence rate and solution accuracy than other optimizers.

#### 1. Introduction

Differential evolution (DE) algorithm, proposed by Storn and Price [1], is a population-based parallel iterative optimization algorithm and outperforms many other optimization methods in terms of convergence speed and robustness over common benchmark functions and real-world problems [2]. Its optimization performance is mainly influenced by three critical control parameters including scaling factor , population size , and crossover rate . Nevertheless, DE algorithm also has immature convergence and search stagnation and other defects, which limit its application range and its ability of optimization, which urgently need to be explored in depth.

Most evolutionary algorithms use a single population of individuals and apply stochastic operators to them as a whole [3]. Some recent studies show that it is easy to implement the parallel computing for those evolutionary algorithms with spatial structure [4]. However, the canonical DE algorithm ignores the spatial structure and the complex interaction among individuals of local groups during evolutionary processes. For example, the population of the canonical DE will not be affected by external disturbances and the individual states. Namely, the death, resurrection, and migration of the individual will never be modified, which is obviously inconsistent with the actual process of biological evolution. Among numerous evolutionary algorithms with spatial structure [5], the cellular evolutionary algorithm (cEA) is a kind of evolutionary algorithms of discrete groups based on spatial structure. It means an individual may interact with its adjacent neighbors. The overlapped neighborhoods of cEA help in exploring the search space while exploitation takes place inside neighborhood by stochastic operators. Dorronsoro and Bouvry [6] proposed a new cellular genetic algorithm (cGA) that automatically manages its neighbors according to the quality of individuals in a population and sets the most reasonable population structure based on the convergence speed. Then the traditional parameters configuration for the population and the neighbor structure becomes less necessary. According to the evolution characteristics of CA in a discrete space and ant optimization, a CA-based optimization method is proposed for solving optimization problems with geometric constraints [7]. Lu et al. [8] proposed a cellular GA with the evolutionary rules and derived the selection of the evolutionary rules, and this GA variant has a better ability to maintain the diversity of the population relative to canonical genetic algorithm. Lorenzo and Glisic [9] presented a novel sequential genetic algorithm (SGA) to optimize the relaying topology in multihop cellular networks aware of the intercell interference and the spatial traffic distribution dynamics. Noman and Iba [10] proposed cellular differential evolution (cDE) algorithm with linear and compact neighbor, which only uses the cellular neighbor but does not consider the dynamic population evolutionary. Noroozi et al. [11] proposed CellularDE to address dynamic optimization problems, which limits the number of individuals in each cell to prevent convergence and maintain diversity of the population. Jia et al. [12] proposed that chaotic local search (CLS) can enhance the performance of DE.

To simulate the natural condition, the paper proposes a CLS-based cDE (ccDE) algorithm to simulate its dynamic process. First of all, the algorithm employs CA with the parallel evolution to balance the exploration and exploitation tradeoff for DE. Next, the CLS helps to avoid immature convergence and search stagnation by utilizing chaotic search with the ergodicity and randomicity. At last, the ccDE employs the orthogonal crossover operator to replace the binomial crossover operator in canonical DE, which contributes to the improvement of the convergence speed and maintenance of the population diversity.

The remainder of this paper is organized as follows. Sections 2 and 3 describe the canonical DE and the ccDE, respectively. Section 4 briefly illustrates the comparison of the performance of the proposed ccDE algorithm with the canonical DE algorithm and four DE variants over a suit of 14 problems. The findings of the paper are demonstrated in Section 5.

#### 2. Canonical DE

DE makes use of a new mutation operator which depends on the differences among randomly selected pairs of individuals instead of predetermined probability distribution function [1]. Its whole evolutionary process consists of two stages: initialization stage and iterative evolutionary stage.

In the initialization stage, the initial individual population is chosen at random in a dimensional search space. A differential evolutionary population of th generation is as follows:wherein denotes th individual of th generation which denotes the generation counter, denotes the population size, denotes the dimension of decision space, and denotes the initial population.

In the iterative evolutionary stage, each individual goes through a succession of iterative processes including mutation, crossover, and selection. Then the individual is evaluated and updated by utilizing the fitness function. The above iterative evolutionary stage is repeated generation after generation until the termination criterion is met.

*(1) Mutation*. The mutant individual of DE is decided from the differences among randomly selected pairs of individuals. For each target individual , a mutant individual is generated as follows:wherein , , and are the best individual or randomly selected individuals from the current population, all of which are different from the target individual . The scaling factor is a predefined constant for scaling the differential individual vector in (2).

*(2) Crossover*. In the canonical version, DE applies binomial crossover operator to generate a trial individual by recombining a target individual and its corresponding mutant individual. The binomial crossover can be outlined as follows:wherein , , and are an element of the trial individual , the mutant individual , and the target individual , respectively. The crossover factor is a predefined constant to adapt to different optimization domain, which is called crossover probability.

*(3) Selection*. The selection scheme in DE adopts one-to-one competitive strategy. The trial individual competes with the target individual , which is decided by the fitness value of their individuals. The selection scheme can be outlined as follows:

#### 3. ccDE Algorithm

##### 3.1. The Evolution Mechanism of Cellular Automata

CA proposed by Neumann [13] is a highly parallel computing model. For the most usual 2D CA, every cell is arranged into a 2D toroidal mesh and follows the same rule and updates on the basis of the local rule synchronously. Different evolutionary rules produce different cellular states. The basic model of the evolution rule is defined as follows:wherein and denote the cellular state of th step and th step. is a set of the numbers of active neighbors needed for the active cell to stay alive and is a set of the numbers of active neighbors in order to resuscitate a dead cell.

The neighborhood structure actually affects the quality of the search, such as the improvement of inferior solutions, the avoidance of local optimum, and the maintenance of the population diversity. The two most common cellular neighbor structures are linear (L) pattern and compact (C) pattern. Figure 1 shows four typical neighborhood structures employed in cEAs.