Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article
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Artificial Intelligence and Its Applications 2014

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Research Article | Open Access

Volume 2015 |Article ID 205709 | 21 pages | https://doi.org/10.1155/2015/205709

A Novel Tournament Selection Based Differential Evolution Variant for Continuous Optimization Problems

Academic Editor: Yudong Zhang
Received19 Sep 2014
Revised02 Dec 2014
Accepted10 Dec 2014
Published04 Oct 2015

Abstract

Differential evolution (DE) is a powerful global optimization algorithm which has been studied intensively by many researchers in the recent years. A number of variants have been established for the algorithm that makes DE more applicable. However, most of the variants are suffering from the problems of convergence speed and local optima. A novel tournament based parent selection variant of DE algorithm is proposed in this research. The proposed variant enhances searching capability and improves convergence speed of DE algorithm. This paper also presents a novel statistical comparison of existing DE mutation variants which categorizes these variants in terms of their overall performance. Experimental results show that the proposed DE variant has significance performance over other DE mutation variants.

1. Introduction

Storn and Price [1] have proposed DE in 1995 which is a stochastic population based evolutionary algorithm to find global optima for any given function [2]. DE has been applied to variety of real life problems such as electrical power systems [3], control systems [4], electromagnetism [5], image processing [6], bioinformatics [7], signal processing [8], chemical engineering [9], and many more. Different state-of-the-art versions of DE algorithm like jDE [10], SaDE [11], JADE [12], EPSDE [13], and so forth are based on parameter selection, parameter adaption, strategy selection, and/or strategy adaption mechanisms. DE state-of-the-art proves to be very powerful and uses conventional DE strategies as a strategy selection pool/strategy adaption and/or uses parameter selection pool/parameter adaption. During the process of optimization, DE algorithm evolves population of potential solutions by exploring the entire search space overtime to locate the optima.

The DE algorithm mutation strategies are formed by the linear combination of existing population members. The trial vector and target vector forms the mutant vector in DE. DE mutation strategies can be formed by any combinations of current vector, random vector(s), better vector, and best vector. In any mutation strategy, the order, number, and name of vector(s) are very important. The behavior of DE algorithm is influenced by the selection of mutation strategy and crossover scheme along with their control parameters: mutation probability “” and crossover rate “CR.” As stated, there are many mutation strategies associated with DE algorithm but current literature does not show any concrete study where all of the popular mutation strategies are analyzed. It has been observed by the authors that some of the DE mutation strategies are more useful than the current dominating ones. Therefore, in this paper, a comparative analysis is also carried out to identify best performer DE mutation variants. Furthermore, this research also proposes a novel DE mutation variant. The proposed DE variant is based on the well known tournament selection criteria. The proposed DE mutation strategy is abbreviated as TSDE in this paper. To generate a mutant vector in DE algorithm, some parents will be selected through tournament selection criteria. The TSDE variant has the ability to increase the convergence speed since it uses the combination of best vectors and random vectors from the population. Tournament selection will be helpful to incorporate some diversity in TSDE since global best vector has less chance of selection as a parent. To generate a mutant vector, the proposed TSDE utilizes two best individuals instead of global best from the whole population which makes it more convenient to escape from the local optima problem. The most state-of-the-art DE algorithms focuses on basic DE mutation strategies and the incorporation of this proposed TSDE mutation variant will be helpful in improving the performance of DE algorithm.

The rest of the paper is organized into the following sections. Section 2 presents literature survey. In Section 3, the DE algorithm operators are discussed. In Section 4, the proposed TSDE variant is presented. Section 5 presents test functions and parameter setting details. In Section 6, experimental results along with discussion are given. Section 7 contains statistical analysis. Section 8 compares proposed TSDE with other well known heuristic algorithms. Finally, conclusion and future work are presented in Section 9.

2. Literature Survey

As stated above, many state-of-the-art versions of the DE algorithms are developed, for example, jDE, SaDE, JADE, and DEGL. Variations in the mutation strategies of DE algorithm have also been the result of intensive research in this area. Qu and Suganthan [29] have introduced the concept of ensemble restricted tournament selection in DE algorithm. They have selected a set of population members and calculated the Euclidean distance of population members from the current individual. The member having smallest distance from current individual is selected as a tournament winner and it became the current individual if its fitness is also better. Epitropakis et al. [30] have introduced two new mutation strategies that incorporate spatial information of neighbors to form new population individual. Their new mutation strategies “DE/nrand/1” and “DE/nrand/2” corresponding to two well known mutation strategies “DE/rand/1” and “DE/rand/2” are important part of the current literature. Qiu et al. [31] have introduced the concept of tournament selection criteria in DE algorithm. They have selected all individuals used in their mutation strategy by using the concept of tournament selection by taking a tournament of size 3. Their research work is applied on “DE/rand/1,” “DE/rand/2,” “DE/current to best/2, and “DE/rand to best/2” DE mutation strategies which proved to be competitive to original DE algorithm and also enhanced jDE algorithm. Neighbourhood based mutation strategy for DE algorithm was introduced by Das et al. [32]. This strategy implements the concepts of local model that works on small neighbourhood of each individual while global model mutates each individual using the neighbourhood concept of entire population. The concept of neighbourhood is implemented on a commonly used DE mutation strategy “DE/target-to-best/1/bin. Sutton et al. [33] have introduced the concept of focus search by imposing a selective pressure for DE mutation strategies. The selective pressure is implemented by using the concept of rank based selection of vectors used in generating donor vector in DE mutation. A bias is imposed on selecting a parent using rank based selection criteria that is based on the fitness of population members. Epitropakis et al. [34] have introduced the concept of proximity based mutation operator in DE algorithm. Their new framework utilizes the information of neighbours in evolving the population. Each population member is assigned a selection probability that is inversely proportional to the distance from mutated vector. Proximity based mutation is applied to “DE/rand/1,” “DE/best/1,” “DE/rand/2,” “DE/best/2,” “DE/rand-to-best/1, and “DE/rand-to-best/2” mutation strategies. Research result shows the significance of proximity based mutation operator in DE algorithm.

DE algorithm has a number of conventional mutation strategies where the performances of these strategies may differ from each other. Zaharie [35] has analyzed the influence of crossover variant on the choice of control parameters of the DE algorithm. He has used “DE/rand/1” for different parameter settings of population size, mutation probability, and the crossover rate by binomial and exponential schemes. Babu and Munawar [36] have applied various DE strategies for the optimal design of shell and tube heat problem which performed better than “DE/rand” for the same problem. They have used different DE variants for the selection of optimal design using various control parameter settings. They conclude that “DE/best” strategies have better performance than “DE/rand” for shell and tube heat problem. Jeyakumar and Velayutham [24] present empirical performance analysis of DE variants for unconstrained global optimization problems where they have shown that binomial variants have superior performance compared to the exponential variants. Mezura-Montes et al. [37] have performed comparative study of DE mutation variants for continuous function optimization application. They found through the empirical study that “DE/best/1/bin” has produced good results. Ali et al. [38] have compared Mixed Strategies Differential Evolution (MSDE) with the five different DE variants for function optimization. They have shown that performance of MSDE is better than five variants of DE algorithm. Mutation operators are also used in other well known algorithms like PSO [39].

Qin et al. [11] introduced both control parameter adaption and strategy adaption mechanisms in DE algorithm (SaDE). In strategy adaption scheme, they used a strategy candidate pool of four conventional strategies: “DE/rand/1/bin,” “DE/rand/2/bin,” “DE/rand-to-best/2/bin, and “DE/current-to-rand/1. Each target vector generates a trial vector based on the learning period (LP) over previous generations based on the success rate. In parameter adaption, SaDE adjusts control parameter CR based on the median value of CR that is calculated based on previous CR values that have successfully generated trial vector. Zhang and Sanderson [12] introduced new parameter adaption method and have used “DE/current to pbest/1” conventional strategy that is based on “DE/current to best/1. Mallipeddi et al. [13] introduced ensemble based crossover and mutation DE strategy (EPSDE) and their corresponding control parameter scheme. They have used a pool of different crossover and mutation strategies and a pool of values for each associated control parameter. Each target vector generates a trial vector based on the assigned strategy and the parameter values. Successful combination of the mutation and crossover strategy and associated parameters values are stored in the pool. EPSDE uses “DE/current-to-rand/1/bin” and JADE mutation strategy along with binomial and exponential crossover. Islam et al. [40] have introduced modified mutation strategy, a modification in conventional binomial crossover scheme, and a parameter adaption scheme in DE algorithm. They have used “DE/current-to-gr_best/1” and call it a less greedy version of “DE/current-to-best/1. Their mutation strategy “DE/current-to-gr_best/1” uses best vector from population of individuals to generate the trial vector for each target vector. In pbest crossover mutant vector can swap p-top ranked individuals of current generation instead of current parent using binomial crossover scheme. In parameter adaption scheme scale factor adaption is based on the Cauchy distribution and crossover probability adaption is based on the Gaussian distribution. Wang et al. [41] have introduced composite DE (CoDE) variant. In this scheme, they have used a pool of three trial vector generation strategies and a pool of three parameter setting combinations. The trial vector strategies used are “rand/1/bin,” “rand/2/bin, and “current-to-rand/1” while the parameter setting combinations are [, ], [, ], and [, ]. To generate a new solution in CoDE, each strategy is coupled with a randomly chosen parameter setting. Gong et al. [42] have introduced a new strategy adaptation mechanism (SaM) in their research work. They have combined SaM with JADE and named it as SaJADE. In this strategy, parameter is used to control the selection of any strategy from the strategy pool. They have chosen four various DE strategies “DE/current-to-pbest” without archive, “DE/rand-to-pbest” without archive, “DE/current-to-pbest” with archive, and “DE/rand-to-pbest” with archive to form a strategy pool.

Table 1 shows different binomial and exponential mutation strategies used with algorithm. Current literature shows variation in naming in these strategies, which is already indicated in Table 1; for example, Four different names are used in the literature for mutation strategy . Throughout this paper, denotes the target vector (or current vector), represents the trial vector, and as a mutant vector. In this paper, states the th random vector for th generation, will refer to th component of donor vector at th generation, states the best vector at th generation, and refers to current vector at th generation.


Mutation strategy numberBinomial/exponential
Variant nameEquation

DE/rand/1 [1]

DE/best/1 [14]

DE/rand/2 [14]

DE/best/2 [14]

DE/current to rand/1 [15]

DE/Current-to-rand/1 [16]

DE/current to best/1 [1] & DE/rand to best/1 [17] &
DE/current to best/2 [18] & DE/rand to best/2 [19]

DE/current to best/1 [20] & DE/rand to best/1 [21]

DE/rand to best/1 [22] & DE/rand to best/2 [20]

DE/rand to best/1 [23]

DE/rand to best/1 [24]

DE/current to best/2 [15] & DE/rand to best/2 [25]

DE/current to rand/2 [15]

DE/rand to best/2 [26]

DE/rand to best/2 [26]

DE/rand to current/2 [27]

DE/rand to best and current/2 [27]

DE/mid-to-better/1 [28]

DE/rand/3 [27]

DE/best/3 [27]

3. DE Algorithm

DE algorithm has three different parameters: a population of size NP, crossover control parameter CR, and difference vector amplification parameter . Each population member in DE is represented as a -dimensional parameter vector. In DE algorithm, population is initialized randomly which is supposed to cover the entire search space. Each vector in the DE is represented by , where and is generation number. New offspring in DE algorithm are generated by mutation, crossover, and selection operators. Three different terminologies of vectors donor vector, trial vector, and target vector are used in DE algorithm. Donor vector is a vector that is created in the mutation operation, trial vector is created in the crossover operation, and target vector is the current vector of population. The detail of these operators with index , is as follows.

Mutation. In mutation operation, mutant vector also called donor vector is created. Donor vector of th population member is calculated by adding the weighted difference of two vectors to third vector:where random indices , , and is mutation probability parameter.

Crossover. DE crossover strategies control the number of inherited components from the mutant vector to form a target vector. Binomial and exponential are main crossover schemes. The DE crossover rate parameter (CR) influences the size of perturbation of the base (target) vector to ensure the population diversity [43]. Following are the binomial and exponential crossover schemes.

Binomial Crossover. In crossover operation of DE algorithm, a trial vector is formed. In binomial crossover scheme, the trail vector is generated by the following equation: where is a randomly chosen integer in the range , is a random number in , and is the donor vector. is crossover control parameter in the range . Due to the range of , is always different from and index , .

Exponential Crossover. In exponential crossover scheme, the trail is created as follows:where , , and denotes the modulo function with modulus . The starting index is chosen at random from . is also a randomly generated number from . The parameters and are regenerated for each trial vector .

Selection. In DE algorithm, new population members are formed using selection operation. Selection operator uses the greedy approach by comparing fitness of trial vector with the fitness of target vector ; the vector having best fitness is selected as a member of new population. For selection, the following equation is used:where fitness function calculates the fitness value of objective function.

There are several DE algorithm mutation variants/strategies that are formed by the linear combination of existing population members. The trial vector and target vector form the mutant vector in DE. Throughout this paper, denotes the target vector (or current vector), is the running index, represents the trial vector, and is a mutant vector. In DE algorithm, different mutation schemes are used to create the trial vector by using any combination of current, best, and random vectors. The behavior of DE algorithm is influenced by the selection of mutation strategy and crossover scheme along with their control parameters: mutation probability “” and crossover rate “CR” [44].

Difference vector (DV) is the difference of two mutating vectors that is used to form offspring in the population [45]. To form the mutant vector in DE, some researcher uses a random value as a coefficient multiplier with the first difference vector [1] and mutation probability “” as a coefficient multiplier with the other difference vector(s) [17]. Some researchers have used only “” as a coefficient multiplier with the difference vector(s) to form the mutant vector [44]. To reduce the number of control parameters of DE algorithm, we use [28]. A list of binomial and exponential mutation strategies that are used in this research is given in Table 1.

4. Proposed Tournament Selection Based DE (TSDE) Variant

Tournament selection is one of the famous selection approaches used in genetic algorithms [46]. The parameter associated with this scheme is the size of tournament that selects number of individuals to participate in the competition of tournament. To decrease the risk of premature convergence, the loss of diversity should be kept as low as possible in the population [47]. To enhance the performance of population based algorithms, various researches have used tournament selection in their research work [48, 49].

The DE algorithm sometimes faces the problem of slow and/or premature convergence [32]. Exploration and exploitation are very important aspects that are helpful in improving the convergence acceleration and solution quality of evolutionary algorithms [50]. The mutation strategies “DE/rand/1” and “DE/rand/2” are helpful for exploration [11] and “DE/best/1,” “DE/rand-to-best, and “DE/best/2” are more helpful for exploitation than exploration due to less population diversity [14]. To balance the exploration and exploitation ability of proposed TSDE variant, both random and best vectors are used.

A novel DE mutation strategy based on the selection of parent is introduced in this research work. The new mutation strategy utilizes the knowledge of some best performing vectors and some random vectors in creating the mutant vector. Like mutation strategies “DE/rand/1” and “DE/rand/2, the proposed mutation strategy uses random vector as a base vector along with two difference vectors. To generate a mutant vector in DE, each difference vector utilizes the knowledge of one distinct best vector that will be helpful in incorporating diversity and improving convergence in the proposed mutation strategy. The two difference vectors and two distinct best vectors selected through tournament selection criteria will be helpful in balancing the exploration and exploitation.

To minimize the loss diversity, a tournament of small size is selected so that less fit individuals may have a chance of selection as a parent. TSDE generates two best vectors using tournament selection mechanism. The selection of parents is based on the tournament selection criteria by taking a tournament of size 3 of randomly selected population members. The equation of the proposed variant TSDE is as follows: The equation of this variant uses two best vectors and and two random vectors and that are selected from the current population. Vectors and are selected by using tournament selection mechanism by taking tournament of any size, say . Vector is the best individual from first tournament and vector is the second best individual from the second tournament. The vectors used in each tournament are selected randomly from the current population.

The proposed TSDE variant has the ability to increase the convergence speed since it uses the combination of best vectors and random vectors from the population. Tournament selection may incorporate some diversity in TSDE since global best vector has less chance of selection as a parent because tournament of small size is used instead of whole population. Selection of two best individuals instead of global best from the whole population makes it more opportune to escape from the local optima problem that accordingly increases the solution quality and convergence speed of DE algorithm. The most state-of-the-art DE algorithm focuses on basic DE mutation strategies and the incorporation of this proposed mutation strategy will be helpful in improving the performance of DE algorithm.

The proposed mutation strategy has the following features.(1)A random number will be used as a base vector since mutation strategies “DE/rand/1” and “DE/rand/2” use random base vectors that maintain better exploration ability but have slow convergence speed [11].(2)Although DE mutation strategies “DE/best/1,” “DE/rand-to-best, and “DE/best/2” have fast convergence [14], these are biased towards global best value that may have a chance to stick in local optima problem [11]. So to improve the convergence speed of proposed TSDE, two best vectors (best1, best2) are selected using tournament selection criteria.(3)The equation of proposed mutation strategy is where is scaling factor of each difference vector. The motivation of this mutation strategy is an existing DE variant “DE/rand to best/1” [23, 51] that places perturbation of random vector towards global best vector and contains two difference vectors with equation Two best vectors (best1, best2) in TSDE will be helpful in escaping local optima problem and will be helpful in improving the solution quality.(4)Two difference vectors are used where each difference vector utilizes one best vector selected through tournament criteria, since mutation strategies having two difference vectors (DV) produce better perturbation mode than one difference vector mutation strategies [41].

The members of tournament are selected randomly which almost makes it equal chance of selection of higher or lower performing individuals from the population. This variant maintains randomness since members of the tournament are selected randomly and we select the best performing amongst the tournament to ignore the poor performing individuals. The proposed variation is simple since it uses two best individuals selected using the tournament selection technique and other random vector selected randomly from the current population. The performance of TSDE is compared with other DE variants in Section 6.

A graphical representation of proposed mutation strategy is shown in Figure 1, where mutant vector is generated using four vectors, , , , and , shown in filled circles and directed arrow shows difference vector.

Algorithm 1 shows the pseudocode of the proposed DE algorithm (TSDE). The proposed method is also implemented through computer simulation which will be discussed in a later part in the paper.

    (1) Generate the initial population for generation and randomly initialize
    each population member where
    (2) FOR to NP
          Calculate fitness for each population member
       END FOR
    (3) WHILE the stopping criterion is not true
                   /*Start of TSDE vectors selection */
        Step  3.1.  TSDE vectors selection
           FOR to number of TSDE vectors
              FOR to Tournment_size
                 Select th tournament member with its fitness randomly from current population
              END FOR
              Select best of best member from the current tournament as   
              Return member index to be used as one of TSDE vectors in proposed mutation strategy
           END FOR
                   /*End of TSDE vectors selection */
     Step  3.2.  Mutation Step
              FOR to NP
               For the th target vector generate a donor vector with the
               specified mutation strategy (From Table 1 strategies or proposed TSDE strategy)
              END FOR
     Step  3.3.  Crossover Step
           FOR to NP
               For the th target vector generate a trial vector with the
               specified crossover scheme (Equation (2) or Equation (3))
           END FOR
    Step  3.4.  Selection Step
        FOR to NP
      Evaluate the trial vector against the target vector with fitness function f
      IF , THEN ,
                 IF , THEN ,
                 END IF
      END IF
        END FOR
    Step  3.5.  increment generation number
END WHILE

5. Test Functions and Parameter Setting

In order to evaluative the performance of the proposed variant and existing variants of DE algorithm, a comprehensive set of 37 -dimensional benchmark functions is used. These benchmark functions are commonly used for multidimensional global optimization problems. These functions are given in Table 2 along with the equation, search space, dimension, and optimum value of each function. -dimensional functions have been used for extensive comparison of DE algorithm variants for various dimensions.


FunctionName of function (type)EquationSearch spaceOptima

Sphere model
(separable, multimodal)
0

Axis parallel hyperellipsoid
(separable, unimodal)
0

Schwefel’s problem 1.2
(nonseparable, unimodal)
0

Rosenbrock’s valley
(nonseparable, unimodal)
0

Rastrigin’s function
(Separable, Multimodal)
0

Griewank’s function
(nonseparable, multimodal)
0

Sum of different power
(nonseparable, multimodal)
0

Ackley’s path function
(nonseparable, multimodal)
0

Levy function
(separable, multimodal)
0

Zakharov function
(nonseparable, multimodal)
0

Schwefel’s problem 2.22
(nonseparable, unimodal)
0

Step function
(separable, unimodal)
0

De Jong’s function-4 (no noise)
(separable, unimodal)
0

Alpine function
(separable, multimodal)
0

Levy and Montalvo Problem
(separable, multimodal)