OPSODE: Opposition based particle swarm optimization instilled with differential evolution

Particle Swarm Optimization (PSO) is a very powerful global optimization technique. Differential Evolution (DE) is another fast and emerging algorithm of evolutionary computing. PSODE is hybrid of PSO and DE that incorporates diversity in the PSO algorithm. In this research a new opposition based version of PSODE (OPSODE) is proposed that incorporates some more diversity by employing the opposition based learning in the PSODE algorithm. Some standard benchmark functions are used to access the performance of the OPOSDE algorithm. The proposed version is then compared with the PSO, OPSO, and PSODE algorithm. The research result shows that the new version OPSODE has significance performance.


Introduction
*The theory of evolutionary algorithms is borrowed from the Darwin's theory of evolution that describes the survival of fittest through natural selection and the fitness improvement of individual species. The idea of successful survival in the evolutionary computing is inherited from reproduction and fitness of natural evolution process (Palit and Popovic, 2005). Evolutionary computing is an optimization paradigm based on mechanisms of evolution such as biological genetics and natural selection (Eberhart et al., 2001). The key objective of the optimization models is to get the best possible choice among the candidate solutions (Engelbrecht, 2005). Best choice refers to an acceptable or satisfactory solution that can be absolute best or any of the best solution over a set of candidate solutions. The advantage of evolutionary computing over other types of numerical methods is their ability to escape from local minima (Engelbrecht, 2005;Haupt and Haupt, 2004), however, EC algorithms do not guarantee to find the exact global optima (Engelbrecht, 2005;2007).
Particle Swarm Optimization (PSO) is one of the well-known population based stochastic evolutionary algorithm that is proposed by Kennedy and Eberhart (1995). PSO has been applied to solve a variety of optimization problems like other popular stochastic search techniques Genetic algorithms, Differential Evolution, Simulated Annealing etc. (Engelbrecht, 2005;2007;Price et al., 2005). PSO has shown itself to have a good performance in solving many test as well as real life optimization problems.
Differential Evolution (DE) is one of the famous population based stochastic evolutionary algorithm that is proposed by Storn and Price (1995). It is an easy, a powerful and one of the simple evolutionary algorithms that are widely used in global optimization. The small number of control parameters associated with DE algorithm makes it suitable for global optimization problems. The major advantage of DE algorithm over other evolutionary algorithms is that the diverse nature of control parameters and mutation strategies of DE algorithm increases the probability of finding optima for function optimization problems and other optimization problems (Wang et al., 2014;Yildiz, 2013;Ali et al., 2005;Engelbrecht, 2007;Storn and Price 1995;1997;Zamee et al., 2016). The varying nature of DE algorithm parameters enables it to escape from local optima problem (Pant et al., 2009;Wang et al., 2015;Li and Yin, 2016). The single objective as well as multi-objective versions of DE algorithm is successfully applied to many real life problems (Adeyemo et al., 2010). EC algorithms can be improved by enhancing their exploration search ability and incorporate some diversity in these algorithms that can improve the global optima finding ability.
In the rest of this paper, section 2 contains the background of PSO and DE, section 3 describes the proposed technique, benchmark functions used in this research are given in section 4, the results and convergence are discussed in section 5, section 6 discusses the conclusion of this research work.

Particle swarm optimization
PSO algorithm is inspired from behaviour of different species due to their social and cooperative nature (Engelbrecht, 2007). PSO consists of population members of solutions termed as particles. The particles in PSO are moved throughout the search space in search of optimal solution. The particles in PSO evolve their position by using their flying direction termed as velocity. The particles of PSO keep track of their position and velocity. Personal best is referred to as a best position so for a particle has reached and gbest maintains the record of best of best particle in the swarm. Every particle in PSO has two attributes namely, position and velocity. Position shows that position of any particle in the n-dimensional space and velocity shows the step size that the PSO particle will use to update its position. The velocity of any i th particle in Ddimensional space is represented by = ( 1 , 2 , … … … . , ). The direction of i th particle is calculated by using the following Eq. 1.
In D-dimensional search space the position of the i th particle is represented by = ( 1 , 2 , … … . . , ) for D dimensional space. The position of each particle is updated by using the following Eq. 2.
In Eqs 1 and 2 is the particle i personal best position, is the global best position, is the current position of a particle, c1 and c2 are social factors, r1 and r2 are uniform random numbers in the range(0, 1), inertia weight is denoted by w.

Differential evolution
DE is one of the well-known stochastic population based algorithm where n dimensional search space is used to randomly initialize the potential solutions. All potential solutions are equally likely to be selected as parent in DE algorithm. The candidate solutions evolve themselves by using a specified objective function to locate the optima by exploring the search space overtime (Yao et al., 1999). The amplified difference vector is added in random, best or current population member that depends on the mutation strategy being used (De Oliveira and Saramago, 2007).
To generate a mutation vector in DE algorithm, different vectors used in the mutation strategy are selected from the existing population. DE algorithm candidate solutions are represented as an Ndimensional vector of the population NP. The randomly initialized population members are supposed to be scattered over the entire search space. , represents i th population member where = 1,2,3, … , at G th generation in DE algorithm. The population of DE algorithm is evolved with the help of selection, mutation and crossover operators.
Here = 1,2,3, … … . is dimension and [1, ] will return an integer in the given range. After creating a trial vectors , +1 its fitness value ( , +1 ) is evaluated using given fitness function that is then compared with the fitness value of the target vector ( , ) and the vector having best fitness is moved to the next generation of DE algorithm. The selection is done based on the greedy approach by using Eq. 5. , +1 = { , +1 ( ( , +1 ) < ( , )) , ℎ Fitness is calculated against a given fitness function.

Related works in hybrid PSO and DE
Number of researchers has done their research work on the hybrid of PSO algorithm and DE algorithm. The detail of the research work on the hybrid of PSO and DE on various problems and applications is discussed in this section. Research work done by other researcher's shows the importance of the research work about the hybrid of PSO and DE algorithms. Abdullah et al. (2011) have discussed the limitation of trapping into local optima problem. They have incorporated DE operators in PSO algorithm to improve the local best capability of the PSO algorithm in solving complex optimization problems. The numerical analysis on the benchmark functions shows that the proposed technique PSODE outperforms both PSO and DE algorithms. Social and cognitive experience evolution in PSO using DE algorithm is considered in a research work of (Epitropakis et al., 2011). They have used DE in PSO algorithm to enhance the convergence of PSO and to efficiently guide the evolution process in PSO. The hybrid variant shows the promising results over a test suit of multimodal benchmark functions. Fu et al. To prior crossover is used to incorporate some diversity in the population that increases the probability to reach to the global optimal point. Research results are generated using a suit of 5 benchmark functions that shows that prior crossover based PSODE outperforms PSO and DE algorithms.

Proposed version
The proposed algorithm is the infusion of the opposition based particle swarm optimization and differential evolution algorithm (OPSODE) shown in Fig. 1. Different researchers work on the hybrid of PSO and DE that is discussed in section 2 of this paper.
In this research an effort is made to improve the performance of the hybrid of PSO and DE algorithm by integrating the concept of the opposition based population initialization scheme. This opposition based scheme in the hybrid of PSO and DE will incorporate the diversity in the population to find the optimum value over the continuous range of search problems. Tizhoosh (2005) has introduced the oppositionbased learning (OBL) scheme in the machine learning. Opposition-based learning scheme uses the concept of opposite points and opposite numbers. OBL generates the opposite number for any number over the specified interval [ , ] by using the following Eq. 6. = + − (6)

Test functions
A test suite of 15 benchmark test functions is used to in this research work. All test functions are the common test functions picked up from the literature (Ali et al., 2005;Brest et al., 2006;Xu and Li, 2007;Ali et al., 2009aAli et al., , 2009bAli et al., , 2009cAbbas et al., 2015). These test functions are given in Table 1 along with the other details like the name of the function, search space and the optimum value.

Experimental settings, results and convergence graphs
The parameters used by the PSO and DE algorithm to obtain the results are given as: the value of the r1 and r2 is taken from (0, 1) and c1=c2=1.49 (Poli et al., 2007).
The range (0.4, 0.9) is used for the value of inertia weight (Poli et al., 2007); Population size = is used as a size of population, The small as well as big dimensions are used during the experimentation such as 2, 4, 10, 20 and 30; Crossover rate and mutation probability have values CR=0.9 and F= 0.5 respectively (Brest et al., 2008;Rahnamayan et al., 2008).
Average fitness experimental results are obtained by taking100 runs and 2000 iterations of DE algorithm. The results obtained against this parameter setting are reported in Table 2.
The convergence graphs PSO, OPSO, PSODE and OPSODE are generated that are shown in Fig. 2. The graphs in Fig. 2 are generated for some selected functions and selected dimensions.
To assess the convergence performance of proposed algorithm, the graph showing the convergence is plotted in Fig. 2     The x-axis is the number of iterations and the yaxis shows the performance (average fitness value). Research results are generated using a test suit of 15 benchmark functions reported in Table 1. The average fitness value and the standard deviation of test suit of functions are reported in Table 2 of this paper. The results are generated using different dimensions 2D, 4D, 10D, 20D and 30D to measure the performance of proposed version with the existing ones. The functions f11, f12 are the 4 dimensional functions; functions f13-f15 are 2 dimensional functions and remaining are ndimensional functions. The results are generated for a suit of test functions given in Table 1. The values that are reported as bold faces in Table 2 are the best values vales among the comparing algorithms. The research result shows that the proposed OPSODE has significant performance.

Conclusion
In this research a new variation of the PSODE is proposed and implemented on a test suit of benchmark functions. Diversity enhances the global search capability of heuristic algorithms. Diversity is incorporated by utilizing the concept of opposition based concept.
The algorithms are run 100 times with fix number of iterations and then average fitness is reported in the result section. Most of the functions are n-dimensional functions, however, some small dimensional functions (f13, f15) with 2 dimensions and (f11, f12) with 4 dimensions are used to generate the experimental results. Results are generated with parameter setting given in the result section for various dimensions (2, 4, 10, 20, 30) for the functions in Table 1.
The research results are reported in Table 2 showing better results as bold faces. It is obvious from the experimental results that the overall performance of OPSODE is better than the performance of PSO, OPSO, PSODE and OPSODE. in a two-area conventional and renewable energy based nonlinear power system. In the 4 th International Conference on the Development in the in Renewable Energy Technology, IEEE, Dhaka, Bangladesh: 1-6. https://doi.org/10.1109/ ICDRET.2016.7421476.