Improvement of Diferential Evolution with Multipopulation-based Ensemble of Mutation Strategies

Abstract

Due to its advantages, DE has been one of the most used evolutionary algorithms (EA) for solving complex optimization problems. Many DE variants have been proposed to enhance its performance. Diferential evolution with a multipopulation-based ensemble of mutation strategies (MPEDE) has been considered as one of the most eficient DE variants. Mutation strategies used in MPEDE are DE/rand/1, current-to-rand/1, current-to-pbest /1. An ameliorated multipopulation-based ensemble DE (AMPEDE) is proposed in the present work where we try to improve the performance of MPEDE by utilizing a new ensemble of mutation strategies and presenting a new mutation strategy called current-to-mpbest /1. In this strategy, we are interested in escaping from the local optimum, where individuals are attracted to the center of gravity of the pbest solutions. In our experiments, a comparison of AMPEDE with MPEDE and other algorithms on Black Box Optimization Benchmarking (BBOB) tool has been achieved and the results show that AMPEDE is very competitive and provides an excellent performance in dealing with some optimization problems.

Keywords

• eol>Evolutionary algorithms
• Diferential evolution
• Multipopulation
• Ensemble of mutation strategies
• Numerical optimization

1. Introduction

Diferential evolution (DE) has been introduced initially by Storm and Price [ 1 ] to find the global optimum of non-linear, non-convex, multi-modal and non-diferentiable functions defined in the continuous parameter space [ 2 ]. It is considered as one of the most popular evolutionary algorithms due to its advantages (e.g., simplicity, few control parameters, eficiency in dealing with complex optimization problems, etc.) [ 3, 4 ].

DE is a population-based stochastic optimization algorithm, which moves the population toward the global optimum by using mutation, crossover, and selection operations at each generation [ 5, 6 ]. Its performance depends on the configuration of mutation strategy and control parameters, such as the population size NP, the scaling factor F and the crossover rate Cr [ 7 ].

Many optimization problems exist in the real world [ 4 ]. For a given optimization problem, during the search process the most suitable control parameters and the proper mutation strategy may not be the same [ 4 ]. Some mutation strategies are applied to improve exploitation while others are used to enhance exploration ability [ 8 ]. It is noteworthy that some strategies can achieve a trade-of between exploration and exploitation [ 9 ]. Concerning control parameter settings, some can speed up the convergence [ 9 ] while others are efective for separable functions [ 10 ].

In classical DE, only a single mutation strategy is used and the control parameters are fixed, therefore the performance of DE may vary greatly for diferent optimization problems [ 4 ]. To address this issue, researchers have been more attracted to automatically tune parameters and construct excellent ensemble of mutation strategies [ 4 ]. Consequently, many improved DE variants have been proposed such as SaDE [ 11 ] (ensemble of two mutation strategies and self-adapted parameters), JADE [ 12 ] (to develop an excellent mutation strategy and self-adapted parameters), jDE [ 13 ] (self-adaptive tuning parameters), CODE [ 14 ] (composition of mutation strategies with their own parameters), MPEDE [ 7 ] (ensemble of multiple mutation strategies), MMRDE [ 4 ] ( multiple mutation strategies based on roulette wheel selection).

In our work, we are interested in MPEDE [ 4 ], which is a new variant of DE with a multipopulationbased ensemble of mutation strategies. In MPEDE, the whole population is divided into four sub-populations, three of them have the same size and they are called indicator sub-populations and the fourth one has a diferent size, it is called reward sub-population. Three mutation strategies are used in MPEDE (DE/rand/1, DE/current-to-rand/1, and DE/current-to-pbest/1 with an archive). Each one of them is afected randomly to an indicator sub-population. At each generation, the sub-populations are randomly sampled from the entire population. The reward sub-population is afected to the best mutation strategy after a certain number of generations.

Nevertheless, to the best of our knowledge, MPEDE sufers from the following limitations: 1) the waste of calculation time and the non-stability of the population due to resampling of sup-populations at each generation, 2) the lack of exploitation with the mutation strategies regrouping and 3) the risk of premature convergence to a local optimum due to the attraction to the best current solution or one among the group containing the pbest solutions. To overcome these limitations, we propose a new ensemble of mutation strategies (DE/rand/1, DE/target-tobest /1, DE/current-to-mpbest /1 without archive) providing a better balance between exploration and exploitation. To reduce the waste of time and the non-stability of the population , the sub-populations were resampled after a certain number of generations instead of doing it at each generation. To reduce the risk of stagnation at a local minimum, we propose replacing current-to-pbest /1 by a new mutation strategy: current-to-mpbest /1. In this strategy, individuals are attracted to the center of gravity of the pbest solutions instead of being attracted to one of them only.

The paper is organized as follows. In section 2, we explain the basic diferential evolution. Section 3 is devoted to research findings in relation with the present work. The proposed approach is described in section 4. Section 5 presents the experiment design and discusses the obtained results in comparison to those of some state-of-the-art algorithms. Finally, section 6 provides a conclusion and some future research directions. Diferential evolution (DE) belongs to the evolutionary algorithms (EA) family [ 3 ]. It provides eficiently outstanding solutions for complex numerical optimization problems [ 3 ]. In DE, the evolutionary process consists of four basic steps: initialization, mutation, crossover, and selection [ 2 ]. Excluding the initialisation, the remaining steps are repeated until the satisfaction of a given stop criteria, for instance reaching the maximum number of function evaluations [ 2 ]. In the remainder of this section, we will present more details about the aforementioned steps.

2.1. Initialization

The initial population in DE is randomly generated of -dimensional real-valued decision vectors [ 2 ]. Where is the population size and is the dimensionality of problem. A decision vector represents an individual. Each individual is composed of decision variables. Each one is defined randomly and uniformly in the range [ ,]. So the initial population can be expressed as shown in equation 1:

= + (0, 1) * ( − )

Where (0, 1) is a uniformly distributed random number between 0 and 1 [ 7 ]. 2. Basic Diferential Evolution (DE) 2.2. Mutation "DE/rand/1" [ 16 ] : "DE/rand/2" [ 17 ] : "DE/best/1" [ 16 ] : "DE/best/2" [ 16 ] : 1–14 (1) (2) (3) (4) (5) (6) V, = X, + .(X1 , −

X2 ,) + .(X3 , −

X4 ,)

Where the indices 1, 2, 3, 4 and 5 are randomly chosen within the range [1, ]. These indices are diferent from [ 2 ]. is a positive integer chosen randomly within the range [ 0,1 ]. It is used to scale the diference vectors. X is the best vector according to its fitness value in generation (current generation). After the initialization step, a donor/mutant vector is created at each generation by using one of the mutation strategies for each parent vector . These strategies include the following: "DE/current-to-rand/1" [ 15 ] :

U, = X, + .(X1 , −

X,) + .(X2 , −

X3 ,) V, = X1 , + .(X2 , −

X3 ,) V, = X1 , + .(X2 , −

X3 ,) + .(X4 , −

X5 ,) V, = X, + .(X1 , −

X2 ,)

2.3. Crossover

After mutation, the crossover step is executed to enhance diversity by generating new trial vectors [ 4 ] from the parent vector X,, and its corresponding mutant vector V,. Two types of crossover operation exist: the binomial crossover and the exponential crossover [ 2 ]. The binomial crossover is generally used in DE and it is defined as follows:

U, = {︃V

, if ( [ 0, 1 ] ≤ ) or ( = ), = 1, 2, ..., }︃

X, ℎ Where is a control parameter called crossover rate. = 1, 2, ..., , = 1, 2, ..., where is the population size and is the dimensionality of problem. = is used to ensure that U, is diferent from the target vector X, at least in one individual.

2.4. Selection

According to the fitness value, the selection step chooses between the parent vector and the trial vector the one which will survive in the next generation. It is defined as shown in equation 8 X,+1 = ︂{ U, if (U, ≤

X,

X,)︂}

3. Related Works

DE can be considered as one of the best evolutionary algorithms (EA) [ 4 ], mainly because of its simplicity and robustness [ 7 ]. Nevertheless, despite these advantages, it sufers from stagnation, premature convergence, and long calculation time [ 4 ]. The performance of DE depends on the used mutation strategies and control parameters [ 18 ] and choosing suitable strategies and parameters is far from being an easy task. It requires much expertise [ 19, 20 ].

Many studies have been proposed to determine the appropriate setting of the control parameters of DE (e.g., size , Crossover , Scale factor , etc.) based on the properties of the problem [ 7 ]. In DE, three main control parameters are used: , and . (population size) has a big efect on the convergence speed of DE. For separable and uni-modal functions, we have to choose a smaller value of to speed up the convergence [ 19 ]. While in multimodal function a larger value of is recommended to avoid premature convergence. is a positive control parameter used to scale the diference between vectors. We have to choose carefully the initial value of . In [ 21 ], 0.9 is claimed to be the best value. Therefore, typical values of are generated randomly in the range [0.4, 0.95] [ 21 ]. is called crossover rate (crossover probability). It is used to control the ratio of genes from the mutant vectors which will participate in the crossover step. A good choice of is within the range [0.3, 0.9], while a good initial value is 0.1 [ 22 ].

As we have seen in the aforementioned studies, many conclusions were proposed for the manual tuning of control parameters. Nevertheless, during the evolution process of DE, it is worthy to adapt the parameters. In fact, the control parameters can be partitioned into 1–14 (7) (8) three categories: deterministic, adaptive, and self-adaptive [ 23 ]. In adaptive parameter, a fuzzy adaptive diferential evolution (FADE) has been introduced by Liu and Lampinen where and are dynamically adapted [ 24 ]. For multiobjective optimization, Zaharie and Petcu have designed an adaptive Pareto DE algorithm and analyzed its parallel implementation [ 25 ]. Concerning self-adaptive parameters, Omran et al. proposed (SDE), where a normal distributed N(0.5,0.15) is used to generate the for each individual [ 26 ]. Brest et al. [ 27 ] proposed the jDE which considered as new variant adaptive of DE. and are adaptive during the execution of DE.

The used mutation strategy has a big influence on the performance of DE [ 4 ]. Thus, to enhance DE performance, some works are interested in introducing a new mutation strategy [ 28 ]. For instance, a new variant of DE/current-to-best, called DE/current-to-pbest, with an optional external archive is proposed by Zhang et al [ 12 ], where ∈ (0, 1] and any of the top 100.% individuals can be chosen to be the best. DE/current-to-pbest ofers a good balance between exploration and exploitation [ 7 ]. DE/current-to-gr-best/1 [ 29 ] is also a variant of of DE/current-to-best/1 in which the best solution is chosen from a group that contains % of the population (randomly chosen).

Instead of spearing the control parameter tuning and the choice of the suitable mutation strategy, researchers proposed to incorporate ensemble strategies and parameters into evolutionary algorithms [ 7 ]. Qin et al. [ 17 ] proposed a self-adaptive DE algorithm (SaDE) in which mutant vector generation strategies and their own control parameters are self adapted based on their previous experiences in generating promising solutions. A hybrid mutation strategy have been proposed by Yi et al. [ 30 ] as a new variant of DE. It uses two types of mutation strategies and self-adapting control parameters. To solve constrained optimization problems, Wang and co-workers [ 31 ] proposed ICDE which uses multiple mutation strategies and the binomial crossover to generate the trial vectors.

4. The Proposed Approach

4.1. Multipopulation-based ensemble DE (MPEDE)

MPEDE [ 7 ] is a variant of DE based on multipopulation. It uses multiple strategies and partitions the whole population into four sub-populations. Three mutation strategies are used in MPEDE DE/rand/1, DE/current-to-rand/1, DE/current-to-pbest with archive. Besides that, there are three equal and small-sized sub-populations (indicator sub-populations) and a largesized sub-population (reward sub-population). After a given number of generations, the best performing mutation strategy is allocated to the reward sub-population and it can win more computational resources. Let determine the entire population and represent the jth sub-population. we can define as:

Let be the size of . The size of is and it is calculated as follows: =

⋃︁ 1–14 (11) (12) ∑︁ = 1 is the portion between and , and 1 = 2 = 3.

As we have seen before, after a certain number of generations the best performing mutation strategy is rewarded by more computational resource (4). The performance of each mutation strategy is calculated by the proportion of fitness improvements ( Δ ) and consumed function evaluations during the last generations(. ). The index of the best mutation strategy can be expressed as shown in equation 12 :

︂( = 1<⩽3 ︂(

Δ )︂ .

4.2. The Limitations of MPEDE and the Adopted Ensemble of Mutation

Strategies

As described above, in MPEDE sub-populations are randomly resampled from the entire population at each generation, This can lead to a loss of computation time. Three mutation strategies are used in MPEDE. First DE/rand/1 which is considered as the most commonly used mutation strategy. It has a big exploration capacity [ 11 ]. Second current-to-rand/1, it is very efective in solving multiobjective optimization problems. It is considered as a rotation-invariant strategy [ 7 ]. Third current-to-pbest /1 which is very useful in solving complex optimization problems . With this strategy, problems like premature convergence can be solved due to its ability to diversify the population [ 12 ]. Current-to-rand/1 is used without crossover while DE/rand/1 and current-to-pbest are used with the combination of binomial crossover [ 7 ]. We find there is a lack of exploitation in this ensemble of mutation strategies.

In this paper, we propose an improved variant of MPEDE (AMPEDE) in which we try to overcome the limitations that we find, and to enhance MPEDE results. In AMPEDE we divided the entire population into three large equally-sized sub-populations (indicator sub-population) and one small-sized sub-population (reward sub-population). Three mutation strategies are used in our approach. DE/rand/1 to ensure a good exploration ability [ 11 ]. DE/target-to-best/1 [ 15 ] to promote exploitation in our ensemble of mutation strategies. Current-to-mpbest/1 without archive is used not to be trapped into a local optimum. In this strategy instead of choosing X randomly one of the top 100.% individuals in the current population with ∈ (0, 1], we take X from the mean of the top 100.% individuals in the current population with ∈ (0, 1]. The three mutation strategies are used with the binomial crossover. As in MPEDE [ 7 ], after every certain number of generations, the best mutation strategy performing is determined using the ratios of fitness improvements and consumed function evaluations during the previous generations. According to our experimental results, as illustrated in the next section, we find that it is not useful to resample sub-populations from the entire population at each generation. Therefore, in our approach we choose resampling sub-populations after every generations to enhance the stability and the adaptability of the algorithm.

4.3. Parameter Adaptation

Because parameters during the evolution process of DE cannot be the same, we need to choose the appropriate control parameters for diferent mutation strategies to enhance DE performance [ 8 ]. Studies like [ 22, 11, 12 ] have proposed diferent approaches to parameter adaptation. In our work, we used the technique defined in [ 12 ], which is the same as that used in MPEDE. A Cauchy distribution is used to generate the scaling factor , of individual X that uses the th mutation strategy. The scaling factor formula is shown in equation 13

, = , ( , 0.1) Where in this Cauchy distribution, represents the location parameters, and the scale parameter is 0.1. After the update, if , is greater than 1, , will be truncated to 1, and it will be regenerated to a feasible value if it is smaller than 0. The initial value of is set to 0.5 and it is updated as follows:

= (1 − ) . + . (, ) Where , is the set of , used by the th mutation strategy and assists this strategy to generate better solutions at generation g. is the Lehmer mean; it is calculated as below: = ∑︀ ∈ 2 ∑︀ ∈ The crossover probability , of individual X uses the mutation strategy is generated according to a normal distribution. The crossover probability equation is defined as follows:

Where in this normal distribution represents the mean value and 0.1 is the standard deviation value. Initial value of is 0.5 and it is updated after each generation as: = (1 − ) . + . (, )

Where , is the set of , used by the th mutation strategy and assists this strategy to generate better solutions at generation . is a positive constant within the range [ 0,1 ], and is the arithmetic mean value of elements in the collection , . The schema of AMPEDE is given in Algorithm1.

5. Experiment and Result Analysis

Our proposed algorithm is tested on the Comparing Continuous Optimizer [ 32 ] (COCO) platform. This platform is used for the BBOB workshops. It uses 24 noiseless test functions for testing real parameters optimization problems with single objective functions. According to their features, the functions are partitioned into 5 sub-groups. The BBOB functions are shown in . Algorithm 1: pseudo code of AMPEDE 1 Set = 1.0, = 0.5, Δ = 0 and Δ = 0 for each j = 1, ..., 4; 2 Initialize, NP, ng, for each j = 1,..., 4; 3 Initialize, the pop randomly distributed in the solution space; 4 Initial and set, = .NP; 5 Randomly partition pop into 1, 2, 3 and 4 with respect to their sizes.; 6 Randomly select a sub-population ( j = 1, 2, 3) and combine with 4. Let = ∪ 4 and = + 4 ;

Calculate and ; Calculate , and , for each individual in ;

Perform the jth mutation strategy and related crossover operators over 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 7 Set g = 0; 8 while g → MaxG do g = g + 1; for j=1 →3 do

subpopulation ;

Set , = ∅ , = ∅; for i=1 →NP do if (,) ≤ (,) then

+1, = ,; else +1,= ,; , → ,; , → ,; pop=⋃︀=1...3 ; if mod(g,ng)==0 then k= ︁( 1<≤ 3 ︁(

Δ ︁) .

; Δ=0; Randomly partition pop into 1, 2, 3 and 4;

Let = ∪ 4 and = + 4;

5.1. Experiment Design

In the experiments, we compared our proposed approach (AMPEDE) to DE-PSO, GA, JADE, CMAES, and MPEDE. Each of the algorithms was run on 15 instances of all the 24 functions in dimensions 5, 10, 20. The evaluation budget was set to 104. function evaluations for each run. Feasible solutions during the execution are within [ -5, 5 ]. Running algorithms is continued until a stop criterion is satisfied: reaching the maximum number of function evaluations or getting a solution close to the best known solution of the problem with a precision greater than 10− 8 . Many variants with diferent parameters values are tested in our experiments. The parameters values used by the best variant are: population size = 150, generation gap =30 which is f1: Sphere function f2: Ellipsoidal Function 1 Separable Functions f3: Rastrigin Function f4: Buche-Rastrigin Function f5: Linear Slope f6: Attractive Sector Function 2 Low or moderate conditioning ff78:: SRtoespenEbllriopcskoiFdualnFctuinonctioorniginal f9: Rosenbrock Function rotated f10: llipsoidal Function f11: Discus Function 3 Unimodal with high conditioning f12: Bent Cigar Function f13: Sharp Bridge Function f14: Diferent Power Function f15: Rastrigin Function f16: Weierstrass Function 4 Adequate global structure with Multi-modal f17: Schafers F7 function f18: Schafers F7 Functions moderately ill-conditioned f19: Composite Griewank-RosenbrockFunction F8F2 f20: Schwefel Function f21: Gallagher’s Guassian 101-me PeaksFunction 5 Multi-modal function with weakglobal structure f22: Gallagher’s Guassian 21-hi PeaksFunction f23: Katsuura Function f24: Lunacek bi-Rastrigin Function used to specify the best mutation strategy periodically, proportion between indicator population and entire population 1 (as 1= 2= 3)=0.3, initial value of =1.0 and =0.5. The value of p in the “Current-to-mpbest/1” is 0.1.

5.2. Result Analysis

Our experimental results are presented in Figure1 and Figure2. Despite the global superiority of CMA-ES and JADE, results obtained show that AMPEDE has a good performance in separable functions, functions with low or moderate conditioning, functions with high conditioning and unimodal, especially f19 where AMPEDE shows better convergence rate and outperformed all algorithms like GA, JADE, CMAES, MPEDE in 5D. According to the results presented in Figure 1 and Figure2, the changes we made in MPEDE were very useful. As we see AMPEDE has an excellent performance compared to MPEDE in all the benchmark function in 5D and 10D. Despite that AMPEDE sufers in dealing with Multi-modal functions with adequate global structure like f15,f16,f19 in 10D and the Multi-modal functions with weak global structure like f23 and f24.

6. Conclusion

In this paper, we have presented a new DE variant, named amelioration multipopulation ensemble DE (AMPEDE). AMPEDE is an improvement of MPEDE where we have introduced a new ensemble of mutation strategies instead of the grouping of mutation strategies in MPEDE and proposed a new mutation strategy.

Mutation strategies used in AMPEDE are first DE/rand/1 , second target-to-rand/1 , third current-to-mpbest which is a new mutation strategy that we have proposed. In current-tompbest individuals are attracted to the center of gravity of the pbest solutions instead of being attracted to the a solution chosen randomly from the pbest solutions. Based on the results of our experiments, current-to-mpbest outperforms current-to-pbest.

IMPEDE is compared to MPEDE and others algorithms on BBOB. The experimental results show that AMPEDE provides an obvious performance improvement in comparison to the original MPEDE especially in 2D, 3D, 5D and 20D.

In our future work, we plan to add an adaptive strategy to AMPEDE to enhance its performance and to solve more optimization problems. In addition, we intend to combine AMPEDE with other existing metaheuristics to improve our results.

Acknowledgments

This work was partially supported by the LABEX-TA project MeFoGL:"Méhodes Formelles pour le Génie Logiciel".

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