New adaption based mutation operator on differential evolution algorithm

Abstract

Among meta-heuristics algorithms, differential evolution (DE) is powerful nature-inspired algorithm used to solve the nonlinear problems. However, at higher generations there is an increase in the computational cost. In this paper, a new approach has been proposed from a new adaption based mutation operator in which the variations of a particular element is kept constant. In the same way, to keep an element constant called as “diversity” in DE, new adaption based mutation operator has been incorporated. The proposed variants are proliferated new adaption based operator with one more vector, named as new adaption based mutation operator in the existing mutation vector to provide more diversity for selecting effective mutant solutions. The proposed approach provides more promising solutions to explorer the evolution and helps DE evade the circumstance of stagnation. Comparisons with other DE variants such as CPI-DE, TSDE, ToPDE, MPEDE and JADEcr establishes that the proposed new adaption based mutation operator is able to improve the performance of DE.

Keywords

Adaptation optimization evolutionary algorithm

1. Introduction

Global optimization problem solving techniques is suitable for every complex problem which provide the best test solution in terms of true minimum optimum, fast convergence rate, and minimum number of control parameters, which are easy to use for real applications. Storn and Price introduced the “Differential Evolution” algorithm in 1995, which is a population based stochastic search algorithm for constraint and unconstrained continuous optimization problems. Further, DE is a preferred optimization technique due to following reasons [1, 16]:

1.
DE is easy to implement for different applications which suffer from certain bottleneck problems that its come across while solving engineering as statistics.
2.
DE has only three parameters to i.e. F (Mutation), Cr (Crossover) and NP (Population Size) to control the algorithm and to be controlled for improving the performance of DE.
3.
DE has the ability to reduce the computation cost.

The basic differential evolution algorithm [16] is described in algorithm 1. DE process goes through four steps i.e., initialization, mutation, crossover and selection.

Basic Differential Evolution [1] DEA

Population initialization

Evaluate the fitness value iteration $=$ 1 while $\textit{FEs}<\textit{MaxFEs}$ Apply donor vector Apply trial vector Apply selection Strategy iteration $=$ iteration $+$ 1 end while

Therefore to prevent solutions on converging towards local optimal solutions and to improve the convergence rate of the algorithm, a new variant of DE named new adaption based mutation operator on differential evolution (NABDE) is proposed. In new adaption based mutation strategy is used. New adaption based mutation operator is to maintain the diversity (candidate solution) of the search space with respect to the variation in different environment. The purpose of this operator is to improve the convergence rate of the algorithm and to maintain the same diversity which has created during initial generations by multiplying new adaption based vectors (NABVs) in vectors on the select better candidate solution in the search space. This strategy has used in the mutation operator of the differential evolution. To check the performance of proposed new adaption based mutation operator variants, these algorithms are compared with other standard algorithms. Performance analysis shows that this variants better performs other variants of DE algorithm.

The rest of this paper is organized as follows, related work is given Section 2, in Section 3 proposed approach using new adaption based mutation operator is explained, Section 4 describes experimental result and analysis with comparing variants of DE, and finally in Section 5 conclusion and future work of this paper are described.
2. Related works

In 1995, Storn and Price [1, 16, 20], proposed Differential evolution performs well in “standard benchmark test functions and real-world optimization problems” [14] such as multi-objective, non-convex, non-differential, non-linear constraints and dynamic components [1, 14, 28]. DE faced the following main problems:

1.
Mutation strategies and control parameters need to be decided well before the application of DE in problem solving.
2.
Every optimization problem need a particular choice of mutation factor and control parameters [2, 9, 14, 19].

For enhancing the performance and application of DE, many variants of DE for continuous, Single-objective and Multi-objective optimization problems and with different choice rules have been developed which are briefly described below.

In 1999, Price [10] proposed DE/current-to-rand/us-ing arithmetic recombination, which replaced binomial crossover strategy with rotationally invariant arithmetic line recombination operator that is used to generate the trial vector using linear combination of the target vector and its corresponding donor vector. In 2003, Fan and Lampienen [5], proposed a variant of DE (i.e. DE/rand/1) using trigonometric mutation operator with a probability of Mt $=$ 0.05 and the mutation strategy with a probability of (1-Mt). Noman and Iba [12], have proposed “a crossover based adaptive local search operation for enhancing the performance of DE”. To reduce the computational time and improve the performance of DE, Tasoulis et al. [3] proposed its parallelization in a virtual parallel environment, in 2004. The proposed model maps a total subpopulation to a processor, allowing different subpopulations to create independently towards a solution. In 2005, Omran et al. [11] introduced SDE with a scaling factor parameter was generated by self-adaptive parameter. Islam et al. [13] proposed a modified DE with p-best crossover $\textit{MDE}\_p\textit{BX}$ , which is the implementation and helps in alleviating the tendency of premature convergence and/or stagnation”. Gong et al. [15] observed that in DE, “the trial vector is directly related to its binary string, but not directly related to the crossover rate” and proposed a crossover rate repair technique for the adaptive DE algorithms. In 2014, Zhou et al. [17], proposed Role Assignment (RA) strategy to dynamically select mutation strategies and parameter settings for different individuals of the population. To obtain self-adaptive control parameter settings, new version of DE proposed by Brest et al. [8] in 2006. The idea “Evolution of the evolution” has been used to find the values of control parameters F and CR. The proposed algorithm has been shown “to be better than or at least comparable to” the DE algorithms. In 2015, Brown et al. [21], proposed a new mutation operator ‘current-by-rand-to-best’ for unconstrained continuous optimization problems. In 2016, Hu et al. [22], suggested that DE cannot guarantee global convergence of multimodal functions in the search space. In 2016, Wang et al. in the work [23], empirically reviewed the selection solution for mutation in DE. In 2016, Wang et al. [24], introduced a cumulative population distribution information based DE framework called CPI-DE. In 2017, Singh and Kumar [29], proposed a HMBDE is inspired from the process of homeostasis, whose primary focus is to maintain the internal environment of the body in spite of the changing external factors. In 2018, Singh and Kumar [31], proposed a $\textit{MODE}-\textit{HBM}\_\textit{SCE}$ is multiobjective problems solved bi-objective and tri-objective. This algorithm is applied to solve software cost estimation problems. In 2014, Wang et al. [25], proposed a covariance matrix learning based differential evolution algorithm. This algorithm is balancing the exploration and exploitation abilities of DE. In 2017, Singh and Kumar [32], proposed a $\textit{DEA}\_\textit{PBHDE}$ is 24 single objective problems solved unimodal and multi-modal functions. This algorithm proposed independent difference random vectors help in avoiding stagnation problem of multi-modal functions.
3. Proposed approach: New adaption based mutation operator on differential evolution algorithm (NABDE)

Sometimes, DE suffers from the problem of stagnation in local optima in the search space. At higher generations, it has been seen that the solutions provided by DE is stagnated. The new propose variant that is created uses mutation operator which gets updated as algorithm progress over time. The factor which is responsible for diversity among solution is the one which needs to be changed. Diversity is an important factor in determining the possible solutions from a search space. It cannot assume about an obtained solution for its correctness unless and until whole search space is correctly explored initially or in the latter stages of the evaluation. The major responsibility of diversity maintenance in DE goes to mutation operator to which proposed approach have given attention. A new approach is proposed named as new adaption based mutation operator to resolve the issue related to loss of diversity in mutation operator. The operator new adaption is applied to the vectors (whole search space) to maintain the diversity among solutions. These vectors (i.e. current populations) is multiplied by new adaption based vectors (NABVs) in the search space which helps in gaining the diversity as well as the convergence rate is improved. This process is very similar to adaption phenomena as it keeps the maintain the diversity of DE algorithm constant.

Similar to these phenomena, the process of adaptive learning has been used to create the proposed optimization algorithm. NABDE, in best of our knowledge, is the process of adaptive learning which has been applied to design optimization algorithm. The theory of adaptive learning was given in [26, 30].

3.1 New adaption based mutation operator (NABMO)

NABMO process is defined as the phenomena which maintains the population (internal environment) of the search space by multiplying different parameter values depending on the nature of the problem and available counterbalancing resources.

NABMO: NABMO maintain the diversity and convergence rate when it is stuck in local optimum problem, then sufficient requirement diversity is achieved with taking new adaption based mutation from new adaption based vectors (NABVs), this vector is chosen according to best search space to improve the diversity and convergence rate of vectors in the search space.

New adaption based vectors (NABVs): Similar to above process, it has created new adaption based vector (NABVs) and multiplied each vector as mutation strategy. This is done to maintain the diversity provided during initial generations of until the end. In some strategies, these vectors are multiplied to provide new areas for exploration while for others these vectors may be used for exploitation. These improved mutation strategies are named by their basic mutation strategies. This is the common procedure for creating new adaption based mutation variant of DE.

The purpose of this NABVs is to improve the convergence rate of the algorithm and to maintain the same diversity which has created during initial generations by new adaption based mutation operator in vectors on the select better candidate solution in the best search space. This process has used in the new adaption based vector1 (NABV1) and new adaption based vector2 (NABV2). This strategy has used in the mutation operator of the differential evolution. This process $\textit{New\_Vector}1$ and $\textit{New\_Vector}2$ explain here:

$\displaystyle\textit{New\_Vector}1=\sum\nolimits_{i=1}^{\textit{NP}}\vec{% \alpha}_{r_{1}^{i},G}*\textit{NABV}1$ (1) $\displaystyle\textit{New\_Vector}2=\sum\nolimits_{i=1}^{\textit{NP}}\vec{% \alpha}_{r_{2}^{i},G}*\textit{NABV}2$ (2)

where NABV1 and NABV2 are new adaption based vectors depending upon the nature of better solution in respect to best search space. $i=1,2,3\ldots\textit{NP}$ (i.e., the current population according to whole search space in respect to best choose the population) and $G$ denotes generation. $\vec{\alpha}_{r_{1}^{i},G}$ and $\vec{\alpha}_{r_{2}^{i},G}$ will be generated from the entire search space.

NABDE-R [1] Set initial values of parameters as $\delta_{1}$ $=$ 0.56 $C r$ $=$ 0.24; PopSize $=$ 100* $D$ Initialize all individual of population randomly iteration = 1 while $\textit{FEs}<\textit{ MaxFEs}$ Apply new adaption based mutation operator and generate donor vector using equation number 4

Apply trail vector Apply selection operator iteration $=$ iteration $+$ 1 end while

NABDE-B [1] Set initial values of parameters as $\delta_{1}$ $=$ 0.55 $C r$ $=$ 0.34; PopSize $=$ 100* $D$ Initialize all individual of population randomly iteration = 1 while $\textit{FEs}<\textit{MaxFEs}$ Apply new adaption based mutation operator and generate donor vector:

$\vec{\gamma}_{i,G}=\vec{\alpha}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},G}+% \delta_{1}\cdot(\sum_{i=1}^{NP}\vec{\alpha}_{r_{1}^{i},G}*\textit{NABV}1-\sum_% {i=1}^{NP}\vec{\alpha}_{r_{2}^{i},G}*\textit{NABV}2)$

Apply trail vector Apply selection operator iteration $=$ iteration $+$ 1 end while

NABDE-CB [1] Set initial values of parameters as $\delta_{1}$ $=$ 0.33 $\delta_{2}$ $=$ 0.12 $C r$ = 0.2; PopSize = 100* $D$ Initialize all individual of population randomly iteration $=$ 1 while $\textit{FEs}<\textit{MaxFEs}$ Apply new adaption based mutation operator and generate donor vector:

$\vec{\gamma}_{i,G}=\mathrm{\alpha}_{i,G}+\delta_{1}\cdot(\mathrm{\alpha}_{% \mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},G}-\sum_{i=1}^{NP}\mathrm{\alpha}_{i,% G})+\delta_{2}\cdot(\mathrm{\sum_{i=1}^{NP}\alpha}_{r_{2}^{i},G}*\textit{NABV}% 1-\mathrm{\sum_{i=1}^{NP}\alpha}_{r_{3}^{i},G}*\textit{NABV}2)$

Apply trail vector Apply selection operator iteration $=$ iteration $+$ 1 end while To generate new adaption based mutation operator of $\textit{DE}/\textit{rand}/1$ , it chooses the better vectors $\textit{New\_Vector}1$ and $\textit{New\_Vector}2$ in the best search space. The new mutant vector is created and explains here:

$\displaystyle\vec{\gamma}_{i,G}=\vec{\alpha}_{\mathrm{rand},G}+\delta_{1}\cdot% (\textit{New\_Vector}1-\textit{New\_Vector}2)$ (3)

Where $\textit{New\_Vector}1=\sum_{i=1}^{\textit{NP}}\vec{\alpha}_{r_{1}^{i},G}*% \textit{NABV}1$ and $\textit{New\_Vector}2=\sum_{i=1}^{\textit{NP}}\vec{\alpha}_{r_{2}^{i},G}*% \textit{NABV}2$ are new adaption based vectors.

Generate the new donor vector:

$\displaystyle\vec{\gamma}_{i,G}=\vec{\alpha}_{\mathrm{rand},G}+\delta_{1}\cdot% \bigg{(}\sum\nolimits_{i=1}^{\textit{NP}}\vec{\alpha}_{r_{1}^{i},G}*\textit{% NABV}1-\sum\nolimits_{i=1}^{\textit{NP}}\vec{\alpha}_{r_{2}^{i},G}*\textit{ % NABV}2\bigg{)}$ (4)

Where $\vec{\alpha}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}$ denotes the random vector of current population. $\vec{\alpha}_{r_{1}^{i},G}$ and $\vec{\alpha}_{r_{2}^{i},G}$ will be generated from the entire search space, $\textit{NABV}1$ and $\textit{NABV}2$ are new adaption based vectors depending upon the nature of candidate solution.

Crossover: There is no change in crossover operator. This is done so that all areas near good solutions can be exploited reasonably.

Selection: There is no change in selection strategy. It implemented the same strategy which was implemented in $D E$ . First, the solution with higher fitness will be moved to the new population. In case of a good solution is selected.

New adaption based mutation operator (NABMO) process is explained in Algorithm 2 (NABDE-R (random)). This algorithm is applied to the other two variants which are NABDE-B (best) and NABDE-CB (current-to-best). These strategies are discussed in Algorithm 3 (NABDE-R) and Algorithm 4 (NABDE-CB).

4. Experimental results

The COmparing Continuous Optimisers (COCO) platform has been used for the Black-Box-Optimizati-on-Benchmarking (BBOB) 24 noise-less test functions [4, 6], and explained in Table 2. These variants are verified in COCO framework with respect to given criteria:

•
Group based benchmark function comparing proposed variants NABDE (NABDE-R (random), NABDE-B (best) and NABDE-CB (current-to-best)) with CPI-DE [24], TSDE [28], ToPDE [27], JADEcr [7] and MPEDE [18] on the dimensions 10D, 20D & 40D in (f1–f5), (f6–f9), (f10–f14), (f15–f19) and (f20–f24) functions.

4.1 Testing framework and benchmark functions

NABDE is tested using COCO framework. The search space is [ $-$ 5, 5] ${}^{D}$ , i.e., vector can have donor and trail vector within domain [ $-$ 5, 5]. Majority of benchmark functions have optima in [ $-$ 4, 4] domain. The problem is tested for minimization. Fifteen instances of each function were taken. The terminal conditions are either function evaluations or a precision of more than 10 ${}^{-8}$ . NABDE uses the various parameters, the value of which is explained as population size, is taken as 100. The population size is fixed. Control parameter is explained in Table 1. The maximum number of functional evaluations (FEs) is 10000* $D$ , where $D$ is the dimension.

The experiments were done on a PC with Intel Core i5 CPU with a speed of 3.40 GHz, installed memory (RAM) 4 GB, operating system Windows 7 Pro 64 bit & $\times$ 64 based processor.

4.1.1 Expected running time (ERT) loss ratio

It shows the ERT loss ratio for 20-D. Figure 4 displays plotted log10 of ERT loss ratio versus given budget $\textit{FEvals}=\textit{FEs}$ in log10. Box-Whisker plot shows 25–75%-ile (box) with median, 10–90%-ile (caps), and minimum and maximum ERT loss ratio (points) [4].

4.1.2 Empirical cumulative distribution functions (ECDF)

ECDF, plotting the fraction of trials with an outcome no larger than the respective value on the $x$ -axis for 20-D. Figure 5 includes ECDF of the number of function evaluations (FEvals) divided by search space dimension D, to fall below $f_{\mathrm{o}\mathrm{p}\mathrm{t}}+\Delta f$ with $\Delta f$ $=$ 10 ${}^{k}$ , where k is the first value in the legend. The thick red line represents the most difficult target value $f_{\mathrm{opt}}$ $+$ 10 ${}^{-8}$ [4].

Table 1
Control parameters

Sr. No.	Parameter	Type
1	Population size (popsize (NP))	100
2	Mutation (scale factor)	$\delta 1$ and $\delta 2$ [0–2]
3	Crossover rate	Cr [0–1]
4	Dimension	D
5	Dimension	[10, 20 & 40]
6	FEs	Function evaluations
7	MaxFEs	Maximum function evaluations

Table 2

Benchmark function

Group	f#	Function name
Separable	f1	Sphere
	f2	Ellipsoidal
	f3	Rastrigin
	f4	Büche-rastrigin
	f5	Linear slope
Low or moderate conditioning	f6	Attractive sector
	f7	Step ellipsoidal
	f8	Rosenbrock, original
	f9	Rosenbrock, rotated
Unimodal with high	f10	Ellipsoidal
conditioning	f11	Discus
	f12	Bent cigar
	f13	Sharp ridge
	f14	Different Powers
Multi-modal with adequate	f15	Rastrigin
global structure	f16	Weierstrass
	f17	Schaffers F7
	f18	Schaffers F7,
		moderately ill-conditioned
	f19	Composite
		Griewank-rosenbrock F8F2
Multi-modal with	f20	Schwefel
weak global structure	f21	Gallagher’s
		Gaussian 101-me peaks
	f22	Gallagher’s
		Gaussian 21-hi peaks
	f23	Katsuura
	f24	Lunacek bi-rastrigin

Table 3

Position of nabde variants on five state-of-ART algorithms over BBOB benchmark functions

Group	Dimension	Proposed HMBDE variants	DE variants
Separable (f1–f5)	10	1-NABDE-CB	4-ToPDE
	20	1-NABDE-B	4-ToPDE
	40	2-NABDE-CB	1-ToPDE
Low or moderate conditioning (f6–f9)	10	4-NABDE-R	1-JADEcr
	20	2-NABDE-CB	1-JADEcr
	40	2-NABDE-CB	1-JADEcr
Unimodal with high conditioning (f10–f14)	10	4-NABDE-CB	1-JADEcr
	20	3-NABDE-B	1-JADEcr
	40	3-NABDE-R	1-JADEcr
Multi-modal with (f15–f19)	10	1-NABDE-B	4-MPEDE
	20	4-NABDE-CB	1-ToPDE
	40	1-NABDE-CB	4-MPEDE
Multi-modal with (f20–f24)	10	3-NABDE-CB	1-ToPDE
	20	2-NABDE-CB	1-MPEDE
	40	2-NABDE-R	1-MPEDE
All Function (f1–f24)	10	2-NABDE-R	1-ToPDE
	20	1-NABDE-B	4-MPEDE
	40	1-NABDE-R	3-MPEDE

4.2 Result analysis

Proposed variant NABDE is compared with standard DE algorithms JADEcr [7], ToPDE [27], MPED-E [18], TSDE [28] and CPI-DE [24] on the dimensions 10D, 20D & 40D, which justifies the result analysis for improvement in the convergence rate for all functions (f1–f24). The result of proposed algorithm is found better than other existing algorithms in terms of minimum number of function evaluations, in order to reach a target function value 10 ${}^{-8}$ . The statistical values including “Function Evaluations (FEs)” and “Best Search” are shown in Table 3.

Figure 1.

Comparison of NABDE variants with other state-of-the-art algorithms in 10-D.

Figure 2.

Comparison of NABDE variants with other state-of-the-art algorithms in 20-D.

Figure 3.

Comparison of NABDE variants with other state-of-the-art algorithms in 40-D.

Figure 4.

ERT of NABDE algorithms in 20-D.

Figure 5.

ECDF of NABDE algorithms on 20-D.

Comparison of function evaluations (FEs), best search and target function on 10-D:

–

During the analysis of group based function evaluations (FEs) and target function on 10-D, I have obtained better convergence rate has been obtained for NABDE in benchmark functions (f1–f5) and (f15–f19). Other standard algorithms have obtained better convergence rate of group benchmark functions like (f20–f24) and (f1–f24) in ToPDE and (f6–f9) and (f10–f14) in JADEcr on 10-D, as evident from Fig. 1.

Comparison of function evaluations(FEs), best search and target function on 20-D:

–

During the analysis of group based function evaluations (FEs) and target function on 20-D, I have obtained better convergence rate has been obtained for NABDE in benchmark functions like (f1–f5), (f1–f24). Other standard algorithms have obtained better convergence rate of group benchmark functions like (f15–f19) in ToPDE, (f6–f9) and (f10–f14) in JADEcr and (f20–f24) in MPEDE on 20-D, as evident from Fig. 2.

Comparison of function evaluations (FEs), best search and target function on 40-D:

–

During the analysis of group based function evaluations (FEs) and target function on 40-D, I have obtained better convergence rate has been obtained for NABDE in benchmark functions (f15–f19) and (f1–f24). Other standard algorithms have obtained better convergence rate of group benchmark functions like (f1–f5) in ToPDE, (f6–f9) and (f10–f14) in JADEcr, and (f20–f24) in MPEDE on 40-D, as evident from Fig. 3.

The new adaption based mutation operator (NABM-O) operator has ts can see that better performance enhancement as compared to JADEcr, ToPDE, MPEDE, TSDE and CPI-DE. Our experimental results and extensive calculations show that DE without NABMO operator may lead to stagnation problem due to less generation of diversity but after multiplying with the vector (NABVs), it can see an upsurge in diversity and convergence rate thereby improving the overall result as shown in Figs 1–3.

From the above discussion I have seen that proposed new adaption based mutation operator is a very important operator for this algorithm. This operator maintains the diversity when it is stuck in local optimum problem. The required threshold diversity is achieved with taking new adaption based vectors (NABVs) from best entire space. NABVs is sufficient for capturing targeted solutions in all functions. This is due to the design of new adaption based mutation operator. A sufficient number of generations of around 100 may work with any number of functions (unimodal and multimodal). The value of control parameters, mutation factor and crossover rate are used for controlling the exploitation and exploration power of algorithm.

5. Conclusion

In this paper, proposed the new adaption based mutation operator helped to improve convergence rate at higher dimensions in the search space. In DE algorithm, its start with the maximum generation of function evaluations and apply the new adaption based mutation approach. The approach provides diversity in local and global search spaces. This strategy verifies COCO platform for real-parameter and global optimizers with CEC’2015 in all 24 benchmark functions [4]. Proposed NABDE variants is compared with standard DE algorithms like CPI-DE, TSDE, ToPDE, MPEDE and JADEcr and results are verified as shown in Table 3 in respect to 10, 20 and 40 dimensions. Our proposed new adaption based mutation operator is able to enhance the performance of other DE algorithms. Future scope adjust the control parameters by dynamic selection of whole population.

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New adaption based mutation operator on differential evolution algorithm

Abstract

Keywords

1. Introduction

3.1 New adaption based mutation operator (NABMO)

4.1.1 Expected running time (ERT) loss ratio

4.1.2 Empirical cumulative distribution functions (ECDF)

Table 1 Control parameters

References

Table 1
Control parameters