Modifications for the Differential Evolution Algorithm

Abstract

Differential Evolution (DE) is a method of optimization used in symmetrical optimization problems and also in problems that are not even continuous, and are noisy and change over time. DE optimizes a problem with a population of candidate solutions and creates new candidate solutions per generation in combination with existing rules according to discriminatory rules. The present work proposes two variations for this method. The first significantly improves the termination of the method by proposing an asymptotic termination rule, which is based on the differentiation of the average of the function values in the population of DE. The second modification proposes a new scheme for a critical parameter of the method, which improves the method’s ability to better explore the search space of the objective function. The proposed variations have been tested on a number of problems from the current literature, and from the experimental results, it appears that the proposed modifications render the method quite robust and faster even in large-scale problems.

1. Introduction

The location of the global minimum of a continuous and differentiable function $f:S\to R,S\subset R^n$ is formulated as:

$$x^{*}=\arg \underset{x\in S}{min}f\left(x\right)$$

(1)

where the set S is defined as:

$$S=\left[a_1,b_1\right]\otimes \left[a_2,b_2\right]\otimes \dots \left[a_n,b_n\right]$$

In the recent literature, there are a plethora of real-world problems that can be formulated as global optimization problems, such as problems from physics [1,2,3,4], chemistry [5,6,7], economics [8,9], etc. Furthermore, global optimization methods are used in many symmetry problems [10,11,12,13,14].There are a variety of proposed methods to handle the global minimum problem, such as Adaptive Random Search [15], Competitive Evolution [16], Controlled Random Search [17], Simulated Annealing [18,19,20], Genetic Algorithms [21,22], Bee optimization [23,24], Ant Colony Optimization [25], Particle Swarm Optimization [26], Differential Evolution [27], etc. Recently, many works have appeared that take advantage of the GPU processing units to implement parallel global optimization methods [28,29,30]. This work introduces two major modifications for the Differential Evolution (DE) method that aim to speed up the algorithm and reduce the total number of function evaluations required by the method. The DE method initially creates a population of candidate solutions and, through a series of iterations, creates new solutions by combining the previous ones. The method does not require any prior knowledge of the derivative and is, therefore, quite fast with low memory requirements. Moreover, the method has been used in various symmetry problems from the relevant literature, such as community detection [31], structure prediction of materials [32], motor fault diagnosis [33], automatic clustering techniques [34], etc.

After a literature review, it was found that differential evolution is used in several areas and many modifications of the original algorithm have been introduced in the recent literature. More specifically, in the research of Zongjun et al. [35], genetic and differential calculus algorithms were used to optimize the parameters of two models aimed at estimating evapotranspiration in three regions, and it was found that the performance of evolution algorithms was better than the genetic algorithm. Other research focused on a case study of a cellular neural network aimed at generating fractional classes of neurons. The best solutions from differential calculus and accelerated particle swarm optimization (APSO) are presented concretely in the work of Tlelo-Cuautle et al. [36]. Another article [37] proposes a regeneration framework based on space search adaptation (ARSA), which can be integrated into different variants of different evolutions to address the problems of early convergence and population stability faced by differential calculus. Another interesting variation of the method is the Bernstain Search Differential Evolution Algorithm [38] for optimizing numerical functions.

The differential evolution method was also applied to energy science. Specifically, the article of Liang et al. [39] evaluates the parameters of solar photovoltaic models through a self-adjusting differential evolution. Similarly, in the study of Peng et al. [40], differential evolution is used for the prediction of electricity prices. Furthermore, differential evolution was also incorporated in a neural architecture search [41]. The DE method has been applied with success to neural network training [42,43,44], to the Traveling Salesman Problem [45,46], training of RBF neural networks [47,48,49], and optimization of the Lennard Jones potential [50,51]. The DE method has also been successfully combined with other techniques for machine learning applications, such as classification [52,53], feature selection [54,55], deep learning [56,57], etc.

The rest of this article is organized as follows: In Section 2, the base DE algorithm, as well as the proposed modifications, are presented. In Section 3, the test functions used in the experiments are presented along with the experimental results. Finally, in Section 4, some conclusions are presented.

2. Modifications

This section starts with a detailed description of the DE method and continues with the modifications suggested in this article. The first modification is a new stopping rule, which measures the difference of the mean of the function values between the iterations of the algorithm. The second modification suggests a new scheme for a critical parameter of the DE algorithm called Differential Weight.

2.1. The Base Algorithm

The DE algorithm has been studied by various researchers in the recent literature, such as the Compact Differential Evolution [58], a self adaptive DE [59] where the critical parameters of the method are adapted from previous generations, a fuzzy adaptive DE method [60] where fuzzy login is employed to adapt the parameters of the method, parallel Differential Evolution [61] with a self adaptation mechanism for the critical parameters of the DE method, etc. A survey of the recent advances in differential evolutions can be found in the work of Das et al. [62]. The base DE algorithm has the steps described in Algorithm 1.

Algorithm 1: DE algorithm.

Set the population size $\mathrm{NP}\ge$ 4, usually $\mathrm{NP}=10n$, where n is the dimension of the input problem. Set the crossover probability $\mathrm{CR}\in [0,1]$. A typical value for this parameter is 0.9. Set the differential weight $F\in [0,2].$ A typical value for this parameter is 0.8. Initialize all members of the population in the search space. The members of the population are called agents. Until some stopping criterion is met, repeat: (a) For$i=1\dots \mathrm{NP}$do. Setx as the agent i. Pick randomly three agents $a,b,c.$ Pick a random index $R\in \left\{1,\dots ,n\right\}.$ Compute the trial vector $y=\left[y_{1,}{y}_2,\dots ,y_n\right]$ as follows. For$j=1,\dots ,n$do: Set$r_i\in [0,1]$ a random number. If$r_j<\mathrm{CR}$or$j=R$then$y_j=a_j+F\times \left(b_j-c_j\right)$else$y_j=x_j$. If$f\left(y\right)\le f\left(x\right)$ then $x=y$. EndFor. (b) EndFor. Return the agent $x_{\mathrm{best}}$ in the population with the lower function value $f\left(x_{\mathrm{best}}\right)$.

2.2. The New Termination Rule

Typically, the DE method is terminated when a predefined number of iterations is reached. This can be extremely inefficient in some problems and, in others, it can lead to premature termination, i.e., termination before the total minimum is found. In the work of Ali et al. [63], a different termination rule is proposed i.e., terminate when:

$$f_{\max }-f_{\min }\le ϵ$$

(2)

where $f_{\max }$ is the function value of the worst agent in the population, $f_{\min }$ is the function value of the best agent, and $ϵ$ is a small positive number.

In the proposed termination rule, the average function value of the population is calculated in each iteration. If this value does not change significantly for a repetitive number of iterations, then it is very likely that the method may not discover a new global minimum and should therefore be terminated. Hence, in every generation t, we measure the quantity:

$${\delta }^{\left(t\right)}=\left|\sum _{i=1}^{\mathrm{NP}}\left|f_i^{\left(t\right)}\right|-\sum _{i=1}^{\mathrm{NP}}\left|f_i^{(t-1)}\right|\right|$$

(3)

and the termination rule is defined as: terminate if ${\delta }^{\left(t\right)}\le ϵ$ for a predefined number of M generations.

2.3. The New Differential Weight

The differential weight initially proposed in the DE algorithm was a static value, which means that some tuning is required in order to discover the global minimum in every optimization function. Ali et al. [63] proposed an adaptation mechanism for this parameter in that the algorithm should search in larger spaces in the the first generations, and become more focused in later generations. The mechanism proposed is expressed as:

$$F=\left\{\begin{array}{ccc}max\left(l_{\min },1-\left|\frac{f_{\max }}{f_{\min }}\right|\right),& \mathrm{if}& \left|\frac{f_{\max }}{f_{\min }}\right|\le 1\\ max\left(l_{\min },1-\left|\frac{f_{\min }}{f_{\max }}\right|\right),& \mathrm{otherwise}\end{array}\right.$$

(4)

The current work proposes a stochastic mechanism similar to the crossover operation of the Genetic algorithms. The proposed scheme is expressed as:

$$F=-\frac{1}{2}+2\times R$$

(5)

where $R\in [0,1]$ is a random number. The proposed scheme, as with the randomness introduced by the method DE will be able to better explore the search space of the objective function and find with greater accuracy and speed the global minimum. In addition, this scheme has been used successfully in Genetic Algorithms.

3. Experiments

In order to determine the effectiveness of the proposed modifications, a series of experiments were performed on known functions from the relevant literature [64,65]. The choice of these functions was made as they are widely used in the literature by many researchers [66,67,68,69], they have quite a complex structure, and in many cases, they have a large number of dimensions that make them ideal for studying and testing.

The experiments were divided into two major categories. In the first category, all the schemes for the Differential Weight were tested using the termination rule of Equation (2), and in the second category, the same schemes were tested using the proposed termination criterion. Furthermore, after every successful termination, the local optimization method BFGS [70] was applied in order to get even closer to the global minimum.

3.1. Test Functions

The descriptions of the test functions used in the experiments are as follows:

• Bf1 (Bohachevsky 1) function defined as:

  $$f\left(x\right)=x_1^2+2x_2^2-\frac{3}{10}cos\left(3\pi x_1\right)-\frac{4}{10}cos\left(4\pi x_2\right)+\frac{7}{10}$$

  with $x\in {[-100,100]}^2$. The value of global minimum is 0.0.
• Bf2 (Bohachevsky 2) function defined as:

  $$f\left(x\right)=x_1^2+2x_2^2-\frac{3}{10}cos\left(3\pi x_1\right)cos\left(4\pi x_2\right)+\frac{3}{10}$$

  with $x\in {[-50,50]}^2$. The value of the global minimum is 0.0.
• Branin function. The function is defined by $f\left(x\right)={\left(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos\left(x_1\right)+10$ with $-5\le x_1\le 10,0\le x_2\le 15$. The value of global minimum is 0.397887.with $x\in {[-10,10]}^2$. The value of global minimum is −0.352386.
• CM function. The Cosine Mixture function is given by the equation:

  $$f\left(x\right)=\sum _{i=1}^n{x}_i^2-\frac{1}{10}\sum _{i=1}^{n}cos\left(5\pi x_i\right)$$

  where $x\in {[-1,1]}^n$. For our experiments we used $n=4$.
• Camel function. The function is given by:

  $$f\left(x\right)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4,x\in {[-5,5]}^2$$
• Easom function. The function is given by the equation:

  $$f\left(x\right)=-cos\left(x_1\right)cos\left(x_2\right)exp\left({\left(x_2-\pi \right)}^2-{\left(x_1-\pi \right)}^2\right)$$

  with $x\in {[-100,100]}^2.$
• Exponential function, defined as:

  $$f\left(x\right)=-exp\left(-0.5\sum _{i=1}^n{x}_i^2\right),-1\le x_i\le 1$$
  The global minimum is located at $x^{*}=(0,0,...,0)$ with value $-1$. In our experiments we used this function with $n=2,4,8,16,32$.
• Goldstein and Price function
  The function is given by the equation:

  $$\begin{array}{ccc}\hfill f\left(x\right)& =& \left[1+{\left(x_1+x_2+1\right)}^2\right.\hfill \\ & & \left(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2\right)]\times \hfill \\ & & [30+{\left(2x_1-3x_2\right)}^2\hfill \\ & & \left(18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2\right)]\hfill \end{array}$$

  with $x\in {[-2,2]}^2$. The global minimum is located at $x^{*}=(0,-1)$ with value 3.0.
• Griewank2 function. The function is given by:

  $$f\left(x\right)=1+\frac{1}{200}\sum _{i=1}^2{x}_i^2-\prod _{i=1}^2\frac{cos\left(x_i\right)}{\sqrt{\left(i\right)}},x\in {[-100,100]}^2$$
  The global minimum is located at the $x^{*}=(0,0,...,0)$ with value 0.
• Gkls function. $f\left(x\right)=\mathrm{Gkls}(x,n,w)$, is a function with w local minima, described in [71] with $x\in {[-1,1]}^n$ and n a positive integer between 2 and 100. The value of the global minimum is −1 and in our experiments we have used $n=2,3$ and $w=50,100$.
• Hansen function. $f\left(x\right)={\sum }_{i=1}^{5}icos\left[(i-1)x_1+i\right]{\sum }_{j=1}^{5}jcos\left[(j+1)x_2+j\right]$, $x\in$${[-10,10]}^2$.
• Hartman 3 function. The function is given by:

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^3{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  with $x\in {[0,1]}^3$ and $a=\left(\begin{array}{ccc}3& 10& 30\\ 0.1& 10& 35\\ 3& 10& 30\\ 0.1& 10& 35\end{array}\right),c=\left(\begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array}\right)$ and

  $$p=\left(\begin{array}{ccc}0.3689& 0.117& 0.2673\\ 0.4699& 0.4387& 0.747\\ 0.1091& 0.8732& 0.5547\\ 0.03815& 0.5743& 0.8828\end{array}\right)$$
• Hartman 6 function.

  $$f\left(x\right)=-\sum _{i=1}^4{c}_{i}exp\left(-\sum _{j=1}^6{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$$

  with $x\in {[0,1]}^6$ and $a=\left(\begin{array}{cccccc}10& 3& 17& 3.5& 1.7& 8\\ 0.05& 10& 17& 0.1& 8& 14\\ 3& 3.5& 1.7& 10& 17& 8\\ 17& 8& 0.05& 10& 0.1& 14\end{array}\right),c=\left(\begin{array}{c}1\\ 1.2\\ 3\\ 3.2\end{array}\right)$ and

  $$p=\left(\begin{array}{cccccc}0.1312& 0.1696& 0.5569& 0.0124& 0.8283& 0.5886\\ 0.2329& 0.4135& 0.8307& 0.3736& 0.1004& 0.9991\\ 0.2348& 0.1451& 0.3522& 0.2883& 0.3047& 0.6650\\ 0.4047& 0.8828& 0.8732& 0.5743& 0.1091& 0.0381\end{array}\right)$$
• Potential function. The molecular conformation corresponding to the global minimum of the energy of N atoms interacting via the Lennard-Jones potential [72] is used as a test case here. The function to be minimized is given by:

  $$V_{LJ}\left(r\right)=4ϵ\left[{\left(\frac{\sigma }{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6\right]$$

  (6)
  In the current experiments three different cases were studied: $N=3,4,5.$
• Rastrigin function. The function is given by:

  $$f\left(x\right)=x_1^2+x_2^2-cos\left(18x_1\right)-cos\left(18x_2\right),x\in {[-1,1]}^2$$
  The global minimum is located at $x^{*}=(0,0)$ with value −2.0.
• Rosenbrock function.
  This function is given by:

  $$f\left(x\right)=\sum _{i=1}^{n-1}\left(100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right),-30\le x_i\le 30.$$
  The global minimum is located at the $x^{*}=(0,0,...,0)$ with $f\left(x^{*}\right)=0$. In our experiments we used this function with $n=4,8,16$.
• Shekel 7 function.

$$f\left(x\right)=-\sum _{i=1}^7\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\\ 2& 9& 2& 9\\ 5& 3& 5& 3\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\\ 0.6\\ 0.3\end{array}\right)$. The value of global minimum is −10.342378.

• Shekel 5 function.

$$f\left(x\right)=-\sum _{i=1}^5\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\end{array}\right)$. The value of global minimum is −10.107749.

• Shekel 10 function.

$$f\left(x\right)=-\sum _{i=1}^{10}\frac{1}{(x-a_i){(x-a_i)}^T+c_i}$$

with $x\in {[0,10]}^4$ and $a=\left(\begin{array}{cccc}4& 4& 4& 4\\ 1& 1& 1& 1\\ 8& 8& 8& 8\\ 6& 6& 6& 6\\ 3& 7& 3& 7\\ 2& 9& 2& 9\\ 5& 5& 3& 3\\ 8& 1& 8& 1\\ 6& 2& 6& 2\\ 7& 3.6& 7& 3.6\end{array}\right),c=\left(\begin{array}{c}0.1\\ 0.2\\ 0.2\\ 0.4\\ 0.4\\ 0.6\\ 0.3\\ 0.7\\ 0.5\\ 0.6\end{array}\right)$. The value of global minimum is −10.536410.

• Sinusoidal function. The function is given by:

  $$f\left(x\right)=-\left(2.5\prod _{i=1}^{n}sin\left(x_i-z\right)+\prod _{i=1}^{n}sin\left(5\left(x_i-z\right)\right)\right),0\le x_i\le \pi .$$
  The global minimum is located at $x^{*}=(2.09435,2.09435,...,2.09435)$ with $f\left(x^{*}\right)=-3.5$. In our experiments we used $n=4,8,16,32$ and $z=\frac{\pi }{6}$ and the corresponding functions are denoted by the labels SINU4, SINU8, SINU16 and SINU32, respectively.
• Test2N function. This function is given by the equation:

  $$f\left(x\right)=\frac{1}{2}\sum _{i=1}^n{x}_i^4-16x_i^2+5x_i,x_i\in [-5,5].$$
  The function has $2^n$ in the specified range and in our experiments we used $n=4,5,6,7$. The corresponding values of global minimum is −156.664663 for $n=4$, −195.830829 for $n=5$, −234.996994 for $n=6$ and −274.163160 for $n=7$.
• Test30N function. This function is given by:

  $$f\left(x\right)=\frac{1}{10}{sin}^2\left(3\pi x_1\right)\sum _{i=2}^{n-1}\left({\left(x_i-1\right)}^2\left(1+{sin}^2\left(3\pi x_{i+1}\right)\right)\right)+{\left(x_n-1\right)}^2\left(1+{sin}^2\left(2\pi x_n\right)\right)$$

  with $x\in [-10,10]$, with ${30}^n$ local minima in the searc space. For our experiments we used $n=3,4$.

3.2. Experimental Results

The experiments were performed 30 times with different seeds for the random generator each time for every test function and the avarage function was measured and reported. The code was implemented in ANSI C++ and the random generator used was the function drand48() of the C programming languages. The execution environment was an Intel Xeon E5-2630 multi-core machine. The parameters used in the experiments are shown in

Table 1. Experimental parameters.

Parameter	Value
NP	10n
F	0.8
CR	0.9
M	20
$ϵ$	${10}^{-4}$

. The experiments where the stopping rule of Equation (2) was used are outlined in

Table 2. Experiments with the termination rule of Ali.

Function	Static	Ali	Proposed
BF1	1142	1431	847
BF2	1164	1379	896
BRANIN	984	816	707
CM4	3590	7572	2079
CAMEL	1094	18,849	685
EASOM	1707	2014	1327
EXP2	532	323	449
EXP4	2421	1019	1494
EXP8	15,750	3670	5632
EXP16	160,031	15,150	21,416
EXP32	320,039	152,548	77,936
GKLS250	784	944	614
GKLS2100	772	1531	599 (0.97)
GKLS350	1906 (0.93)	3263	1275 (0.93)
GKLS3100	1883	3539	1373
GOLDSTEIN	988	818	769
GRIEWANK2	1299 (0.97)	1403	883 (0.93)
HANSEN	2398	2968	1400
HARTMAN3	1448	836	1050
HARTMAN6	9489(0.97)	4015(0.97)	4667(0.80)
POTENTIAL3	90,027	89,776	21,824
POTENTIAL4	120,387 (0.97)	120,405 (0.33)	45,705 (0.97)
POTENTIAL5	150,073	150,104	83,342
RASTRIGIN	1246	1098 (0.93)	871
ROSENBROCK4	6564	9695	4499
ROSENBROCK8	44,240	72,228	13,959
ROSENBCROK16	160,349 (0.90)	160,538 (0.60)	53,594
SHEKEL5	5524	3810	3057 (0.83)
SHEKEL7	5266	3558	2992 (0.87)
SHEKEL10	5319	3379	3076
TEST2N4	4200	1980	2592
TEST2N5	7357	2957	4055
TEST2N6	12,074	4159	5836
TEST2N7	18,872	5490	7904
SINU4	3270	1855	2216
SINU8	23,108	6995	8135
SINU16	160,092	36,044	30,943
SINU32	213,757 (0.70)	160,536 (0.53)	83,369 (0.80)
TEST30N3	1452	1732	959
TEST30N4	1917	2287	1378
Total	1,564,515 (0.97)	1,062,714 (0.96)	506,404 (0.98)

, and the experiments with the proposed stopping rule are listed in

Table 3. Experiments with the proposed termination rule.

Function	Static	Ali	Proposed
BF1	996	1124	889
BF2	926	1026	816
BRANIN	878	900	730
CM4	1148 (0.70)	1991	1103
CAMEL	1049	904 (0.93)	846
EASOM	447	448	446
EXP2	470	461	467
EXP4	915	903	892
EXP8	1797	3558	1796
EXP16	3578	7082	3521
EXP32	7082	14,125	7022
GKLS250	498	576	493
GKLS2100	533	884 (0.97)	515
GKLS350	823	1130 (0.93)	814 (0.97)
GKLS3100	858	1495 (0.97)	829 (0.93)
GOLDSTEIN	945	993	915
GRIEWANK2	947	921	826
HANSEN	2104	1949	1479
HARTMAN3	1017	1005	952
HARTMAN6	4679 (0.90)	3744 (0.97)	3128 (0.87)
POTENTIAL3	21,473	2284	8197
POTENTIAL4	44,191 (0.43)	3098 (0.33)	24,659 (0.97)
POTENTIAL5	75,910	3443	52,664
RASTRIGIN	841	994	777
ROSENBROCK4	4934	7192	3300
ROSENBROCK8	29,583	49,696	10,907
ROSENBCROK16	160,349	160,538 (0.60)	38,315
SHEKEL5	4389 (0.97)	4266	2839 (0.83)
SHEKEL7	3905	3685	2668
SHEKEL10	4049	3548	2629
TEST2N4	2785	2275	2221
TEST2N5	4481	3170	3122
TEST2N6	6852	4286	4296
TEST2N7	11971	5701	6267
SINU4	2322	1987	1755
SINU8	9990	6156	5113
SINU16	6892	3628 (0.97)	16,905
SINU32	7235 (0.80)	7438 (0.83)	7218
TEST30N3	1033	1098	951
TEST30N4	1355	1444	1285
Total	432,610 (0.98)	321,166 (0.96)	224,567 (0.99)

. The numbers in the cells represent average function calls. The fraction in parentheses stands for the fraction of runs where the global optimum was found. If this number is missing then the global minimum was discovered in every independent run (100% success). The column STATIC represents the static value for the differential weight $(F=0.8)$, the column ALI stands for the mechanism given in the Equation (4), and lastly, the column PROPOSED stands for the proposed scheme given in Equation (5).

From the experiments, we observe that the two proposed variations drastically reduce the required number of function calls. Moreover, the proposed changes did not seem to affect the average performance of the method, as it remained high in all cases. The effect of the proposed scheme for the differential weight is presented graphically in Figure 1, where we plot the average function calls for the functions ROSENBROCK4, ROSENBROCK8, and ROSENBROCK16 using the three schemes of differential weights. Furthermore, in the plot of Figure 2, the average calls for the same functions are shown with both the proposed scheme for the differential weights and the proposed termination rule. It is evident that the combination of both modifications reduced, even more, the average number of function calls required to locate the global minimum of the test functions. To show the effectiveness of the modifications, Figure 3 presents the total time for 30 executions on an I7 computer with LINUX DEBIAN and 16 GB of memory. The comparison was made between the initial method with the ALI termination criterion, the proposed termination criterion (first modification), and the proposed termination criterion together with the proposed weight scheme. The proposed modifications significantly reduced the number of calls and also the required execution time. To compare the proposed scheme for the differential weight with the other two methods, the Wilcoxon signed-rank test was used. The results obtained with this statistical test are shown in Figure 4.

4. Conclusions

In this text, two main additions to the DE method are presented. In the first, an asymptotic termination rule was introduced and used successfully, even in multidimensional problems. This rule is based on the observation that from one point onwards the average of the functional values of the agents does not change. This means that either the algorithm has already found the global minimum or its further continuation will have no meaning.

In the second case, a stochastic scheme was used to produce the differential weight. This scheme helped the algorithm to better explore the search space of the objective function without the need for more calls to the objective function.

The proposed modifications significantly speed up the original method in terms of function calls, in most cases. Each of the proposed modifications can be used separately or together in the DE method. If used together, there is a large reduction in the number of required function calls that reach up to 80% without problems, and in the reliability of the method and its ability to successfully find the total minimum.

The DE method can also be used in cases of optimization problems with constraints as long as there is a change in the original problem so that the constraints are included in the function with the usage for example Langrage multipliers.