Differential evolution dynamics analysis by complex networks

Abstract

Differential evolution is a simple yet efficient heuristic originally designed for global optimization over continuous spaces that has been used in many research areas. The question how to improve its performance is still popular and during the years, many successful methods dealing with optimal setting or hybridization of the control parameters were proposed. In this paper, we propose a novel approach based on modeling of the differential evolution dynamics by complex networks. In each generation, the individuals are mapped to the nodes and the relationships between them are modeled by the edges of the graph. Thanks to this simple visualization, the interconnection between the differential evolution convergence speed and the weighted clustering coefficients has been revealed. As a consequence, we have focused on the parents selection in the mutation step where the individuals are not selected randomly as usual but on the basis of their weighted clustering coefficients. Our enhancement has been incorporated in the classical differential evolution, self-adaptive differential evolution (jDE) and differential evolution with composite trial vector generation strategies and control parameters. Finally, a set of well-known benchmark functions (including 21 functions) has been used to test and evaluate the performance of the proposed enhancement of the differential evolution. The experimental results and statistical analysis indicate that the enhanced algorithms perform better or at least comparable to their original versions and the analysis of the differential evolution dynamics with the aim of the complex network might be an effective tool to improve the differential evolution convergence in the future.

Appendix 1: Benchmark functions

1.1 Sphere Model

$$\begin{aligned}&f_1(x) = \sum _{i=1}^{30}{x_{i}^{2}}, \nonumber \\&-100 \le x_i \le 100, \min (f_1) = f_1(0, \ldots , 0) = 0. \end{aligned}$$

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1.2 Schwefel’s Problem 2.22

$$\begin{aligned}&f_2(x) = \sum _{i=1}^{30}{|x_i|} + \prod _{i=1}^{30}{|x_i|}, \nonumber \\&-10 \le x_i \le 10, \min (f_2) = f_2(0, \ldots ,0) = 0. \end{aligned}$$

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1.3 Schwefel’s Problem 1.2

$$\begin{aligned}&f_3(x) = \sum _{i=1}^{30}{\left( \sum _{j=1}^{i}{x_j}\right) ^2}, \nonumber \\&-100 \le x_i \le 100, \min (f_3) = f_3(0, \ldots ,0) = 0. \end{aligned}$$

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1.4 Schwefel’s Problem 1.21

$$\begin{aligned}&f_4(x) = \max _{i} \{|x_i|, 1\le i \le 30\}, \nonumber \\&-100 \le x_i \le 100, \min (f_4) = f_4(0, \ldots ,0) = 0. \end{aligned}$$

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1.5 Generalized Rosenbrock’s Function

$$\begin{aligned}&f_5(x) = \sum _{i=1}^{29}{[100(x_{i+1}-x_{i}^2)^2 + (x_i-1)^2]}, \nonumber \\&-30 \le x_i \le 30, \min (f_5) = f_5(1, \ldots ,1) = 0. \end{aligned}$$

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1.6 Step Function

$$\begin{aligned}&f_6(x) = \sum _{i=1}^{30}{(\left\lfloor x_i + 0.5\right\rfloor )^2}, \nonumber \\&-100 \le x_i \le 100, \min (f_6) = f_6(0, \ldots ,0) = 0. \end{aligned}$$

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1.7 Quartic Function i.e. Noise

$$\begin{aligned}&f_7(x) = \sum _{i=1}^{30}{ix_i^4} + \text {random}[0,1), \nonumber \\&-1.28 \le x_i \le 1.28, \min (f_7) = f_7(0, \ldots ,0) = 0. \end{aligned}$$

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1.8 Generalized Schwefel’s Problem 2.26

$$\begin{aligned}&f_8(x) = -\sum _{i=1}^{30}{x_i \sin {(\sqrt{|x_i|})}}, \nonumber \\&-500 \le x_i \le 500, \min (f_8)\nonumber \\&\quad = f_8(420.9687, \ldots , 420.9687) = -12569.5. \end{aligned}$$

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1.9 Generalized Rastrigin’s Function

$$\begin{aligned}&f_9(x) = \sum _{i=1}^{30}{[x_i^2 - 10\cos {(2 \pi x_i)} + 10]}, \nonumber \\&-5.12 \le x_i \le 5.12, \min (f_9) = f_9(0, \ldots , 0) = 0. \end{aligned}$$

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1.10 Ackley’s Function

$$\begin{aligned} f_{10}(x)= & {} -20 \exp \left( -0.2 \sqrt{\frac{1}{30}\sum _{i=1}^{30}{x_i^2}} \right) \nonumber \\&- \exp \left( \frac{1}{30} \sum _{i=1}^{30}{\cos (2\pi x_i)}\right) + 20 + e, \nonumber \\&-32 \le x_i \le 32, \min (f_{10})\nonumber \\ {}= & {} f_{10}(0, \ldots , 0) = 0. \end{aligned}$$

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1.11 Generalized Griewangk Function

$$\begin{aligned}&f_{11}(x) = \frac{1}{4000} \sum _{i=1}^{30}{x_{i}^2} - \prod _{i=1}^{30}{\cos \left( \frac{x_i}{\sqrt{i}}\right) + 1}, \nonumber \\&-600 \le x_i \le 600, \min (f_{11}) = f_{11}(0, \ldots , 0) = 0. \end{aligned}$$

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1.12 Generalized Penalized Functions

$$\begin{aligned} f_{12}(x)= & {} \frac{\pi }{30} \left( 10 \sin ^{2}(\pi x_1) + \sum _{i=1}^{29}(y_i-1)^2\right. \nonumber \\&\qquad \qquad \left. \times \, [1 + 10 \sin ^2(\pi y_{i+1})] + (y_n-1)^2 \right) \nonumber \\&+\sum _{i=1}^{30}{u(x_i,10,100,4)}, -50 \le x_i \le 50, \min (f_{12})\nonumber \\= & {} f_{12}(1, \ldots , 1) = 0.\end{aligned}$$

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$$\begin{aligned} f_{13}(x)= & {} 0.1 \left( \sin ^{2}(3 \pi x_1) + \sum _{i=1}^{29}(x_i-1)^2[1 + \sin ^2(3\pi x_{i+1})]\right. \nonumber \\&\qquad \qquad \left. +\, (x_n-1)^2[1+ \sin ^2(2 \pi x_{30})] \right) \nonumber \\&+ \sum _{i=1}^{30}{u(x_i,5,100,4)},-50 \le x_i \le 50, \min (f_{13})\nonumber \\= & {} f_{13}(1, \ldots , 1) = 0. \end{aligned}$$

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where

$$\begin{aligned}u(x_i,a,k,m)= & {} \left\{ \begin{array}{l l} k(x_i - a)^m &{}\quad x_i >a,\\ 0 &{} \quad -a \le x_i \le a,\\ k(-x_i - a)^m &{} \quad x_i < -a. \end{array} \right. \\ y_i= & {} 1 + \frac{1}{4}(x_i+1). \end{aligned}$$

1.13 Shekel’s Foxholes Function

$$\begin{aligned}&f_{14}(x) = \left[ \frac{1}{500} + \sum _{i=1}^{25}{\frac{1}{j + \sum _{i=1}^{2}{(x_i - a_{i,j})^6}}}\right] ^{-1} \nonumber \\&-65.536 \le x_i \le 65.536, \min (f_{14}) = f_{14}(-32, 32) \approx 1.\nonumber \\ \end{aligned}$$

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where

$$\begin{aligned} a_{i,j} = \begin{bmatrix} -32&-16&0&16&32&-32 \ldots 0&16&32\\ -32&-32&-32&-32&-32&-16 \ldots&32&32&32\\ \end{bmatrix} \end{aligned}$$

1.14 Kowalik’s Function

See Table 6.

$$\begin{aligned}&f_{15}(x) = \sum _{i=1}^{11}{\left[ a_i - \frac{x_1(b_i^2 + b_i x_2)}{b_i^2 + b_i x_3 + x_4}\right] ^2}, \nonumber \\&-5 \le x_i \le 5, \min (f_{15}) \approx f_{15}(0.1928, 0.1908,\nonumber \\&\quad 0.1231, 0.1358) \approx 0.0003075. \end{aligned}$$

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Table 6 Kowalik’s function $f_{15}$

1.15 Six-Hump Camel-Back Function

$$\begin{aligned}&f_{16}(x) = 4x_1^2 - 2.1 x_1^4 + \frac{1}{3}x_1^6 + x_1 x_2 - 4x_2^2 + 4x_2^4, -5 \le x_i \le 5, \nonumber \\&x_{\text {min}}(0.08983, -0.7126),(-0.08983, 0.7126),\nonumber \\&\min (f_{16}) = -1.0316285 \end{aligned}$$

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1.16 Branin Function

$$\begin{aligned}&f_{17}(x) = \left( x_2 - \frac{5.1}{4\pi ^2} x_1^2 + \frac{5}{\pi } x_1 - 6\right) ^2 + 10 \left( 1- \frac{1}{8 \pi }\right) \cos (x_1)\nonumber \\&\quad + 10,-5 \le x_1 \le 10, \nonumber \\&0 \le x_2 \le 15, x_{\text {min}}(-3.142, 12.275),(3.142, 2.275),\nonumber \\&\quad (9.425, 2.425), \min (f_{17})= 0.398. \end{aligned}$$

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1.17 Goldstein–Price Function

$$\begin{aligned}&\!\!\!\!f_{18}(x) = [1 + (x_1 + x_2 +1)^2 (19-14x_1 + 3x_1^2 - 14x_2 \nonumber \\&\qquad \qquad +\, 6x_1 x_2 + 3x_2^2)] [30 + (2x_1 - 3x_2)^2 \times (18 - 32x_1 \nonumber \\&\qquad \qquad +\, 12 x_1^2 + 48 x_2 - 36x_1x_2 + 27x_2^2)],\nonumber \\&-2 \le x_i \le 2, x_{\text {min}}(0,-1), \min (f_{18}) = 3. \end{aligned}$$

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1.18 Shekel’s Family

$$\begin{aligned} f_{19}(x) = -\sum _{i=1}^{m}{[(x - a_i)(x-a_i)^T + c_i]^{-1}}, \end{aligned}$$

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where $m = 5$, 7, and 10 for $f_{19}$, $f_{20}$, $f_{21}$. The function $f_{19}(x)$ has 10, $f_{20}(x)$ 7 and $f_{21}(x)$ 10 local minima, $x_{\text {local}\_\text {opt}}\approx 1/{c_i}$ for $1 \le i \le m$. The coefficients are defined by Table 7.

Table 7 Shekel functions $f_{19}$, $f_{20}$ and $f_{21}$