Three modified versions of differential evolution algorithm for continuous optimization

Abstract

Differential evolution (DE) is one simple and effective evolutionary algorithm (EA) for global optimization. In this paper, three modified versions of the DE to improve its performance, to repair its defect in accurate converging to individual optimal point and to compensate the limited amount of search moves of original DE are proposed. In the first modified version called bidirectional differential evolution (BDE), to generate a new trial point, is used from the bidirectional optimization concept, and in the second modified version called shuffled differential evolution (SDE), population such as shuffled frog leaping (SFL) algorithm is divided in to several memeplexes and each memeplex is improved by the DE algorithm. Finally, in the third modified version of DE called shuffled bidirectional differential evolution (SBDE) to improve each memeplex is used from the proposed BDE algorithm. Three proposed modified versions are applied on two types of DE and six obtained algorithms are compared with original DE and SFL algorithms. Experiments on continuous benchmark functions and non-parametric analysis of obtained results demonstrate that applying bidirectional concept only improves one type of the DE. But the SDE and the SBDE have a better success rate and higher solution precision than original DE and SFL, whereas those are more time consuming on some functions. In a later part of the comparative experiments, a comparison of the proposed algorithms with some modern DE and the other EAs reported in the literature confirms a better or at least comparable performance of our proposed algorithms.

Appendix: the set of benchmark functions

For all functions listed below, N is considered to be the function dimension.

1.1 High-dimensional benchmark functions

HF₁: Griewangk’s function

$$ \begin{gathered} f(x) = 1 + \sum\limits_{i = 1}^{N} {{\frac{{x_{i}^{2} }}{4,000}}} - \prod\limits_{i = 1}^{N} {\cos (x_{i} } /\sqrt i ) \hfill \\ x_{i} \in [ - 512,511],\,i = 1, \ldots ,N \hfill \\ \end{gathered} $$

It has a unique global minima f = 0 located at point $(0, \ldots ,0)$.

HF₂: extended Schaffer 2 (EF10) function

$$ \begin{gathered} f(x) = \sum\limits_{i = 1}^{N - 1} {\left( {x_{i}^{2} + x_{i + 1}^{2} } \right)^{0.25} } \left[ {\sin^{2} \left( {50\left( {x_{i}^{2} + x_{i + 1}^{2} } \right)^{0.1} } \right) + 1} \right] \hfill \\ x_{i} \in [ - 100,100],i = 1, \ldots ,N \hfill \\ \end{gathered} $$

HF₃: Ackley’s function:

$$ \begin{gathered} f(x) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{N}} \sum\limits_{i = 1}^{N} {x_{i}^{2} } } \right) - \exp \left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {\cos 2\pi x_{i} } } \right) \hfill \\ x_{i} \, \in \left[ { - 32,32} \right],\,i = 1, \ldots ,N \hfill \\ \end{gathered} $$

It has a global minima $f = - 20 - \text{e}$ located at point $(0, \ldots ,0)$.

HF₄: generalized penalized 2 function

$$ \begin{gathered} f(x) = {\frac{\pi }{N}}\{ \sum\limits_{i = 1}^{N - 1} {[(x_{i} } - 1)^{2} [1 + 10\sin^{2} (3\pi x_{i + 1} ) + 10\sin^{2} (\pi x_{i} )] \hfill \\ + (x_{N} - 1)^{2} [1 + 10\sin^{2} (3\pi x_{N} )]\} + \sum\limits_{i = 1}^{N} {u(x_{i} } ,5,100,4) \hfill \\ u(x_{i} ,5,100,4) = \left\{ \begin{gathered} 100(x_{i} - 5)^{4} \quad x_{i} > 5 \hfill \\ 0\quad - 5 \le x_{i} \le \quad 5 \hfill \\ 100( - x_{i} - 5)^{4} \quad x_{i} < - 5 \hfill \\ \end{gathered} \right. \hfill \\ x_{i} \in [ - 50,50],i = 1, \ldots ,N \hfill \\ \end{gathered} $$

It has a unique global minima f = 0 located at point $(1, \ldots ,1)$.

HF₅: Zakharov’s function

$$ \begin{gathered} f(x) = \sum\limits_{i = 1}^{N} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{N} {0.5ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{N} {0.5ix_{i} } } \right)^{4} \hfill \\ x_{i} \, \in \,[ - 5,10],i = 1, \ldots ,N \hfill \\ \end{gathered} $$

It has a unique global minima f = 0 located at point $(0, \ldots ,0)$.

HF₆: Rosenbrock’s function

$$ \begin{gathered} f(x) = \sum\limits_{i = 1}^{N - 1} {\left[ {100\left( {x_{i}^{2} - x_{i + 1} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]}^{2} \hfill \\ x_{i} \in \left[ { - 100,100} \right],i = 1, \ldots ,N \hfill \\ \end{gathered} $$

It has a unique global minima f = 0 located at point $(1, \ldots ,1)$.

1.2 The low-dimensional benchmark functions

LF₁: modified Shubert function

$$ \begin{gathered} f(x) = \left\{ {\sum\limits_{j = 1}^{5} {j\cos [(j + 1)x_{1} } + j]} \right\}\left\{ {\sum\limits_{j = 1}^{5} {j\cos [(j + 1)x_{2} + j]} } \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x_{i} \in [ - 10,10],i = 1,2 \hfill \\ \end{gathered} $$

It has one global minima f = −186.73909 located at point (−1.42513,−0.80032).

LF₂: modified Longerman function

$$ \begin{gathered} f(x) = - \sum\limits_{j = 1}^{5} {c_{j} } \cos (d_{j} /\pi )\exp ( - \pi d_{j} ) \hfill \\ d_{j} = \sum\limits_{i = 1}^{N} {(x_{i} } - a_{ij} )^{2} , \hfill \\ x_{i} \in [0,10],N = 5 \hfill \\ \end{gathered} $$

$$ A = \left( {\begin{array}{*{20}c} { 9. 6 8 1} & {0. 6 6 7} & { 4. 7 8 3} & { 9.0 9 5} & { 3. 5 1 7} \\ { 9. 400} & { 2.0 4 1} & { 3. 7 8 8} & { 7. 9 3 1} & { 2. 8 8 2} \\ { 8.0 2 5} & { 9. 1 5 2} & { 5. 1 1 4} & { 7. 6 2 1} & { 4. 5 6 4} \\ { 2. 1 9 6} & {0. 4 1 5} & { 5. 6 4 9} & { 6. 9 7 9} & { 9. 5 10} \\ { 8.0 7 4} & { 8. 7 7 7} & { 3. 4 6 7} & { 1. 8 6 7} & { 6. 70 8} \\ \end{array} } \right)\quad C = \left( {\begin{array}{*{20}c} {0. 80 6} & {0. 5 1 7} & {0. 1} & {0. 90 8} & {0. 9 6 5} \\ \end{array} } \right) $$

It has a unique minima f = −0.965000 located at point (8.074, 8.777, 3.467, 1.867, 6.708).

LF₃: Easom function

$$ \begin{gathered} f(x) = - \cos x_{1} \cos x_{2} \exp ( - ((x_{1} - \pi )^{2} + (x_{2} - \pi )^{2} )) \hfill \\ \,\,x_{i} \in [ - 100,100],i = 1,2 \hfill \\ \end{gathered} $$

It has a unique global minima f = −1 located at the point $(\pi ,\pi )$.

LF₄: Goldstein and Price function

$$ \begin{gathered} f(x) = [1 + (x_{1} + x_{2} + 1)^{2} \times (19 - 14x_{1} + 3x_{1}^{2} - 14x_{2} + 6x_{1} x_{2} + 3x_{2}^{2} )] \times \hfill \\ \,\,\,\,[30 + (2x_{1} - 3x_{2} )^{2} ) \times (18 - 32x_{1} - 12x_{1}^{2} + 48x_{2} - 36x_{1} x_{2} + 27x_{2}^{2} )] \hfill \\ x_{i} \in [ - 2,2],i = 1,2 \hfill \\ \end{gathered} $$

It has a unique global minima f = 3 located at the point (0,−1).

LF₅: (Kwon et al. 2003)

$$ \begin{gathered} f(x) = \left[ {0.002 + \sum\limits_{j = 1}^{25} {\left( {j + \sum\limits_{i = 1}^{2} {(x_{i} - a_{ij} )^{6} } } \right)}^{ - 1} } \right]^{ - 1} \hfill \\ x_{i} \in [ - 65.536,65.536],\,i = 1,2 \hfill \\ \end{gathered} $$

Where A is given by

$$ A =\left( \begin{array}{lllllllllllllllllllllllll} { - 32} & { - 16} & 0 & {16} & {32} & { - 32} & { - 16} & 0 & {16} & {32} & { - 32} & { - 16} & 0 & {16} & {32} & { - 32} & { - 16} & 0 & {16} & {32} & { - 32} & { - 16} & 0 & {16} & {32} \\ { - 32} & { - 32} & { - 32} & { - 32} & { - 32} & { - 16} & { - 16} & { - 16} & { - 16} & { - 16} & 0 & 0 & 0 & 0 & 0 & {16} & {16} & {16} & {16} & {16} & {32} & {32} & {32} & {32} & {32} \\ \end{array}\right) $$

It has one global minima f = 0.9980 located at point (−31.9772,−31.9783).

LF₆: Himmelblau function

$$ \begin{gathered} f(x) = (x_{1}^{2} + x_{2} - 11)^{2} + (x_{1} + x_{{_{2} }}^{2} - 7)^{2} + 0.1[(x_{1} - 3)^{2} + (x_{2} - 2)^{2} ] \hfill \\ \,x_{i} \in [ - 6,6],\,i = 1,2 \hfill \\ \end{gathered} $$

It has a unique global minima f = 0 located at the point (3,2).