Large-Scale Evolutionary Optimization Using Multi-Layer Strategy Differential Evolution

Abstract

This paper proposes The Multi-Layer Strategies Differential Evolution (MLSDE) algorithm, which finds optimal solutions for large scale problems. To solve large scale problems were grouped different strategies together and applied them to date set. Furthermore, these strategies were applied to selected vectors to strengthen the exploration ability of the algorithm. Extensive computational analysis were also carried out to evaluate the performance of the proposed algorithm on a set of well-known CEC 2015 benchmark functions. This benchmark was utilized for the assessment and performance evaluation of the proposed algorithm.

Appendix: The Benchmark Functions Suite

CEC 2015 The benchmark functions that used in experimental study.

Name	Test functions	S
Sphere	$f_{01} = \sum\nolimits_{i = 1}^{D} {x_{i}^{2} }$	[−100, 100]D
Schwefel 2.22	$f_{02} = \sum\nolimits_{i = 1}^{D} {\left| {x_{i} } \right|} + \prod\nolimits_{i = 1}^{D} {\left| {x_{i} } \right|}$	[−10, 10]D
Schwefel 1.2	$f_{03} = \sum\nolimits_{i = 1}^{D} {\left( {\sum\nolimits_{j = 1}^{i} {x_{j} } } \right)^{2} }$	[−100, 100]D
Schwefel 2.21	$f_{04} = \max_{i} \left\{ {\left| {x_{i} } \right|,1 \le i \le D} \right\}$	[−100, 100]D
Rosenbrock	$f_{05} = \sum\nolimits_{i = 1}^{D - 1} {\left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]}$	[−30, 30]D
Step	$f_{06} = \sum\nolimits_{i = 1}^{D - 1} {\left( {\left\lfloor {x_{i} + 0.5} \right\rfloor } \right)^{2} }$	[−100, 100]D
Quartic	$f_{07} = \sum\nolimits_{i = 1}^{D} {x_{i}^{4} + random[0,1)}$	[−1.28, 1.28]D
Schwefel 2.26	$f_{08} = \sum\nolimits_{i = 1}^{D} {\left( { - x_{i} \sin \left( {\sqrt {\left| {x_{i} } \right|} } \right)} \right) + 418.98288727243369 \times D}$	[−500, 500]D
Rastrigin	$f_{09} = \sum\nolimits_{i = 1}^{D} {(x_{i}^{2} - 10\;\cos \left( {2\uppi\,x_{i} } \right) + 10)}$	[−5.12, 5.12]D
Ackley	$f_{10} = - 20\,\exp ( - 0.2\sqrt {\frac{1}{D}\sum\nolimits_{i = 1}^{D} {x_{i}^{2} } } ) - \exp (\frac{1}{D}\sum\nolimits_{i = 1}^{D} {\cos (2\uppi\,x_{i} )} ) + 20 + \exp (1)$	[−32, 32]D
Griewank	$f_{11} = \frac{1}{4000}\sum\nolimits_{i = 1}^{D} {x_{i}^{2} - \prod\nolimits_{i = 1}^{D} {\cos \,(\frac{{x_{i} }}{\sqrt i }) + 1} }$	[−600, 600]D
Penalized 1	$\begin{aligned} f_{12} & = \frac{\pi }{D}\left\{ {10\,\sin^{2} (\uppi\,y_{i} ) + \sum\nolimits_{i = 1}^{D - 1} {\left( {y_{i} - 1} \right)^{2} \cdot \left[ {1 + 10\,\sin^{2} (\uppi\,y_{i} + 1)} \right] + (y_{D} - 1)^{2} } } \right\} \\ & + \sum\nolimits_{i = 1}^{D} {u(x_{i} ,10,100,4)} \\ \end{aligned}$	[−50, 50]D
Penalized 2	$\begin{aligned} f_{13} & = \frac{1}{10}\left\{ {\sin^{2} (3\uppi\,x_{1} ) + \sum\nolimits_{i = 1}^{D - 1} {\left( {x_{i} - 1} \right)^{2} \cdot \left[ {1 + \sin^{2} (3\uppi\,x_{i} + 1)} \right] + (x_{D} - 1)^{2} [1 + \sin^{2} (2\uppi\,x_{D} )]} } \right\} \\ & + \sum\nolimits_{i = 1}^{D} {u(x_{i} ,5,100,4)} \\ \end{aligned}$	[−30, 50]D
Neuniaier 3	$f_{14} = \sum\nolimits_{i = 1}^{D} {(x_{i} - 1)^{2} + \sum\nolimits_{i = 2}^{D} {x_{i} x_{i - 1} + \frac{D(D + 4)(D - 1)}{6}} }$	[−D2, D2]D
Salomon	$f_{15} = 1 - \cos \left( {2\uppi\left\| x \right\|} \right) + 0.1\left\| x \right\|,{\text{where}}\left\| x \right\| = \sum\nolimits_{i = 1}^{D} {x_{i} }$	[−100, 100]D
Alpine	$f_{16} = \sum\nolimits_{i = 1}^{D} {\left| {x_{i} \,\sin x_{i} + 0.1x_{i} } \right|}$	[−10, 10]D