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A multi-population cultural algorithm with adaptive diversity preservation and its application in ammonia synthesis process

  • LSMS2010 and ICSEE 2010
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Abstract

A multi-population cultural differential evolution (MCDE) algorithm is proposed. Each of the populations is managed by its private cultural differential evolution algorithm, in which a center individual is introduced into the belief space and selection function follows a new method to select the offspring for the next generation. To accelerate the convergence speed, the populations exchange their knowledge with each other every given generations. An adaptive mechanism of population diversity preservation is put forward to prevent the populations from being trapped in local optima. In the adaptive mechanism, the idea of culture fusion between populations is used to know the convergence status, so that the diversity of populations is kept along the evolutionary process. The performance evaluation on MCDE using eleven constrained optimization problems shows that MCDE is a competitive approach. MCDE is further applied to a practical optimization problem in an ammonia synthesis system with the objective to maximize the net value of ammonia. The results achieved by MCDE are compared with those by two traditional differential evolution algorithms, which indicate that MCDE has more excellent performance and better effectiveness.

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Acknowledgments

We are very grateful to the editors and anonymous reviewers for their valuable comments and suggestions to help improve our paper. This work is supported by National High Technology Research and Development Program of China (863 Program) (No. 2009AA04Z141), National Natural Science Foundation of China (Grant no. 61174040), Shanghai Commission of Science and Technology (Grant no. 08JC1408200), Shanghai Leading Academic Discipline Project (Grant no. B504), and the Specialized Research Fund for Doctoral Program of Higher Education of China (No. 200802510010).

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Authors and Affiliations

  1. Key Laboratory of Advanced Control and Optimization for Chemical Processes, East China University of Science and Technology, Ministry of Education, Shanghai, 200237, China

    Wei Xu, Raofen Wang, Lingbo Zhang & Xingsheng Gu

  2. State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, 310027, China

    Wei Xu

Authors
  1. Wei Xu
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  2. Raofen Wang
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  3. Lingbo Zhang
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  4. Xingsheng Gu
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Corresponding author

Correspondence to Xingsheng Gu.

Appendix: Test constrained optimization problems

Appendix: Test constrained optimization problems

Problem g01

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = 5\sum\limits_{i = 1}^{4} {x_{i} } - 5\sum\limits_{i = 1}^{4} {x_{i}^{2} } - \sum\limits_{i = 5}^{13} {x_{i} } } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {g_{1} (x) = 2x_{1} + 2x_{2} + x_{10} + x_{11} - 10 \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = 2x_{1} + 2x_{3} + x_{10} + x_{12} - 10 \le 0,} \hfill \\ {} \hfill & {g_{3} (x) = 2x_{2} + 2x_{3} + x_{11} + x_{12} - 10 \le 0,} \hfill \\ {} \hfill & {g_{4} (x) = - 8x_{1} + x_{10} \le 0,} \hfill \\ {} \hfill & {g_{5} (x) = - 8x_{2} + x_{11} \le 0,} \hfill \\ {} \hfill & {g_{6} (x) = - 8x_{3} + x_{12} \le 0,} \hfill \\ {} \hfill & {g_{7} (x) = - 2x_{4} - x_{5} + x_{10} \le 0,} \hfill \\ {} \hfill & {g_{8} (x) = - 2x_{6} - x_{7} + x_{11} \le 0,} \hfill \\ {} \hfill & {g_{9} (x) = - 2x_{8} - x_{9} + x_{12} \le 0,} \hfill \\ {} \hfill & {0 \le x_{i} \le 1\,(i = 1, \ldots ,9),0 \le x_{i} \le 100\,(i = 10,11,12),0 \le x_{13} \le 1.} \hfill \\ \end{array} $$
(26)

Problem g02

$$ \begin{array}{*{20}c} {\max } \hfill & {f(x) = \left| {\frac{{\sum\nolimits_{i = 1}^{n} {\cos^{4} (x_{i} )} - 2\prod\nolimits_{i = 1}^{n} {\cos^{2} (x_{i} )} }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {ix_{i}^{2} } } }}} \right|} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {g_{1} (x) = 0.75 - \prod\limits_{i = 1}^{n} {x_{i} } \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = \sum\limits_{i = 1}^{n} {x_{i} } - 7.5\quad n \le 0, \, } \hfill \\ {} \hfill & {0 \le x_{i} \le 10\,(i = 1, \ldots ,20). \, } \hfill \\ \end{array} $$
(27)

Problem g03

$$ \begin{array}{*{20}c} {\max } \hfill & {f(x) = \left( {\sqrt n } \right)^{n} \prod\limits_{i = 1}^{n} {x_{i} } } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {h_{1} (x) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } - 1 = 0,} \hfill \\ {} \hfill & {0 \le x_{i} \le 1\,(i = 1, \ldots ,10).} \hfill \\ \end{array} $$
(28)

Problem g04

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = 5.3578547x_{3}^{2} + 0.8356891x_{1} x_{5} + 37.293239x_{1} - 40,792.141} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {g_{1} (x) = 85.334407 + 0.0056858x_{2} x_{5} + 0.0006262x_{1} x_{4} - 0.0022053x_{3} x_{5} - 92 \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = - 85.334407 - 0.0056858x_{2} x_{5} - 0.0006262x_{1} x_{4} + 0.0022053x_{3} x_{5} \le 0,} \hfill \\ {} \hfill & {g_{3} (x) = 80.51249 + 0.0071317x_{2} x_{5} + 0.0029955x_{1} x_{2} + 0.0021813x_{3}^{2} - 110 \le 0,} \hfill \\ {} \hfill & {g_{4} (x) = - 80.51249 - 0.0071317x_{2} x_{5} - 0.0029955x_{1} x_{2} - 0.0021813x_{3}^{2} + 90 \le 0,} \hfill \\ {} \hfill & {g_{5} (x) = 9.300961 + 0.0047026x_{3} x_{5} + 0.0012547x_{1} x_{3} + 0.0019085x_{3} x_{4} - 25 \le 0,} \hfill \\ {} \hfill & {g_{6} (x) = - 9.300961 - 0.0047026x_{3} x_{5} - 0.0012547x_{1} x_{3} - 0.0019085x_{3} x_{4} + 20 \le 0,} \hfill \\ {} \hfill & {78 \le x_{1} \le 102,33 \le x_{2} \le 45,27 \le x_{i} \le 45\,(i = 3,4,5).} \hfill \\ \end{array} $$
(29)

Problem g05

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = 3x_{1} + 0.000001x_{1}^{3} + 2x_{2} + (0.000002/3)x_{2}^{3} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & { \, g_{1} (x) = - x_{4} + x_{3} - 0.55 \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = - x_{3} + x_{4} - 0.55 \le 0,} \hfill \\ {} \hfill & {h_{1} (x) = 1,000\sin ( - x_{3} - 0.25) + 1,000\sin ( - x_{4} - 0.25) + 894.8 - x_{1} = 0,} \hfill \\ {} \hfill & {h_{2} (x) = 1,000\sin (x_{3} - 0.25) + 1,000\sin (x_{3} - x_{4} - 0.25) + 894.8 - x_{2} = 0,} \hfill \\ {} \hfill & {h_{3} (x) = 1,000\sin (x_{4} - 0.25) + 1,000\sin (x_{4} - x_{3} - 0.25) + 1294.8 = 0,} \hfill \\ {} \hfill & {0 \le x_{i} \le 1,200\,(i = 1,2), - 0.55 \le x_{i} \le 0.55\,(i = 3,4).} \hfill \\ \end{array} $$
(30)

Problem g06

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = (x_{1} - 10)^{3} + (x_{2} - 20)^{3} \, } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & { \, g_{1} (x) = - (x_{1} - 5)^{2} - (x_{2} - 5)^{2} + 100 \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = (x_{1} - 6)^{2} + (x_{2} - 5)^{2} - 82.81 \le 0,} \hfill \\ {} \hfill & {13 \le x_{1} \le 100,0 \le x_{2} \le 100.} \hfill \\ \end{array} $$
(31)

Problem g07

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = x_{1}^{2} + x_{2}^{2} + x_{1} x_{2} - 14x_{1} - 16x_{2} + (x_{3} - 10)^{2} + 4(x_{4} - 5)^{2} + (x_{5} - 3)^{2} } \hfill \\ {} \hfill & {\quad \quad \quad + 2(x_{6} - 1)^{2} + 5x_{7}^{2} + 7(x_{8} - 11)^{2} + 2(x_{9} - 10)^{2} + (x_{10} - 7)^{2} + 45} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & { \, g_{1} (x) = - 105 + 4x_{1} + 5x_{2} - 3x_{7} + 9x_{8} \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = 10x_{1} - 8x_{2} - 17x_{7} + 2x_{8} \le 0,} \hfill \\ {} \hfill & {g_{3} (x) = - 8x_{1} + 2x_{2} + 5x_{9} - 2x_{10} - 12 \le 0,} \hfill \\ {} \hfill & {g_{4} (x) = 3(x_{1} - 2)^{2} + 4(x_{2} - 3)^{2} + 2x_{3}^{2} - 7x_{4} - 120 \le 0,} \hfill \\ {} \hfill & {g_{5} (x) = 5x_{1}^{2} + 8x_{2} + (x_{3} - 6)^{2} - 2x_{4} - 40 \le 0,} \hfill \\ {} \hfill & {g_{6} (x) = x_{1}^{2} + 2(x_{2} - 2)^{2} - 2x_{1} x_{2} + 14x_{5} - 6x_{6} \le 0,} \hfill \\ {} \hfill & {g_{7} (x) = 0.5(x_{1} - 8)^{2} + 2(x_{2} - 4)^{2} + 3x_{5}^{2} - x_{6} - 30 \le 0,} \hfill \\ {} \hfill & {g_{8} (x) = - 3x_{1} + 6x_{2} + 12(x_{9} - 8)^{2} - 7x_{10} \le 0,\quad - 10 \le x_{i} \le 10\,(i = 1, \ldots ,10).} \hfill \\ \end{array} $$
(32)

Problem g08

$$ \begin{array}{*{20}c} {\max } \hfill & {f(x) = \frac{{\sin^{3} (2\pi x_{1} )\sin (2\pi x_{2} )}}{{x_{1}^{3} (x_{1} + x_{2} )}}} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & { \, g_{1} (x) = x_{1}^{2} - x_{2} + 1 \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = 1 - x_{1} + (x_{2} - 4)^{2} \le 0,} \hfill \\ {} \hfill & {0 \le x_{i} \le 10\,(i = 1,2).} \hfill \\ \end{array} $$
(33)

Problem g09

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = (x_{1} - 10)^{2} + 5(x_{2} - 12)^{2} + x_{3}^{4} + 3(x_{4} - 11)^{2} } \hfill \\ {} \hfill & {\quad \quad \quad + 10x_{5}^{6} + 7x_{6}^{2} + x_{7}^{4} - 4x_{6} x_{7} - 10x_{6} - 8x_{7} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {g_{1} (x) = - 127 + 2x_{1}^{2} + 3x_{2}^{4} + x_{3} + 4x_{4}^{2} + 5x_{5} \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = - 282 + 7x_{1} + 3x_{2} + 10x_{3}^{2} + x_{4} - x_{5} \le 0,} \hfill \\ {} \hfill & {g_{3} (x) = - 196 + 23x_{1} + x_{2}^{2} + 6x_{6}^{2} - 8x_{7} \le 0,} \hfill \\ {} \hfill & {g_{4} (x) = 4x_{1}^{2} + x_{2}^{2} - 3x_{1} x_{2} + 2x_{3}^{2} + 5x_{6} - 11x_{7} \le 0,\quad - 10 \le x_{i} \le 10\,(i = 1, \ldots ,7).} \hfill \\ \end{array} $$
(34)

Problem g10

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = x_{1} + x_{2} + x_{3} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {g_{1} (x) = - 1 + 0.0025(x_{4} + x_{6} ) \le 0,} \hfill \\ {} \hfill & {g_{2} (x) = - 1 + 0.0025(x_{5} + x_{7} - x_{4} ) \le 0,} \hfill \\ {} \hfill & {g_{3} (x) = - 1 + 0.01(x_{8} - x_{5} ) \le 0,} \hfill \\ {} \hfill & {g_{4} (x) = - x_{1} x_{6} + 833.33252x_{4} + 100x_{1} - 83,333.333 \le 0,} \hfill \\ {} \hfill & {g_{5} (x) = - x_{2} x_{7} + 1,250x_{5} + x_{2} x_{4} - 1,250x_{4} \le 0,} \hfill \\ {} \hfill & {g_{6} (x) = - x_{3} x_{8} + 1,25,0000 + x_{3} x_{5} - 2,500x_{5} \le 0,} \hfill \\ {} \hfill & {100 \le x_{1} \le 10,000,1,000 \le x_{i} \le 10,000\,(i = 2,3),10 \le x_{i} \le 1,000\,(i = 4, \ldots ,8).} \hfill \\ \end{array} $$
(35)

Problem g11

$$ \begin{array}{*{20}c} {\min } \hfill & {f(x) = x_{1}^{2} + (x_{2} - 1)^{2} } \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {h(x) = x_{2} - x_{1}^{2} = 0,\quad - 1 \le x_{i} \le 1\,(i = 1,2).} \hfill \\ \end{array} $$
(36)

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Xu, W., Wang, R., Zhang, L. et al. A multi-population cultural algorithm with adaptive diversity preservation and its application in ammonia synthesis process. Neural Comput & Applic 21, 1129–1140 (2012). https://doi.org/10.1007/s00521-011-0749-5

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