A new hybrid algorithm to solve bound-constrained nonlinear optimization problems

The goal of this work is to propose a hybrid algorithm called real-coded self-organizing migrating genetic algorithm by combining real-coded genetic algorithm (RCGA) and self-organizing migrating algorithm (SOMA) for solving bound-constrained nonlinear optimization problems having multimodal continuous functions. In RCGA, exponential ranking selection, whole-arithmetic crossover and non-uniform mutation operations have been used as different operators where as in SOMA, a modification has been done. The performance of the proposed hybrid algorithm has been tested by solving a set of benchmark optimization problems taken from the existing literature. Then, the simulated results have been compared numerically and graphically with existing algorithms. In the graphical comparison, a modified performance index has been proposed. Finally, the proposed algorithm has been applied to solve two real-life problems.

The authors are very much thankful to the anonymous reviewers for their constructive comments. The third and fifth author would like to acknowledge the finical support provided by WBDSTBT (429(Sanc)/ST/P/S&T/16G-23/2018) for continuing the research. The first author is thankful to Mrs. Kabita Garai, Dakshin Changrachak Sukanta Vidyapith, Moyna, Purba Medinipur, West Bengal, India-721644 and Dr. Navonil Bose, Department of Physics, Supreme Knowledge Foundation Group of Institutions, Mankundu, Hooghly, West Bengal, India-712139 for their kind cooperation and suggestions.

Authors and Affiliations

1. Department of Mathematics, Supreme Knowledge Foundation Group of Institutions, Mankundu, Hooghly, West Bengal, 712139, India

   Avijit Duary

2. Department of Mathematics, The University of Burdwan, Burdwan, West Bengal, 713104, India

   Md Sadikur Rahman, Ali Akbar Shaikh & Asoke Kumar Bhunia

3. Department of Industrial Engineering, Sharif University of Technology, P.O. Box 11155, 9414 Azadi Ave., Tehran, 1458889694, Iran

   Seyed Taghi Akhavan Niaki

Corresponding author

Correspondence to Ali Akbar Shaikh.

Publisher's Note

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Appendix

The detailed description of 25 problems that have been used in this paper is given below:

1. 1.

   Ackley’s problem:

   $$\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{1} ({\mathbf{x}}) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {\cos (2\pi x_{i} )} } \right) + 20 + e,$$

   where \(- 32 \le x_{i} \le 32\), with the known global optimum \(f_{1} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

2. 2.

   Cosine mixture problem:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{2} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } - 0.1\sum\limits_{i = 1}^{n} {\cos (5\pi x_{i} )}\), \({\text{where }}\,\, - 1 \le x_{i} \le 1\), with the known global optimum \(f_{2} ({\mathbf{x}}^{*} ) = - \,0.1\,n \,\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

3. 3.

   Exponential problem:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{3} ({\mathbf{x}}) = - \exp \left( { - 0.5\sum\limits_{i = 1}^{n} {x_{i}^{2} } } \right)\),\({\text{where }}\, - 1 \le x_{i} \le 1\), with the known global optimum \(f_{3} ({\mathbf{x}}^{*} ) = - 1 \,\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

4. 4.

   Griewank problem:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{4} ({\mathbf{x}}) = 1 + \frac{1}{4000}\sum\limits_{i = 1}^{n} {x_{i}^{2} } - \prod\limits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right)}\), \({\text{where }}\, - 600 \le x_{i} \le 600\), with the known global optimum \(f_{4} ({\mathbf{x}}^{*} ) = 0 \,\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

5. 5.

   Levy and Montalvo problem 1:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{5} ({\mathbf{x}}) = \frac{\pi }{n}\left( {10\sin^{2} (\pi y_{1} ) + \sum\limits_{i = 1}^{n - 1} {(y_{i} - 1)^{2} } \left[ {1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)} \right] + (y_{n} - 1)^{2} } \right)\)\({\text{where }}\,y_{i} = 1 + \frac{1}{4}(x_{i} + 1),\,\,\, - 10 \le x_{i} \le 10\), with the known global optimum \(f_{5} ({\mathbf{x}}^{*} ) = 0 \,\) at \(x_{i}^{*} = 1\) for \(i = 1,2, \ldots ,n.\)

6. 6.

   Levy and Montalvo problem 2:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{6} ({\mathbf{x}}) = 0.1\left( {\sin^{2} (3\pi x_{1} ) + \sum\limits_{i = 1}^{n - 1} {(x_{i} - 1)^{2} } \left[ {1 + \sin^{2} (3\pi x_{i + 1} )} \right] + (x_{n} - 1)^{2} \left[ {1 + \sin^{2} (2\pi x_{n} )} \right]} \right)\)\({\text{where }}\,\, - 5 \le x_{i} \le 5\), with the known global optimum \(f_{6} ({\mathbf{x}}^{*} ) = 0 \,\) at \(x_{i}^{*} = 1\) for \(i = 1,2, \ldots ,n.\)

7. 7.

   Paviani problem:

   \(\mathop {\hbox{min} }\limits_{x} f_{7} (x) = \sum\limits_{i = 1}^{10} {\left[ {(1n(x_{i} - 2))^{2} + (1n(10 - x_{i} ))^{2} } \right] - \left( {\prod\limits_{i = 1}^{10} {x_{i} } } \right)^{0.2} } ,\;{\text{where }}2 \le x_{i} \le 10\) with the known global optimum \(f_{7} ({\mathbf{x}}^{*} ) = - 45.778 \,\) at \(x_{i}^{*} = 9.351\) for \(i = 1,2, \ldots ,n.\)

8. 8.

   Rastrigin problem:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{8} ({\mathbf{x}}) = 10n + \sum\limits_{i = 1}^{n} {\left[ {x_{i}^{2} - 10\cos (2\pi x_{i} )} \right]}\), \({\text{where }} - 5.12 \le x_{i} \le 5.12\), with the known global optimum \(f_{8} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

9. 9.

   Rosenbrock problem:

   \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{9} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n - 1} {\left[ {100(x_{i + 1} - x_{i}^{2} )^{2} + (x_{i} - 1)^{2} } \right]}\), \({\text{where }} - 30 \le x_{i} \le 30\), with the known global optimum \(f_{9} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 1\) for \(i = 1,2, \ldots ,n.\)

10. 10.

    Schwefel problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{10} ({\mathbf{x}}) = 418.9829n - \sum\limits_{i = 1}^{n} {\left[ {x_{i} \sin (\sqrt {\left| {x_{i} } \right|} )} \right]\,}\), \({\text{where }} - 500 \le x_{i} \le 500\), with the known global optimum \(f_{10} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 420.97\) for \(i = 1,2, \ldots ,n.\)

11. 11.

    Sinusoidal problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{11} ({\mathbf{x}}) = - \left[ {2.5\prod\limits_{i = 1}^{n} {\sin \left( {x_{i} - \frac{\pi }{6}} \right)} + \prod\limits_{i = 1}^{n} {\sin \left( {5\left( {x_{i} - \frac{\pi }{6}} \right)} \right)} } \right]\), \({\text{where }}\,\,0 \le x_{i} \le \pi\), with the known global optimum \(f_{11} ({\mathbf{x}}^{*} ) = - 3.5\) at \(x_{i}^{*} = 2\pi /3\) for \(i = 1,2, \ldots ,n\).

12. 12.

    Zakharov’s problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{12} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{n} {\frac{i}{2}x_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{n} {\frac{i}{2}x_{i} } } \right)^{4}\), \({\text{where}} - 5.12 \le x_{i} \le 5.12\), with the known global optimum \(f_{12} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n\)

13. 13.

    Sphere problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{13} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {x_{i}^{2} }\), \({\text{where}} - 5.12 \le x_{i} \le 5.12\), with the known global optimum \(f_{13} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

14. 14.

    Axis parallel hyper ellipsoid problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{14} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {ix_{i}^{2} }\), \({\text{where}} - 5.12 \le x_{i} \le 5.12\), with the known global optimum \(f_{14} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

15. 15.

    Schwefel double sum problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{15} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {\left( {\sum\limits_{j = 1}^{i} {x_{i} } } \right)^{2} }\), \({\text{where }}\,\, - 65.536 \le x_{i} \le 65.536\), with the known global optimum \(f_{15} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n\).

16. 16.

    Schwefel problem 4:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{16} ({\mathbf{x}}) = \mathop {\hbox{min} }\limits_{i} \{ \,\left| {x_{i} } \right|,\,\,1 \le i \le n\}\), \({\text{where}} - 100 \le x_{i} \le 100\), with the known global optimum \(f_{16} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

17. 17.

    De-Jong problem with noise:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{17} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {x_{i}^{4} + {\text{rand}}\, (0,\,\,1)}\), \({\text{where }} - 10 \le x_{i} \le 10\), with the known global optimum \(f_{17} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 0\) for \(i = 1,2, \ldots ,n.\)

18. 18.

    Ellipsoidal problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{18} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {(x_{i} - i)^{2} }\), \({\text{where }}\, - n \le x_{i} \le n\), with the known global optimum \(f_{18} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = i\) for \(i = 1,2, \ldots ,n.\)

19. 19.

    Generalized penalized problem 1:

    $$\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{19} ({\mathbf{x}}) = \frac{\pi }{n}\left( {10\sin^{2} (\pi y_{1} ) + \sum\limits_{i = 1}^{n - 1} {(y_{i} - 1)^{2} } \left[ {1 + 10\sin^{2} (\pi y_{i + 1} )} \right] + (y_{n} - 1)^{2} } \right) + \sum\limits_{i = 1}^{n} u (x_{i} ,\,10,\,100,\,4),$$

    \({\text{where}}\;y_{i} = 1 + \frac{1}{4}(x_{i} + 1)\,\,{\text{and}}\,\, - 10 \le x_{i} \le 10\), with the known global optimum \(f_{19} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 1\) for \(i = 1,2, \ldots ,n.\)

20. 20.

    Generalized penalized problem 2:

    $$\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{20} ({\mathbf{x}}) = 0.1\left( {\sin^{2} (3\pi x_{1} ) + \sum\limits_{i = 1}^{n - 1} {(x_{i} - 1)^{2} } \left[ {1 + \sin^{2} (3\pi x_{i + 1} )} \right] + (x_{n} - 1)^{2} \left[ {1 + \sin^{2} (2\pi x_{n} )} \right]} \right) + \sum\limits_{i = 1}^{n} {u(x_{i} ,\,10,\,100,\,4)} ,\,\,\,{\text{where }}\, - 5 \le x_{i} \le 5,$$

    with the known global optimum \(f_{20} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = 1\) for \(i = 1,2, \ldots ,n.\)

    In problems number 19 and 20, the penalty function u is given by the following expression:

    $$u(x,\,a,\,k,\,m) = \left\{ {\begin{array}{*{20}l} {k(x - a)^{m} ,} \hfill & {{\text{if}}\;x > a,} \hfill \\ { - k(x - a)^{m} ,} \hfill & {{\text{if}}\;x < - a,} \hfill \\ {0,} \hfill & {{\text{otherwise}} .} \hfill \\ \end{array} } \right.$$
21. 21.

    Easom problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{21} ({\mathbf{x}}) = - \left( {\mathop \prod \limits_{i = 1}^{n} \cos \,(x_{i} )} \right) \times \left( {e^{{ - \sum\limits_{i = 1}^{n} {(x_{i} - \pi )^{2} } }} } \right)\), \({\text{where }}\, - 10 \le x_{i} \le 10\), with the known global optimum \(f_{21} ({\mathbf{x}}^{*} ) = - 1\) at \(x_{i}^{*} = \pi\) for \(i = 1,2, \ldots ,n.\)

22. 22.

    Trid problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{22} ({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {(x_{i} - 1)^{2} } - \sum\limits_{i = 2}^{n} {x_{i} x_{i - 1} }\), \({\text{where }}\,\, - n^{2} \le x_{i} \le n^{2}\), with the known global optimum \(f_{22} ({\mathbf{x}}^{*} ) = - \frac{n(n + 4)(n - 1)}{6}\, \,\) at \(x_{i}^{*} = i\,(n + 1 - i)\) for \(i = 1,2, \ldots ,n.\)

23. 23.

    Dixon and Price problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{23} ({\mathbf{x}}) = \,(x_{1} - 1)^{2} + \,\sum\limits_{i = 2}^{n} {i\,(2x_{i}^{2} - x_{i - 1} )^{2} }\), \({\text{where }} - 10 \le x_{i} \le 10\), with the known global optimum \(f_{23} ({\mathbf{x}}^{*} ) = 0\) at \(x_{i}^{*} = f\left( {2^{{\left( {\frac{{2^{i} - 2}}{{2^{i} }}} \right)}} } \right)\) for \(i = 1,2, \ldots ,n.\)

24. 24.

    Michalewicz’s problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{24} ({\mathbf{x}}) = - \sum\limits_{i = 1}^{n} {\sin (x_{i} )} \left( {\sin \left( {\frac{{ix_{i}^{2} }}{\pi }} \right)} \right)^{2m}\), \(\,\,{\text{where }}\,0 \le x_{i} \le \pi\) and \(m = 10\,\), with the known global optimum \(f_{24} ({\mathbf{x}}^{*} ) = - 9.6602\) for \(i = 1,2, \ldots ,n.\)

25. 25.

    Shekel 10 problem:

    \(\mathop {\hbox{min} }\limits_{{\mathbf{x}}} f_{25} ({\mathbf{x}}) = - \sum\limits_{j = 1}^{m} {\left[ {\sum\limits_{i = 1}^{4} {(x_{i} - a_{ij} )^{2} + c_{j} } } \right]}^{ - 1}\), \(\,\,{\text{where }}\,0 \le x_{i} \le 10\) and \(m = 10\),

    $$a_{ij} = \left[ {\begin{array}{*{20}c} 4 & 1 & 8 & 6 & 3 & 2 & 5 & 8 & 6 & 7 \\ 4 & 1 & 8 & 6 & 7 & 9 & 5 & 1 & 2 & {3.6} \\ 4 & 1 & 8 & 6 & 3 & 2 & 3 & 8 & 6 & 7 \\ 4 & 1 & 8 & 6 & 7 & 9 & 3 & 1 & 2 & {3.6} \\ \end{array} } \right]$$

    \({\text{and}}\,\,c_{j} = [0\begin{array}{*{20}c} {.1} & {0.2} & {0.2} & {0.4} & {0.4} & {0.6} & {0.3} & {0.7} & {0.5} & {0.5} \\ \end{array} ]^{\rm T}\), with the known global optimum \(f_{25} ({\mathbf{x}}^{*} ) = - 10.5364\) at \(x_{i}^{*} = 4\), for \(i = 1,2,3,4.\)

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