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Constrained Laplacian biogeography-based optimization algorithm

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Abstract

Biogeography-based optimization (BBO) is a relatively new nature inspired optimization technique proposed by Dan Simon for unconstrained optimization, which was later generalized and improved by Happing Ma and Dan Simon for constrained optimization, called blended biogeography-based optimization. In an earlier paper, the authors have proposed a Laplacian biogeography-based optimization algorithm (LX-BBO) for unconstrained optimization. The purpose of the present paper is to generalize the LX-BBO from the unconstrained case to the constrained case. This is done by using the Deb’s constrained handling method. In order to evaluate the performance of the proposed constrained LX-BBO for constrained optimization problems, five different constrained optimization problems and popular CEC 2006 benchmark collection is used. Based on the analysis of results it is shown that the proposed Constrained LX-BBO outperforms Blended BBO for constrained optimization.

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Acknowledgements

The first author acknowledges the financial support offered by the MHRD under the grant to conduct this research.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Roorkee, India

    Vanita Garg & Kusum Deep

Authors
  1. Vanita Garg
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  2. Kusum Deep
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Corresponding author

Correspondence to Vanita Garg.

Appendix

Appendix

Test problem COP-1

$$\begin{aligned} & \hbox{min} \quad f\left( X \right) = \left( {x - 2} \right)^{2} + \left( {y - 1} \right)^{2} \\ & s.t. \quad g_{1} \left( X \right) :\; x + y - 2 \le 0 \\ &\quad g_{2} \left( X \right) :\; x^{2} - y + 2 \le 0 \\ &\quad - 5 \le x,y \le 5 \\ \end{aligned}$$

Test problem COP-2

$$\begin{aligned} & \hbox{min} \quad f\left( X \right) = \left( {x_{1}^{2} + x_{2} - 11} \right)^{2} + \left( {x_{2}^{2} + x_{1} - 7} \right)^{2} \\ & s.t.\quad g_{1} \left( X \right) :\; \left( {x_{1} - 0.05} \right)^{2} + \left( {x_{2} - 2.5} \right)^{2} - 4.84 \le 0 \\ &\quad\quad g_{2} \left( X \right) :\; 4.84 - x_{1}^{2} - \left( {x_{2} - 2.5} \right)^{2} \le 0 \\ &\quad\quad 0 \le x_{i} \le 6, \quad i = 1,2 \\ \end{aligned}$$

Test problem COP-3

$$\begin{aligned} & \hbox{min} \quad f\left( X \right) = \left( {x_{1} - x_{2} } \right)^{2} + \left( {x_{2} + x_{3} - 2} \right)^{2} + \left( {x_{4} - 1} \right)^{2} + \left( {x_{5} - 1} \right)^{2} \\ & s.t.\quad h_{1} \left( X \right): x_{1} + 3x_{2} = 0 \\ &\quad\quad h_{2} \left( X \right) :\; x_{3} + x_{4} - 2x_{5} = 0 \\ &\quad\quad h_{3} \left( X \right) :\; x_{2} - x_{5} = 0 \\ &\quad\quad - 10 \le x_{i} \le 10,\quad i = 1,2,3,4,5 \\ &\quad x^{*} = \left( {\frac{ - 33,11,27, - 5,11}{43}} \right),\quad f\left( {x^{*} } \right) = 176/43 \\ \end{aligned}$$

Test problem COP-4

$$\begin{aligned} & \hbox{min} \quad f\left( X \right) = 3x_{1} + x_{1}^{3} *10^{ - 6} + 2x_{2} + \frac{2}{3}x_{2}^{3} *10^{ - 6} \\ & s.t.\quad g_{1} \left( X \right): \left( {x_{3} - x_{4} } \right) - 0.48 \le 0 \\ &\quad\quad g_{2} \left( X \right) :\; \left( {x_{4} - x_{3} } \right) - 0.48 \le 0 \\ &\quad\quad h_{1} \left( X \right) :\;1000({ \sin }( - x_{3} - 0.25) + { \sin }\left( { - x_{4} - 0.25} \right)) - x_{1} + 894.8 = 0 \\ &\quad\quad h_{2} \left( X \right) :\;1000({ \sin }(x_{3} - 0.25) + { \sin }\left( {x_{3} - x_{4} - 0.25} \right)) - x_{2} + 894.8 = 0 \\ &\quad\quad h_{3} \left( X \right) :\;1000({ \sin }(x_{4} - 0.25) + { \sin }\left( {x_{4} - x_{3} - 0.25} \right)) - x_{1} + 1294.8 = 0 \\ &\quad xL \le x_{i} \le xU, \quad i = 1,2,3,4 \\ &\quad xL = \left[ {0,0, - 0.48, - 0.48} \right] \\ &\quad xU = \left[ {1200,1200,0.48,0.48} \right] \\ &\quad x^{*} = \left( {76.159201,925.195139,0.051108, - 0.428810} \right),\quad f\left( {x^{*} } \right) = 5174.412675 \\ \end{aligned}$$

Test problem COP-5

$$\begin{aligned} & \hbox{min} \quad f\left( X \right) = x_{1} + x_{2} + x_{3} \\ & s.t.\quad g_{1} \left( X \right) :\; 0.0025\left( {x_{4} + x_{6} } \right) - 1 \le 0 \\ &\quad g_{2} \left( X \right) :\; 0.0025\left( {x_{5} + x_{7} - x_{4} } \right) - 1 \le 0 \\ &\quad g_{3} \left( X \right): :\;0.01\left( {x_{8} - x_{5} } \right) - 1 \le 0 \\ &\quad g_{4} \left( X \right) :\; - x_{1} x_{6} + 833.33252x_{4} + 100x_{1} - 83333.333 \le 0 \\ &\quad g_{5} \left( X \right) :\; - x_{3} x_{7} + 1250x_{5} + x_{2} x_{4} - 1250x_{4} \le 0 \\ &\quad g_{6} \left( X \right) :\; - x_{3} x_{8} - 2500x_{5} + x_{3} x_{5} \mp 1250000 \le 0 \\ &\quad\quad 100 \le x_{1} \le 10000 \\ &\quad\quad 1000 \le x_{i} \le 10000,\quad i = 2,3 \\ &\quad 10 \le x_{i} \le 1000, i = 4,5,6,7,8 \\ &\quad f\left( {x^{*} } \right) = 7049.2480205 \\ \end{aligned}$$

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Garg, V., Deep, K. Constrained Laplacian biogeography-based optimization algorithm. Int J Syst Assur Eng Manag 8 (Suppl 2), 867–885 (2017). https://doi.org/10.1007/s13198-016-0539-7

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