C-SOMAQI: Self Organizing Migrating Algorithm with Quadratic Interpolation Crossover Operator for Constrained Global Optimization

Abstract

SOMAQI is a variant of Self Organizing Migrating Algorithm (SOMA) in which SOMA is hybridized with Quadratic Interpolation crossover operator, presented by Singh et al. (Advances in intelligent and soft computing. Springer, India, pp. 225–234, 2014). The algorithm SOMAQI has been designed to solve unconstrained nonlinear optimization problems. Earlier it has been tested on several benchmark problems and the results obtained by this technique outperform the results taken by several other techniques in terms of population size and function evaluations. In this chapter SOMAQI has been extended for solving constrained nonlinear optimization problems (C-SOMAQI) by including a penalty parameter free approach to select the feasible solutions. This algorithm also works with small population size and converges very fast. A set of 10 constrained optimization problems has been used to test the performance of the proposed algorithm. These problems are varying in complexity. To validate the efficiency of the proposed algorithm results are compared with the results obtained by C-SOMGA and C-SOMA. On the basis of the comparison it has been concluded that C-SOMAQI is efficient to solve constrained nonlinear optimization problems.

Appendix

Problem 1

$$ \begin{aligned} & \mathop {{{\min}}}\limits_{x} \;f\left( x \right) = - x_{1} - x_{2} , \\ & {\text{Subject to:}} \\ & g_{1} \left( x \right) = \left[ {\left( {x_{1} - 1} \right)^{2} + \left( {x_{2} - 1} \right)} \right]\left[ {{1 \mathord{\left/ {\vphantom {1 {2a^{2} - {1 \mathord{\left/ {\vphantom {1 {2b^{2} }}} \right. \kern-0pt} {2b^{2} }}}}} \right. \kern-0pt} {2a^{2} - {1 \mathord{\left/ {\vphantom {1 {2b^{2} }}} \right. \kern-0pt} {2b^{2} }}}}} \right] \\ & \quad \quad \qquad + \left( {x{}_{1} - 1} \right)\left( {x_{2} - 1} \right)\left[ {{1 \mathord{\left/ {\vphantom {1 {a^{2} - {1 \mathord{\left/ {\vphantom {1 {b^{2} }}} \right. \kern-0pt} {b^{2} }}}}} \right. \kern-0pt} {a^{2} - {1 \mathord{\left/ {\vphantom {1 {b^{2} }}} \right. \kern-0pt} {b^{2} }}}}} \right] - 1 \ge 0. \end{aligned}$$

where a = 2, b = 0.25.

This problem has two global minima one of which is at (1, 0.8729) with f_min = −1.8729. The feasible domain of the problem is disconnected. The bounds on the variables are 0 ≤ x₁, x₂ ≤ 1.

Problem 2

$$ \begin{aligned} & \mathop {{{\min}}}\limits_{x} \;f\left( x \right) = 3x_{1} + x_{2} + 2x{}_{3} + x_{4} - x_{5} , \\ & {\text{Subject to:}} \\ & g_{1} \left( x \right) = 25x_{1} - 40x_{2} + 16x_{3} + 21x_{4} + x_{5} \le 300, \\ & g_{2} \left( x \right) = x_{1} + 20x_{2} - 50x_{3} + x_{4} - x_{5} \le 200, \\ & g_{3} \left( x \right) = 60x_{1} + x_{2} - x_{3} + 2x_{4} + x_{5} \le 600, \\ & g_{4} \left( x \right) = - 7x{}_{1} + 4x_{2} + 15x_{3} - x_{4} + 65x_{5} \le 700. \\ \end{aligned}$$

This problem has global minima at (4, 88, 35, 150, 0) with f _min = 320. The bounds on the variables are 1 ≤ x₁ ≤ 4, 80 ≤ x₂ ≤ 88, 30 ≤ x₃ ≤ 35, 145 ≤ x₄ ≤ 150, 0 ≤ x₅ ≤ 2.

Problem 3

$$ \begin{aligned} & \mathop {{{\min}}}\limits_{x} \;f\left( x \right) = 4.3x_{1} + 31.8x_{2} + 63.3x{}_{3} + 15.8x_{4} + 68.5x_{5} + 4.7x_{6} ,\\ & {\text{Subject to:}} \\ & g_{1} \left( x \right) = 17.1x_{1} + 38.2x_{2} + 204.2x_{3} + 212.3x_{4} + 623.4x_{5} + 1495.5x_{6} - 169x_{1} x_{3} - 3580x_{3} x_{5} \\ & \quad \qquad - 3810x_{4} x_{5} - 18500x_{4} x_{6} - 24300x_{5} x_{6} - 4.97 \ge 0, \\& g_{2} \left( x \right) = 1.88 + 17.9x_{1} + 36.8x_{2} + 113.9x_{3} + 169.7x_{4} + 337.8x_{5} + 1385.2x_{6} - 139x_{1} x_{3} \\ & \quad \qquad - 2450x_{4} x_{5} - 600x_{4} x_{6} - 17200x_{5} x_{6} \ge 0, \\& g_{3} \left( x \right) = 429.08 - 273x_{2} - 70x_{4} - 819x_{5} + 26000x_{4} x_{5} \ge 0, \\ &g_{4} \left( x \right) = 159.9x{}_{1} - 311x_{2} + 587x_{4} + 391x_{5} + 2198x_{6} - 14000x_{1} x_{6} + 78.02 \ge 0. \\ \end{aligned} $$

This problem has global minima at (0, 0, 0, 0, 0, 0.00333) with f_min = 0.0156. The bounds on the variables are 0 ≤ x₁ ≤ 0.31, 0 ≤ x₂ ≤ 0.046, 0 ≤ x₃ ≤ 0.068, 0 ≤ x₄ ≤ 0.042, 0 ≤ x₅ ≤ 0.028, 0 ≤ x₆ ≤ 0.0134.

Problem 4

$$ \begin{aligned} & \mathop {{\max} }\limits_{x} \;f\left( x \right) = 25\left( {x_{1} - 2} \right)^{2} + \left( {x_{2} - 2} \right)^{2} + \left( {x_{3} - 1} \right)^{2} + \left( {x_{4} - 4} \right)^{2} + \left( {x_{5} - 1} \right)^{2} + \left( {x_{6} - 4} \right)^{2} , \\ & {\text{Subject to:}} \\ & g{}_{1}\left( x \right) = x_{1} + x_{2} - 2 \ge 0, \\ & g_{2} \left( x \right) = - x_{1} + x_{2} + 6 \ge 0, \\ & g{}_{3}\left( x \right) = x_{1} - x_{2} + 2 \ge 0, \\ & g{}_{4}\left( x \right) = - x_{1} + 3x_{2} + 2 \ge 0, \\ & g{}_{5}\left( x \right) = \left( {x_{3} - 3} \right)^{2} + x_{4} - 4 \ge 0, \\ & g{}_{6}\left( x \right) = \left( {x_{5} - 3} \right)^{2} + x_{6} - 4 \ge 0, \\ \end{aligned} $$

This problem has 18 global maxima and one global maxima at (5, 1, 5, 0, 5, 10) with f_max = 310. The bounds on the variables are 0 ≤ x₁ ≤ 5, 0 ≤ x₂ ≤ 1, 1 ≤ x₃ ≤ 5, 0 ≤ x₄ ≤ 6, 0 ≤ x₅ ≤ 5, 0 ≤ x₆ ≤ 10.

Problem 5

$$ \begin{aligned} & \mathop {{{\min}}}\limits_{x} \;f\left( x \right) = \left( {x_{1}^{2} + x_{2} - 11} \right)^{2} + \left( {x_{1} + x_{2}^{2} - 7} \right)^{2} , \\ & {\text{Subject to:}} \\ & g{}_{1}\left( x \right) = 4.84 - \left( {x_{1} - 0.05} \right)^{2} - \left( {x_{2} - 2.5} \right)^{2} \ge 0, \\ & g_{2} \left( x \right) = x_{1}^{2} + \left( {x_{2} - 2.5} \right)^{2} - 4.84 \ge 0. \\ \end{aligned} $$

This problem has two decision variables. It has global minima at $x^{*} = \left( {2.246826,\,2.381865} \right)$ with f_min = 13.59085. The bounds on the variables are $0 \le x_{i} \le 6,\;for\;i = 1,\,2.$

Problem 6

$$ \begin{aligned} \mathop {{{\min}}}\limits_{x} \;f\left( x \right) & = 5\sum\limits_{i = 1}^{4} {x_{i} } - 5\sum\limits_{i = 1}^{4} {x_{i}^{2} } - \sum\limits_{i = 5}^{13} {x_{i} } , \\ g{}_{1}\left( x \right) & = 10 - (2x_{1} + 2x_{2} + x_{10} + x_{11} ) \ge 0, \\ g_{2} \left( x \right) & = 10 - (2x_{1} + 2x_{3} + x_{10} + x_{12} ) \ge 0, \\ g{}_{3}\left( x \right) & = 10 - (2x_{2} + 2x_{3} + x_{11} + x_{12} ) \ge 0, \\ g{}_{4}\left( x \right) & = 8x_{1} - x_{10} \ge 0, \\ g{}_{5}\left( x \right) & = 8x_{2} - x_{11} \ge 0, \\ g{}_{6}\left( x \right) & = 8x_{3} - x_{12} \ge 0, \\ g{}_{7}\left( x \right) & = 2x_{4} + x_{5} - x_{10} \ge 0, \\ g{}_{8}\left( x \right) & = 2x_{6} + x_{7} - x_{11} \ge 0, \\ g{}_{9}\left( x \right) & = 2x_{8} + x_{9} - x_{12} \ge 0, \\ x_{i} & \ge 0,\quad i = 1, \ldots ,13, \\ x_{i} & \le 1,\quad i = 1, \ldots 9,13. \\ \end{aligned} $$

This problem has global minima at $x^{*} = \left( {1,\,1,\, \ldots 1,\,3,\,3,\,3,\,1} \right)$ with f_min = −15. The bounds on the variables are $0 \le x_{i} \le u_{i} ,\;i = 1,2, \ldots ,n,$ where $u = \left( {1,\,1, \ldots ,1,\,100,\,100,\,100,\,1} \right).$

Problem 7

$$ \begin{aligned} & \mathop {{{\max}}}\limits_{x} \;f\left( x \right) = \frac{{\sin^{3} \left( {2\pi x_{1} } \right)\sin \left( {2\pi x_{2} } \right)}}{{x_{1}^{3} \left( {x_{1} + x_{2} } \right)}}, \\ & {\text{Subject to:}} \\ & g_{1} \left( x \right) = - x_{1}^{2} + x_{2} - 1 \ge 0, \\ & g_{2} \left( x \right) = - 1 + x_{1} - \left( {x_{2} - 4} \right)^{2} \ge 0. \\ \end{aligned} $$

This problem has global maxima at $x^{*} = \left( {1.2279713,\,4.2453733} \right),\;f\left( {x^{*} } \right) = 0.095$ with f_max = 0.095. The bounds on the variables are $0 \le x_{i} \le 10,\;i = 1,2.$

Problem 8

$$ \begin{aligned} & \mathop {{{\min}}}\limits_{x} \;f\left( x \right) = x_{1}^{2} + \left( {x_{2} - 1} \right)^{2}\\ & {\text{Subject to:}} \\ & h_{1} \left( x \right) = x{}_{2} - x_{1}^{2} = 0. \\ \end{aligned} $$

This problem has global minima at $x^{*} = \pm \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {2^{0.5} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2^{0.5} }$}},{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right),\;f\left( {x^{*} } \right) = 0.75$ with f_min = 0.75. The bounds on the variables are: $- 1 \le x_{i} \le 1,\;i = 1,2.$

Problem 9

$$ \begin{aligned} & \mathop {{{\max}}}\limits_{x} \;f\left( x \right) = \left( {\sqrt n } \right)^{n} \prod\limits_{i = 1}^{n} {x_{i} } , \\ & {\text{Subject to:}} \\ &h_{1} \left( x \right) = \sum\limits_{i = 1}^{n} {x_{i}^{2} } - 1 = 0.\\ \end{aligned} $$

This problem has global maxima at $x^{*} = \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {n^{0.5} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n^{0.5} }$}}, \ldots {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {n^{0.5} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${n^{0.5} }$}}} \right)$ with f_max = 1. The bounds on the variables are: $0 \le x_{i} \le 1,\;i = 1,2, \ldots ,n.$

Problem 10

$$ \begin{aligned} & \mathop {{{\max}}}\limits_{x} \;f\left( x \right) = \left| {\frac{{\sum\nolimits_{i = 1}^{n} {\cos^{4} \left( {x_{i} } \right) - 2\prod\nolimits_{i = 1}^{n} {\cos^{2} \left( {x_{i} } \right)} } }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {ix_{i}^{2} } } }}} \right|\,,\\ & {\text{Subject to:}} \\ & g{}_{1}\left( x \right) = \prod\limits_{i = 1}^{n} {x_{i} } - 0.75 \ge 0, \\ & g_{2} \left( x \right) = 7.5n - \sum\limits_{i = 1}^{n} {x_{i} } \ge 0, \\ \end{aligned} $$

This problem has global maxima at f_max = 0.803619 for n = 20. The bounds on the variables are: $0 \le x_{i} \le 10,\;i = 1,2, \ldots ,n.$