A novel hybrid genetic algorithm for constrained optimization

Abstract

Finding global optimal solution of a non-linear constrained optimization problem with high complexity is a challenge for the researchers. Now days, real coded genetic algorithm (GA) becomes popular to solve them, due to their diversity preserving mechanism. In recent literature it is proved that for solving constrained optimization problem, the real coded GA (LX-PM) that uses Laplace Crossover and Power Mutation, is much efficient. In this paper an attempt is made to improve the performance of LX-PM, hybridizing with Quadratic Approximation. The efficiency and reliability of the design hybrid algorithm is realized through a set of 15 benchmark test problems.

Appendix

Problem 1

(Salkin 1975)

$$ \mathop {Max}\limits_{x} f\left( x \right) = 3x_{1} + x_{2} + 2x_{3} + x_{4} - x_{5} $$

Subject to

$$ \begin{gathered} 25x_{1} - 40x_{2} + 16x_{3} + 21x_{4} + x_{5} \le 300 \hfill \\ x_{1} + 20x_{2} - 50x_{3} + x_{4} - x_{5} \le 200 \hfill \\ 60x_{1} + x_{2} - x_{3} + 2x_{4} + x_{5} \le 600 \hfill \\ - 7x_{1} + 4x_{2} + 15x_{3} - x_{4} + 65x_{5} \le 700 \hfill \\ 1 \le x_{1} \le 4;\,80 \le x_{2} \le 88;\,30 \le x_{3} \le 35;\,145 \le x_{4} \le 150;\,0 \le x_{5} \le 2 \hfill \\ \end{gathered} $$

Solution: x = (4, 88, 35, 150, 0)^T, $f^{*} \left( x \right) = 320$

Problem 2

(Himmelblau 1972)

$$ \begin{gathered} \mathop {Max}\limits_{x} f\left( x \right) = 25\left( {x_{1} - 2} \right)^{2} + \left( {x_{2} - 2} \right)^{2} + \left( {x_{3} - 1} \right)^{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {x_{4} - 4} \right)^{2} + \left( {x_{5} - 1} \right)^{2} + \left( {x_{6} - 4} \right)^{2} \hfill \\ \end{gathered} $$

Subject to

$$ \begin{gathered} x_{1} + x_{2} - 2 \ge 0 \hfill \\ 6 - x_{1} + x_{2} \ge 0 \hfill \\ 2 + x_{1} - x_{2} \ge 0 \hfill \\ 2 - x_{1} + 3x_{2} \ge 0 \hfill \\ \left( {x_{3} - 3} \right)^{2} + x_{4} - 4 \ge 0 \hfill \\ \left( {x_{5} - 3} \right)^{2} + x_{6} - 4 \ge 0 \hfill \\ \end{gathered} $$

$$ \begin{gathered} 0 \le x_{1} \le 5;\,0 \le x_{2} \le 1;\,1 \le x_{3} \le 5;\,0 \le x_{4} \le 6;\, \hfill \\ 0 \le x_{5} \le 5;\,0 \le x_{6} \le 10; \hfill \\ \end{gathered} $$

.

Solution: x = (5, 1, 5, 0, 5, 10)^T, $f^{*} \left( x \right) = 310$

Problem 3

(Schittkowski 1987)

$$ \mathop {Min}\limits_{x} f\left( x \right) = \left( {x_{1}^{2} + x_{2} - 11} \right)^{2} + \left( {x_{1}^{{}} + x_{2}^{2} - 7} \right)^{2} $$

Subject to

$$ \begin{gathered} 4.84 - \left( {x_{1} - 0.05} \right)^{2} - \left( {x_{2} - 2.5} \right)^{2} \ge 0 \hfill \\ x_{1}^{2} + \left( {x_{2} - 2.5} \right)^{2} - 4.84 \ge 0 \hfill \\ \end{gathered} $$

$$ 0 \le x_{1} \le 6;\,0 \le x_{2} \le 6 $$

Solution: x = (2.246826, 2.381865)^T, $f^{*} \left( x \right) = 1 3. 5 90 8 5$

Problem 4

(Michalewicz 1996)

$$ \begin{gathered} \,\mathop {Min}\limits_{x} f\left( x \right) = \left( {x_{1} - 10} \right)^{2} + 5\left( {x_{2} - 12} \right)^{2} + x_{3}^{4} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + 3\left( {x_{4} - 11} \right)^{2} \,\, + 10x_{5}^{6} + 7x_{6}^{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + x_{7}^{4} - 4x_{6} x_{7}^{{}} - 10x_{6} - 8x_{7}^{{}} \hfill \\ \end{gathered} $$

Subject to

$$ \begin{gathered} 127 - 2x_{1}^{2} - 3x_{2}^{4} - x_{3} - 4x_{4}^{2} - 5x_{5} \ge 0 \hfill \\ 282 - 7x_{1} - 3x_{2} - 10x_{3}^{2} - x_{4} + x_{5} \ge 0 \hfill \\ 196 - 23x_{1} - x_{2}^{2} - 6x_{6}^{2} + 8x_{7} \ge 0 \hfill \\ - 4x_{1}^{2} - x_{2}^{2} + 3x_{1} x_{2} - 2x_{3}^{2} - 5x_{6} + 11x_{7} \ge 0 \hfill \\ - 10 \le x_{i} \le 10,i = 1,2 \ldots ,7. \hfill \\ \end{gathered} $$

Solution: x = (2.330499, 1.951372, −0.4775414, 4.365726, −0.6244870, 1.038131, 1.594227)^T $f^{*} \left( x \right) = 6 80. 6 300 5 7 3$

Problem 5

(Floudas and Pardalos 1990)

$$ \mathop {Min}\limits_{x} f\left( x \right) = x_{1}^{0.6} + x_{2}^{0.6} + x_{3}^{0.4} + 2x_{4} + 5x_{5} - 4x_{3} - x_{6} $$

Subject to

$$ \begin{gathered} x_{2} - 3x_{1} - 3x_{4} = 0 \hfill \\ x_{3} - 2x_{2} - 3x_{5} = 0 \hfill \\ 4x_{4} - x_{6} = 0 \hfill \\ x_{1} + 2x_{4} - 4 \le 0 \hfill \\ x_{2} + x_{5} - 4 \le 0 \hfill \\ x_{3} + x_{6} - 6 \le 0 \hfill \\ \end{gathered} $$

$$ \begin{gathered} 0 \le x_{1} \le 3;\,0 \le x_{2} \le 4;\,1 \le x_{3} \le 4;\,0 \le x_{4} \le 2;\, \hfill \\ 0 \le x_{5} \le 2;\,0 \le x_{6} \le 6; \hfill \\ \end{gathered} $$

Solution: x = (0.67, 2, 4, 0, 0, 0)^T, $f^{*} \left( x \right) = - 1 1. 9 6$

Problem 6

(Levy and Montalvo 1985)

$$ \mathop {Min}\limits_{x} f\left( x \right) = - x_{1} - x_{2} $$

Subject to

$$ \begin{gathered} x_{2} \le 2 + 2x_{1}^{4} - 8x_{1}^{3} + 8x_{1}^{2} \hfill \\ x_{2} \le 4x_{1}^{4} - 32x_{1}^{3} + 88x_{1}^{2} - 96x_{1} + 36 \hfill \\ \end{gathered} $$

$$ 0 \le x_{1} \le 3;\,0 \le x_{2} \le 4 $$

Solution: x = (2.3295, 3.1783)^T, $f^{*} \left( x \right) = - 5. 50 7 8$

Problem 7

(Floudas and Pardalos 1990)

$$ \mathop {Min}\limits_{x} f\left( x \right) = - 12x_{1} - 7x_{2} + x_{2}^{2} $$

Subject to

$$ - 2x_{1}^{4} - x_{2} + 2 = 0 $$

$$ 0 \le x_{1} \le 2;\,0 \le x_{2} \le 3 $$

Solution: x = (0.7175, 1.47)^T, $f^{*} \left( x \right) = - 1 6. 7 3 9 1$

Problem 8

(Michalewicz 1996)

$$ \begin{gathered} \mathop {\,Min}\limits_{x} f\left( x \right) = \,\,3x_{1} \, + \,0.000001\,x_{1}^{3} \, + \,2\,x_{2} \, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\left( {0.000002\,/\,3} \right)\,x_{2}^{3} \hfill \\ \end{gathered} $$

Subject to

$$ \begin{gathered} x_{4} - x_{3} + 0.55 \ge 0 \hfill \\ x_{3} - x_{4} + 0.55 \ge 0 \hfill \\ 1000\sin ( - x_{3} - 0.25) + 1000\sin ( - x_{4} - 0.25) + 894.8 - x_{1} = 0 \hfill \\ 1000\sin (x_{3} - 0.25) + 1000\sin (x_{3} - x_{4} - 0.25) + 894.8 - x_{2} = 0 \hfill \\ 1000\sin (x_{4} - 0.25) + 1000\sin (x_{4} - x_{3} - 0.25) + 1294.8 = 0 \hfill \\ \end{gathered} $$

$$ 0 \le x_{i} \le 1200,\,\,i = 1,\,2;\, - 0.55 \le x_{i} \le 0.55,\,\,i = 3,4 $$

Solution: x = (679.9453, 1,026.067, 0.1188764, −0.3962336)^T, $f^{*} \left( x \right) = 5 1 2 6. 4 9 8 1$

Problem 9, 10, 11

(Michalewicz and Naguib 1994)

$$ \mathop {Min}\limits_{x} f\left( x \right) = \left\{ \begin{gathered} f_{1} = x{}_{2} + 10^{ - 5} (x_{2} - x_{1} )^{2} - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,0 \le x_{1} < 2 \hfill \\ f_{2} = \frac{1}{27\sqrt 3 }\left( {\left( {x_{1} - 3} \right)^{2} - 9} \right)x_{2}^{3} \,\,\,\,\,\,\,\,\,if\,\,2 \le x_{1} < 4 \hfill \\ f_{3} = \frac{1}{3}\left( {x_{1} - 2} \right)^{3} + x_{2} - \frac{11}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,4 \le x_{1} \le 6 \hfill \\ \end{gathered} \right\} $$

Subject to

$$ \begin{gathered} \frac{{x_{1} }}{\sqrt 3 } - x_{2} \, \ge \,0 \hfill \\ - x_{1} - \sqrt 3 x_{2} + 6\, \ge \,0 \hfill \\ \end{gathered} $$

$$ 0 \le x_{2} \le 5 $$

Solution: x = (0, 0) ^T, (3, $\sqrt 3$)^T, (4, 0)^T, $f^{*} \left( x \right) = - 1$

Problem 12

(Kim and Myung 1996)

$$ \mathop {Min}\limits_{x} f\left( x \right) = 100\left( {x_{2} - x_{1}^{2} } \right)^{2} + \left( {1 - x_{1} } \right)^{2} $$

Subject to

$$ \begin{gathered} - x_{1} - x_{2}^{2} \le 0 \hfill \\ - x_{1}^{2} - x_{2}^{{}} \le 0 \hfill \\ \end{gathered} $$

$$ - 0.5 \le x_{1} \le 0.5;\;0 \le x_{2} \le 1 $$

Solution: x = (0.5, 0.25) ^T, $f^{*} \left( x \right) = 0. 2 5$

Problem 13

(Kim and Myung 1996)

$$ \mathop {Min}\limits_{x} f\left( x \right) = 0.01 \times \left( {x_{1}^{2} + x_{2}^{2} } \right) $$

Subject to

$$ \begin{gathered} - x_{1} x_{2} + 25 \le 0 \hfill \\ - x_{1}^{2} - x_{2}^{2} + 25 \le 0 \hfill \\ \end{gathered} $$

$$ 2 \le x_{1} \le 50;\,0 \le x_{2} \le 50 $$

Solution: x = $\left( {\sqrt {250} ,\sqrt {250} } \right)^{T}$, $f^{*} \left( x \right) = 0. 5$

Problem 14

(Chootinan and Chen 2006)

$$ \mathop {Max}\limits_{x} f\left( x \right) = \frac{{\sin^{3} \left( {2\pi x_{1} } \right)\sin \left( {2\pi x_{2} } \right)}}{{x_{1}^{3} \left( {x_{1} + x_{2} } \right)}} $$

Subject to

$$ \begin{gathered} x_{1}^{2} - x_{2}^{{}} + 1 \le 0 \hfill \\ 1 - x_{1} + \left( {x_{2} - 4} \right)^{2} \le 0 \hfill \\ \end{gathered} $$

$$ 0 \le x_{1} \le 10;\,0 \le x_{2} \le 10 $$

Solution: x = (1.2279713, 4.2453733)^T, $f^{*} \left( x \right) = 0.0 9 5 8 2 5$

Problem 15

(Michalewicz and Schoenauer 1996)

$$ \mathop {Min}\limits_{x} f\left( x \right) = \left( {x_{1} - 10} \right)^{3} + \left( {x_{2} - 20} \right)^{3} $$

Subject to

$$ \begin{gathered} \left( {x_{1} - 5} \right)^{2} + \left( {x_{2} - 5} \right) - 100 \ge 0 \hfill \\ - \left( {x_{1} - 6} \right)^{2} - \left( {x_{2} - 5} \right) + 82.82 \ge 0 \hfill \\ \end{gathered} $$

$$ 13 \le x_{1} \le 100;\,0 \le x_{2} \le 100 $$

Solution: x = (14.095, 0.84296)^T, $f^{*} \left( x \right) = -6 9 6 1. 8 1 3 8 1$