An efficient chaotic water cycle algorithm for optimization tasks

Abstract

Water cycle algorithm (WCA) is a new population-based meta-heuristic technique. It is originally inspired by idealized hydrological cycle observed in natural environment. The conventional WCA is capable to demonstrate a superior performance compared to other well-established techniques in solving constrained and also unconstrained problems. Similar to other meta-heuristics, premature convergence to local optima may still be happened in dealing with some specific optimization tasks. Similar to chaos in real water cycle behavior, this article incorporates chaotic patterns into stochastic processes of WCA to improve the performance of conventional algorithm and to mitigate its premature convergence problem. First, different chaotic signal functions along with various chaotic-enhanced WCA strategies (totally 39 meta-heuristics) are implemented, and the best signal is preferred as the most appropriate chaotic technique for modification of WCA. Second, the chaotic algorithm is employed to tackle various benchmark problems published in the specialized literature and also training of neural networks. The comparative statistical results of new technique vividly demonstrate that premature convergence problem is relieved significantly. Chaotic WCA with sinusoidal map and chaotic-enhanced operators not only can exploit high-quality solutions efficiently but can outperform WCA optimizer and other investigated algorithms.

Appendix

List of 13 constrained benchmark problems:

Problem G ₁

$$\mathop {\hbox{min} }\limits_{x} \, f(x) = 5\sum\nolimits_{i = 1}^{4} {x_{i} } - 5\sum\nolimits_{i = 1}^{4} {x_{i}^{2} } - \sum\nolimits_{i = 5}^{13} {x_{i} }$$

$$\begin{aligned} {\text{s}}. {\text{t}}. {\text{ g}}_{1} (x) &= 2x_{1} + 2x_{2} + x_{10} + x_{11} - 10 \le 0, \hfill \\ {\text{ g}}_{2} (x) &= 2x_{1} + 2x_{3} + x_{10} + x_{12} - 10 \le 0, \hfill \\ {\text{ g}}_{3} (x) &= 2x_{2} + 2x_{3} + x_{11} + x_{12} - 10 \le 0, \hfill \\ {\text{ g}}_{4} (x) &= - 8x_{1} + x_{10} \le 0, \hfill \\ {\text{ g}}_{5} (x) &= - 8x_{2} + x_{11} \le 0, \hfill \\ \, g_{6} (x) &= - 8x_{3} + x_{12} \le 0, \hfill \\ \, g_{7} (x) &= - 2x_{4} - x_{5} + x_{10} \le 0, \hfill \\ \, g_{8} (x) &= - 2x_{6} - x_{7} + x_{11} \le 0, \hfill \\ \, g_{9} (x) &= - 2x_{8} - x_{9} + x_{12} \le 0, \hfill \\ \, & \quad x_{i} \ge 0, \, i = 1, \ldots ,13, \hfill \\ & \quad x_{i} \le 1, \, i = 1, \ldots 9,13. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (1, 1, 1, 1, 1, 1, 1, 1, 1, 100, 100, 100, 1),\, LB}} = (0, \ldots ,0), \hfill \\ {\text{Global minimum: \textit{x}}}^{best} {\text{ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1), f(\textit {x}}}^{best} ) { = } - 15. \hfill \\ \end{aligned}$$

Problem G ₂

$$\mathop {\hbox{max} }\limits_{x} \, f(x) = \left| {\frac{{\sum\nolimits_{i = 1}^{n} {\cos^{4} (x_{i} ) - 2\prod\nolimits_{i = 1}^{n} {\cos^{2} } (x_{i} )} }}{{\sqrt {\sum\nolimits_{i = 1}^{n} {ix_{i}^{2} } } }}} \right|$$

$$\begin{aligned} {\text{s}}. \quad {\text{t}}. { }g_{1} (x) = - \prod\nolimits_{i = 1}^{n} {x_{i} } + 0.75 \le 0, \hfill \\ \, g_{2} (x) = - \sum\nolimits_{i = 1}^{n} {x_{i} } - 7.5n \le 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound}}\,{\text{restrictions: UB = (10,}} \ldots , { 10)}\,{\text{ and}}\,{\text{ LB = (0,}} \ldots , { 0)}.\hfill \\ {\text{Best}}\,{\text{known}}\,{\text{value: f (}}x^{best} ) { = 0}. 8 0 3 6 1 9 , { }\quad {\text{for \textit{n} = 20}}.\hfill \\ \end{aligned}$$

Problem G ₃

$$\begin{aligned} \mathop {\hbox{max} }\limits_{x} \, f(x) = (\sqrt n )^{n} \prod\nolimits_{i = 1}^{n} {x_{i} } \hfill \\ {\text{s}}. {\text{t}}. \quad { }h_{1} (x) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } - 1 = 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} & {\text{Bound restrictions: UB = (1,}} \ldots , {\text{ 1) and LB = (0,}} \ldots , { 0)}.\hfill \\ & {\text{Global maximum: x}}^{best} = (\frac{1}{\sqrt n }, \ldots ,\frac{1}{\sqrt n }), \, f (x^{best} ) { = 1}.\hfill \\ \end{aligned}$$

Problem G ₄

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) &= 5.3578547x_{3}^{2} + 0. 8 3 5 6 8 9 1x_{1} x_{5} + 3 7. 2 9 3 2 3 9 {\text{x}}_{1} - 4 0 7 9 2. 1 4 1\hfill \\ {\text{s}}. {\text{t}}. \quad g_{1} (x) &= u(x) - 92 \le 0, \hfill \\ \, g_{1} (x) &= - u(x) \le 0, \hfill \\ \, g_{3} (x) &= v(x) - 110 \le 0, \hfill \\ \, g_{4} (x) &= - v(x) + 90 \le 0, \hfill \\ \, g_{5} (x) &= w(x) - 25 \le 0, \hfill \\ \, g_{6} (x) &= - w(x) + 20 \le 0, \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{where }} \hfill \\ \, u(x) &= 85. 3 3 4 4 0 7 { + 0}. 0 0 5 6 8 5 8x_{2} x_{5} { + 0}. 0 0 0 6 2 6 2x_{1} x_{4} - 0. 0 0 2 2 0 5 3x_{3} x_{5} ,\hfill \\ \, v(x) &= 8 0. 5 1 2 4 9 { + 0}. 0 0 7 1 3 1 7x_{2} x_{5} { + 0}. 0 0 2 9 9 5 5x_{1} x_{2} + 0. 0 0 2 1 8 1 3x_{3}^{2} , \hfill \\ \, w(x) &= 9. 3 0 0 9 6 1 { + 0}. 0 0 4 7 0 2 6x_{3} x_{5} { + 0}. 0 0 1 2 5 4 7x_{1} x_{3} + 0. 0 0 1 9 0 8 5x_{3} x_{4}.\hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (102, 45, 45, 45, 45) and LB = (78, 33, 27, 27, 27)}}.\hfill \\ {\text{Global minimum: }}x^{best} = ( 7 8 , { 33, 29}. 9 9 5 2 5 6 0 2 5 6 8 2 , { 45, 36}. 7 7 5 8 1 2 9 0 5 7 8 8 ) , \, { }f(x^{best} ) = - 3 0 6 6 5. 5 3 9.\hfill \\ \end{aligned}$$

Problem G ₅

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) &= 3x_{1} + 10^{ - 6} x_{1}^{3} + 2x_{2} + \frac{2}{3} \times 10^{ - 6} x_{2}^{3} \hfill \\ {\text{s}}. {\text{t}}. \quad { }g_{1} (x) &= x_{3} - x_{4} - 0.55 \le 0, \hfill \\ \, g_{2} (x) &= x_{4} - x_{3} - 0.55 \le 0, \hfill \\ \, h_{1} (x) &= 1000[\sin ( - x_{3} - 0.25) + \sin ( - x_{4} - 0.25)] + 8 9 4. 8- x_{1} = 0, \hfill \\ \, h_{2} (x) &= 1000[\sin (x_{3} - 0.25) + \sin (x_{3} - x_{4} - 0.25)] + 8 9 4. 8- x_{2} = 0, \hfill \\ \, h_{3} (x) &= 1000[\sin (x_{4} - 0.25) + \sin (x_{4} - x_{3} - 0.25)] + 1 2 9 4. 8 = 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (1200, 1200, 0}}. 5 5 , { 0}. 5 5 ) {\text{ and LB = (}}0, \, 0, - 0.55, - 0.55 ).\hfill \\ {\text{Best known solution: }}x^{best} = (679.9453, \, 1026, \, 0.118876, - 0.3962336 ), \, f(x^{best} ) = 5 1 2 6. 4 9 8 1. \hfill \\ \end{aligned}$$

Problem G ₆

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) &= (x_{1} - 10)^{3} + (x_{2} - 20)^{3} \hfill \\ {\text{s}}. {\text{t}}. \quad { }g_{1} (x) &= (x_{1} - 5)^{2} + (x_{2} - 5)^{2} + 100 \le 0, \hfill \\ \, g_{2} (x) &= (x_{1} - 5)^{2} + (x_{2} - 5)^{2} - 8 2. 8 1\le 0, \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (100, 100) and LB = (13, 0)}}.\hfill \\ {\text{Global minimum: }}x^{best} &= ( 1 4. 0 9 5 , { 0}. 8 4 2 9 6 ), \, f(x^{best} ) &= - 6 9 6 1. 8 1 3 8 8.\hfill \\ \end{aligned}$$

Problem G ₇

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) = x_{1}^{2} + x_{2}^{2} + x_{1} x_{2} - 14x_{1} - 16x_{2} + (x_{3} - 10)^{2} + 4(x_{4} - 5)^{2} + (x_{5} - 3)^{2} \hfill \\ \, + 2(x_{6} - 1)^{2} + 5x_{7}^{2} + 7(x_{8} - 11)^{2} + 2(x_{9} - 10)^{2} + (x_{10} - 7)^{2} + 45 \hfill \\ {\text{s}}. {\text{t}}. \quad { }g_{1} (x) = 4x_{1} + 5x_{2} - 3x_{7} + 9x_{8} - 105 \le 0, \hfill \\ \, \, \, g_{2} (x) = 10x_{1} - 8x_{2} - 17x_{7} + 2x_{8} \le 0, \hfill \\ \, g_{3} (x) = - 8x_{1} + 2x_{2} + 5x_{9} - 2x_{10} - 12 \le 0, \hfill \\ \, g_{4} (x) = 3(x_{1} - 2)^{2} + 4(x_{2} - 3)^{2} + 2x_{3}^{2} - 7x_{4} - 120 \le 0, \hfill \\ \, g_{5} (x) = 5x_{1}^{2} + 8x_{2} + (x_{3} - 6)^{2} - 2x_{4} - 40 \le 0, \hfill \\ \, g_{6} (x) = 0.5(x_{1} - 8)^{2} + 2(x_{2} - 4)^{2} + 3x_{5}^{2} - x_{6} - 30 \le 0, \hfill \\ \, g_{7} (x) = x_{1}^{2} + 2(x_{2} - 2)^{2} - 2x_{1} x_{2} + 14x_{5} - 6x_{6} \le 0, \hfill \\ \, g_{8} (x) = - 3x_{1} + 6x_{2} + 12(x_{9} - 8)^{2} - 7x_{10} \le 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} &{\text{Bound restrictions: UB = (10,}} \ldots , {\text{ 10) and LB = }}( - 10, \ldots , - 10). \hfill \\ &{\text{Global minimum: }}x^{best} = ( 2. 1 7 1 9 9 6 , { 2}. 3 6 3 6 8 3 , { 8}. 7 7 3 9 2 6 , { 5}. 0 9 5 9 8 4 , { 0}. 9 9 0 6 5 4 8, 1. 4 3 0 5 7 4 , { 1}. 3 2 1 6 4 4 , { 9}. 8 2 8 7 2 6 , { 8}. 2 8 0 0 9 2 , { 8}. 3 7 5 9 2 7), \, f(x^{best} ) = 2 4. 3 0 6 2 0 9 1.\hfill \\ \end{aligned}$$

Problem G ₈

$$\begin{aligned} \mathop {\hbox{max} }\limits_{x} \, f(x) = \frac{{\sin^{3} (2\pi x_{1} )\sin (2\pi x_{2} )}}{{x_{1}^{3} (x_{1} + x_{2} )}} \hfill \\ {\text{s}}. {\text{t}}.\quad { }g_{1} (x) = x_{1}^{2} - x_{2} + 1 \le 0, \hfill \\ \, g_{2} (x) = 1 - x_{1} + (x_{2} - 4)^{2} \le 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (10, 10) and LB = (0, 0)}}.\hfill \\ {\text{Global maximum: }}x^{best} = ( 1. 2 2 7 9 7 1 3 , { 4}. 2 4 5 3 7 3 3), \, f(x^{best} ) = 0. 0 9 5 8 2 5.\hfill \\ \end{aligned}$$

Problem G ₉

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) = (x_{1} - 10)^{2} + 5(x_{2} - 12)^{2} + x_{3}^{4} + 3(x_{4} - 11)^{2} + 10x_{5}^{6} + 7x_{6}^{2} + x_{7}^{4} - 4x_{6} x_{7} - 10x_{6} - 8x_{7} \hfill \\ {\text{s}}. {\text{t}}. \quad g_{1} (x) = 2x_{1}^{2} + 3x_{2}^{4} + x_{3} + 4x_{4}^{2} + 5x_{5} - 127 \le 0, \hfill \\ \, g_{2} (x) = 7x_{1} + 3x_{2} + 10x_{3}^{2} + x_{4} - x_{5} - 282 \le 0, \hfill \\ \, g_{3} (x) = 23x_{1} + x_{2}^{2} + 6x_{6}^{2} - 8x_{7} - 196 \le 0, \hfill \\ \, g_{4} (x) = 4x_{1}^{2} + x_{2}^{2} - 3x_{1} x_{2} + 2x_{3}^{2} + 5x_{6} - 11x_{7} \le 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (10,}} \ldots , {\text{ 10) and LB = }}( - 10, \ldots , - 10).\hfill \\ {\text{Global minimum: }}x^{best} = (2.330499, \, 1.951372, - 0.4775414, \, 4.365726, \hfill \\ - 0.6244870, \, 1.038131, \, 1.594227) , \, { }f(x^{best} ) = 6 8 0. 6 3 0 0 5 7 3. \hfill \\ \end{aligned}$$

Problem G ₁₀

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) &= x_{1} + x_{2} + x_{3} \hfill \\ {\text{s}}. {\text{t}}. \quad g_{1} (x) &= - 1 + 0.0025(x_{4} + x_{6} ) \le 0, \hfill \\ \, g_{2} (x) &= - 1 + 0.0025( - x_{4} + x_{5} + x_{7} ) \le 0, \hfill \\ \, g_{3} (x) &= - 1 + 0.01( - x_{5} + x_{8} ) \le 0, \hfill \\ \, g_{4} (x) &= 100x_{1} - x_{1} x_{6} + 833.33252x_{4} - 83333.333 \le 0, \hfill \\ \, g_{5} (x) &= x_{2} x_{4} - x_{2} x_{7} - 1250x_{4} + 1250x_{5} \le 0, \hfill \\ \, g_{6} (x) &= x_{3} x_{5} - x_{3} x_{8} - 2500x_{5} + 1250000 \le 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (10000, 10000, 10000, 1000, 1000, 1000, 1000, 1000) and}} \hfill \\ {\text{LB = (100, 1000, 1000, 10, 10, 10, 10, 10)}}.\hfill \\ {\text{Global minimum: }}x^{best} = ( 5 7 9. 3 1 6 7 , { 1359}. 9 4 3 , { 5110}. 0 7 1 , { 182}. 0 1 7 4 , { 295}. 5 9 8 5 ,\hfill \\ 2 1 7. 9 7 9 9 , { 286}. 4 1 6 2 , { 395}. 5 9 7 9), \, f(x^{best} ) = 7 0 4 9. 3 3 0 7.\hfill \\ \end{aligned}$$

Problem G ₁₁

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) = x_{1}^{2} + (x_{2} - 1)^{2} \hfill \\ {\text{s}}. {\text{t}}. \quad { }h_{1} (x) = x_{2} - x_{1}^{2} = 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (1, 1) and LB = }}( - 1, - 1).\hfill \\ {\text{Global minimum:}} \, x^{best} = ( \pm \frac{1}{\sqrt 2 },\frac{1}{2}), \, f(x^{best} ) = 0.75. \hfill \\ \end{aligned}$$

Problem G ₁₂

$$\begin{aligned} &\mathop {\hbox{min} }\limits_{x} \, f(x) = 1 - 0.01\left[ {(x_{1} - 5)^{2} + (x_{2} - 5)^{2} + (x_{3} - 5)^{2} } \right] \hfill \\ &{\text{s}}. {\text{t}}. \quad g_{i,j,k} (x) = (x - i)^{2} + (x_{2} - j)^{2} + (x_{3} - k)^{2} - 0.0625 \le 0, \hfill \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad i, j, k = 1,2, \ldots ,9. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB = (10, 10, 10) and LB = (0, 0, 0)}}.\hfill \\ {\text{Global minimum: }}x^{best} = (5,{ 5, 5}), \, f(x^{best} ) = 1. \hfill \\ \end{aligned}$$

Problem G ₁₃

$$\begin{aligned} \mathop {\hbox{min} }\limits_{x} \, f(x) &= e^{{x_{1} x_{2} x_{3} x_{4} x_{5} }} \hfill \\ {\text{s}}. {\text{t}}. \quad h_{1} (x) &= x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} - 10 = 0, \hfill \\ \, h_{2} (x) &= x_{2} x_{3} - 5x_{4} x_{5} = 0, \hfill \\ \, h_{3} (x) &= x_{1}^{3} + x_{2}^{3} + 1 = 0. \hfill \\ \end{aligned}$$

$$\begin{aligned} {\text{Bound restrictions: UB}} = (2. 3 , { 2}. 3 , { 3}. 2 , { 3}. 2 , { 3}. 2 ) {\text{ and LB = (}} - 2.3, - 2.3, - 3.2, - 3.2, - 3.2 ).\hfill \\ {\text{Global minimum: }}x^{best} = (- 1.717143, \, 1.595709, \, 1.827247, - 0.7636413, - 0.763645 ) , \, f(x^{best} ) = 0. 0 5 3 9 4 9 8. \hfill \\ \end{aligned}$$