A Hybrid CS–GSA Algorithm for Optimization

Abstract

The chapter presents a hybridized population-based Cuckoo search–Gravitational search algorithm (CS–GSA) for optimization. The central idea of this chapter is to increase the exploration capability of the Gravitational search algorithm in the Cuckoo search (CS) algorithm. The CS algorithm is common for its exploitation conduct. The other motivation behind this proposal is to obtain a quicker and stable solution. Twenty-three different kinds of standard test functions are considered here to compare the performance of our hybridized algorithm with both the CS and the GSA methods. Extensive simulation-based results are presented in the results section to show that the proposed algorithm outperforms both CS and GSA algorithms. We land up with a faster convergence than the CS and the GSA algorithms. Thus, best solutions are found with significantly less number of function evaluations. This chapter also explains how to handle the constrained optimization problems with suitable examples.

Appendix: Benchmark Functions

• a. Sphere Model

  $$F_{1} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} {x_{i}^{2} } , \, - 95 \le x_{i} \le 95,{\text{ and min}}\left( {F_{1} } \right) = F_{1} \left( {0, \ldots ,0} \right) = 0$$

• b. Schwefel’s Problem 2.22 [21, 42]

  $$F_{2} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} {\left| {x_{i} } \right| + \prod\nolimits_{i = 1}^{n} {\left| {x_{i} } \right|} } , \, - 12 \le x_{i} \le 12 , {\text{ and min}}\left( {F_{2} } \right) = F_{2} \left( {0, \ldots ,0} \right) = 0$$

• c. Schwefel’s Problem 1.2

  $$F_{3} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} {\left( {\sum\nolimits_{j = 1}^{i} {x_{j} } } \right)^{2} } , { } - 90 \le x_{i} \le 90,{\text{ and min}}\left( {F_{3} } \right) = F_{3} \left( {0, \ldots ,0} \right) = 0$$

• d. Schwefel’s Problem 2.21

  $$F_{4} \left( {\rm X} \right) = \mathop {\hbox{max} }\limits_{i} \left\{ {\left| {x_{i} } \right|,1 \le i \le n} \right\} , { } - 90 \le x_{i} \le 90,{\text{ and min}}\left( {F_{4} } \right) = F_{4} \left( {0, \ldots ,0} \right) = 0$$

• e. Generalized Rosenbrock’s Function

  $$\begin{aligned} & F_{5} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n - 1} {\left[ {100\left( {x_{i + 1} - x_{i}^{2} } \right)^{2} + \left( {x_{i} - 1} \right)^{2} } \right]} , { } - 30 \le x_{i} \le 30 \\ & { \hbox{min} }\left( {F_{5} } \right) = F_{5} \left( {0, \ldots ,0} \right) = 0. \\ \end{aligned}$$

• f. Step Function

  $$F_{6} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} {\left( {\left\lfloor {x_{i} + 0.5} \right\rfloor } \right)^{2} } , { } - 100 \le x_{i} \le 100,{\text{ and min}}\left( {F_{6} } \right) = F_{6} \left( {0, \ldots ,0} \right) = 0.$$

• g. Quartic Function i.e. Noise

  $$\begin{aligned} & F_{7} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} i x_{i}^{4} + \text{random}\left[ {0,1} \right), \, - 1.28 \le x_{i} \le 1.28 \\ & { \hbox{min} }\left( {{\text{F}}_{ 7} } \right) = F_{7} \left( {0, \ldots ,0} \right) = 0 \\ \end{aligned}$$

• h. Generalized Rastrigin’s Function

  $$\begin{aligned} & F_{8} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{n} {\left[ {x_{i}^{2} - 10\cos \left( {2\pi x_{i} } \right) + 10} \right] , { } - 5.12 \le x_{i} \le 5.12} \\ & \hbox{min} \left( {F_{8} } \right) = F_{8} \left( {0, \ldots ,0} \right) = 0. \\ \end{aligned}$$

• i. Ackley’s Function

  $$\begin{aligned} & F_{9} \left( {\rm X} \right) = - 20\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} } } } \right) - \, \exp \left( {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {\cos \left( {2\pi x_{i} } \right)} } \right) + 20 + e \\ & - 32 \le x_{i} \le 32,{\text{ and min}}\left( {F_{9} } \right) = F\left( {0, \ldots ,0} \right) = 0. \\ \end{aligned}$$

• j. Generalized Griewank Function

  $$\begin{aligned} & F_{10} \left( {\rm X} \right) = \frac{1}{4000}\sum\nolimits_{i = 1}^{n} {x_{i}^{2} - \prod\limits_{i = 1}^{n} {\cos \left( {\frac{{x_{i} }}{\sqrt i }} \right) + 1} } , { } - 600 \le x_{i} \le 600 \\ & \hbox{min} \left( {F_{10} } \right) = F_{10} \left( {0, \ldots ,0} \right) = 0. \\ \end{aligned}$$

• k. Generalized Penalized Function 1

  $$\begin{aligned} F_{11} \left( {\rm X} \right) = & \frac{\pi }{n}\left\{ {10\sin \left( {\pi y_{i} } \right) + \sum\nolimits_{i = 1}^{n} {\left( {y_{i} - 1} \right)^{2} } \left[ {1 + 10\sin^{2} \left( {\pi y_{i + 1} } \right)} \right] + \left( {y_{n} - 1} \right)^{2} } \right\} \\ + & \sum\nolimits_{i = 1}^{n} {u\left( {x_{i} ,10,100,4} \right)} , \\ \end{aligned}$$

  where

  $$\begin{aligned} & u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{*{20}c} {k\left( {x_{i} - a} \right)^{m} , \, x_{i} > a} \\ {0,{ - }a < x_{i} < a,} \\ {k\left( { - x_{i} - a} \right)^{m} , \, x_{i} < - a} \\ \end{array} } \right.{\text{ and }}y_{i} = 1 + \frac{1}{4}\left( {x_{i} + 1} \right) \\ & - 50 \le x_{i} \le 50,{\text{ and min}}\left( {F_{11} } \right) = F_{11} \left( {1, \ldots ,1} \right) = 0. \\ \end{aligned}$$

• l. Generalized Penalized Function 2

  $$\begin{aligned} F_{12} \left( {\rm X} \right) = 0.1\left\{ {\sin^{2} \left( {3\pi x_{1} } \right) + \sum\nolimits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} } } \right.\left[ {1 + \sin^{2} \left( {3\pi x_{i} + 1} \right)} \right] + \left( {x_{n} - 1} \right)^{2} \cdot \hfill \\ \, \left. {\left[ {1 + \sin^{2} \left( {2\pi x_{n} } \right)} \right]} \right\} + \sum\nolimits_{i = 1}^{n} {u\left( {x_{i} ,5,100,4} \right)} ,\hfill \\ \end{aligned}$$

  where

  $$\begin{aligned} & u\left( {x_{i} ,a,k,m} \right) = \left\{ {\begin{array}{*{20}c} {k\left( {x_{i} - a} \right)^{m} , \, x_{i} > a} \\ {0,{ - }a < x_{i} < a} \\ {k\left( { - x_{i} - a} \right)^{m} , \, x_{i} < - a} \\ \end{array} } \right. , {\text{ and }}y_{i} = 1 + \frac{1}{4}\left( {x_{i} + 1} \right) \\ & - 50 \le x_{i} \le 50,{\text{ and min}}\left( {F_{12} } \right) = F_{12} \left( {1, \ldots ,1} \right) = 0. \\ \end{aligned}$$

• m. Generalized Schwefel’s Problem 2.26

  $$\begin{aligned} & F_{13} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{30} { - x_{i} \sin \left( {\sqrt {\left| {x_{i} } \right|} } \right)} , \, - 500 \le x_{i} \le 500 \\ & \hbox{min} \left( {F_{13} } \right) = F_{13} \left( {420.9687, \ldots ,420.9687} \right) = - 12569.5. \\ \end{aligned}$$

• n. Shekel’s Foxholes Function

  $$\begin{aligned} & F_{14} \left( {\rm X} \right) = \left( {\frac{1}{500} + \sum\nolimits_{j = 1}^{25} {\frac{1}{{j + \sum\nolimits_{i = 1}^{2} {\left( {x_{i} - a_{\text{ij}} } \right)^{6} } }}} } \right)^{ - 1} ,- 65.536 \le x_{i} \le 65.536 \\ & \hbox{min} \left( {F_{14} } \right) = F_{14} \left( { - 32, - 32} \right) \approx 1, \\ \end{aligned}$$

  where

  $$a_{\text{ij}} = \left( {\begin{array}{*{20}l} { - 32} \hfill & { - 16} \hfill & 0 \hfill & {16} \hfill & {32} \hfill & { - 32} \hfill & \cdots \hfill & 0 \hfill & {16} \hfill & {32} \hfill \\ { - 32} \hfill & { - 32} \hfill & { - 32} \hfill & { - 32} \hfill & { - 32} \hfill & { - 16} \hfill & \cdots \hfill & {32} \hfill & {32} \hfill & {32} \hfill \\ \end{array} } \right).$$

• o. Kowalik’s Function

  $$F_{15} \left( {\rm X} \right) = \sum\nolimits_{i = 1}^{11} {\left[ {a_{i} - \frac{{x_{1} \left( {b_{i}^{2} + b_{i} x_{2} } \right)}}{{b_{i}^{2} + b_{i} x_{3} + x_{4} }}} \right]^{2} , { } - 5 \le x_{i} \le 5}$$

  $\hbox{min} \left( {F_{15} } \right) \approx F_{15} \left( {0.1928,0.1908,0.1231,0.1358} \right) \approx 0.0003075$. The coefficients are displayed in Table 14 [21, 41, 42].

  Table 14 Kowalik’s function F ₁₅

• p. Six-Hump Camel-Back Function

  $$F_{16} \left( {\rm X} \right) = 4x_{1}^{2} - 2.1x_{1}^{4} + \frac{1}{3}x_{1}^{6} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4} , \, - 5 \le x_{i} \le 5$$
  $$\begin{aligned} & X_{\hbox{min} } = \left( {0.08983, - 0.7126} \right),\left( { - 0.08983,0.7126} \right) \\ & \hbox{min} \left( {F_{16} } \right) = - 1.0316285. \\ \end{aligned}$$

• q. Branin Function

  $$\begin{aligned} & F_{17} \left( {\rm X} \right) = \left( {x_{2} - \frac{5.1}{{4\pi^{2} }}x_{1}^{2} + \frac{5}{\pi }x_{1} - 6} \right) + 10\left( {1 - \frac{1}{8\pi }} \right)\cos x_{1} + 10 \\ & - 5 \le x_{1} \le 10, \, 0 \le x_{2} \le 15 \\ \end{aligned}$$
  $$\begin{aligned} & X_{\hbox{min} } = \left( { - 3.142,12.275} \right),\left( {3.142,2.275} \right),\left( {9.425,2.425} \right) \\ & \hbox{min} \left( {F_{17} } \right) = 0.398. \\ \end{aligned}$$

• r. Goldstein-Price Function

  $$\begin{aligned} & F_{18} \left( {\rm X} \right) = \left[ {1 + \left( {x_{1} + x_{2} + 1} \right)^{2} \left( {19 - 14x_{1} + 3x_{1}^{2} } \right.} \right.\left. {\left. {{ - 14}x_{2} + 6x_{1} x_{2} } \right)} \right] \times \left[ {\left( {2x_{1} - 3x_{2} } \right)^{2} } \right. \\ & \quad \quad \quad \quad \times \left( { 1 8 { - 32}x_{1} + 12x_{1}^{2} + 48x_{2} } \right.\left. {\left. { - 36x_{1} x_{2} + 27x_{2}^{2} } \right) + 30} \right] \\ & - 2 \le x_{i} \le 2,{\text{ and }}\hbox{min} \left( {F_{18} } \right) = F_{18} \left( {0, - 1} \right) = 3. \\ \end{aligned}$$

• s. Hartman’s Family

  $$F\left( {\rm X} \right) = - \sum\limits_{i = 1}^{4} {c_{i} \exp \left( { - \sum\limits_{j = 1}^{n} {a_{\text{ij}} \left( {x_{j} - p_{\text{ij}} } \right)^{2} } } \right)} , \, 0 \le x_{j} \le 1, \, n = 3,6$$

  for F ₁₉(X) and F ₂₀(X), respectively. $X_{\hbox{min} } {\text{ of }}F_{19} = \left( {0.114,0.556,0.852} \right)$, and $\hbox{min} \left( {F_{19} } \right) = - 3.86$. $X_{\hbox{min} } {\text{ of }}F_{20} = \left( {0.201,0.150,0.477,0.275,0.311,0.657} \right)$, and $\hbox{min} \left( {F_{20} } \right) = - 3.32$.

  The coefficients are shown in Tables 15 and 16, respectively.

  Table 15 Hartman function F ₁₉

  Table 16 Hartman function F ₂₀

• t. Shekel’s Family $F\left( X \right) = - \sum\nolimits_{i = 1}^{m} {\left[ {\left( {X - a_{i} } \right)\left( {X - a_{i} } \right)^{T} + c_{i} } \right]^{ - 1} , \, m = 5,7,{\text{ and 10, for }}F_{21} ,F_{22} , {\text{ and }}F_{23} }$ ${ 0} \le x_{j} \le 10,\,x_{{{\text{local\_optima}}}} \approx a_{i} ,\,{\text{and}}\,F\left( {x_{{{\text{local\_optima}}}} } \right) \approx {1 \mathord{\left/ {\vphantom {1 {c_{i} }}} \right. \kern-0pt} {c_{i} }}\,{\text{for}}\,1 \le i \le m.$

  These functions have five, seven, and ten local minima for $F_{21} ,F_{22} , {\text{ and }}F_{23}$, respectively. The coefficients are shown in Table 17.

  Table 17 Shekel function F ₂₁, F ₂₂, and F ₂₃