A novel multi-population coevolution immune optimization algorithm

Abstract

A novel multi-population coevolution immune optimization algorithm (MCIA) is proposed to solve numerical and engineering optimization problem in real world. MCIA is inspired by the mechanism that how neuroendocrine system affects T cells and B cells in immune system to eliminate the danger and the main idea of MCIA is to promote three populations, population B, population T and assistant population A, to coevolution through self-adjusted clone operator, the applied dislocation arithmetic crossover, cloud self-adapting mutation operator and local search operator to produce lymphocyte with high affinity. Self-adjusted clone operator and selecting elite elements in the memory population enable the search space be broadened and compressed, cloud self-adapting mutation operator characterized with randomness, stable topotaxis and local search technique enable global and local search be integrated to find the global optima with high population diversity. Therefore, several operators enable MCIA enjoy the capability of broadening the elite search space, boosting the global and local search around elites in search space. The performance comparisons of MCIA with three known immune algorithms and other three optimization algorithms in optimizing twelve benchmark functions indicate that MCIA is an effective algorithm for solving global optimization problems with high precision, good robustness and low time complexity.

Appendix

(A) Rosenbrock function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^{n-1} {( {100( {x_{l+1} -x_l^2 })^2+( {x_l -1})^2})} \\ \quad -30\le x_l \le 30\quad \text{ for }\,\, \text{ all }\,\, l\,\,=1,2,\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {1\ldots 1} ] \\ \end{array} \end{aligned}$$

(B) Step function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {( {\lfloor {( {x_l +0.5})^2} \rfloor })} \\ \quad -100\le x_l \le 100 \quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0}] \\ \end{array} \end{aligned}$$

(C) Quadric function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {\left( {\sum \limits _{j=1}^l {x_j } }\right) ^2} \\ \quad -10\le x_l \le 10 \quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0} ] \\ \end{array} \end{aligned}$$

(D) Schwefels function

$$\begin{aligned} \begin{array}{l} \min f( x)=418.9829-\sum \limits _{l=1}^n {\left( {x_l \sin \sqrt{| {x_l } |} }\right) } \\ \quad -500\le x_l \le 500\quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {420.9687\ldots 420.9687}] \\ \end{array} \end{aligned}$$

(E) Rastrigin function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {( {x_l^2 -10\cos ( {2\pi x_l })+10})} \\ \quad -5.12\le x_l \le 5.12 \quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0}] \\ \end{array} \end{aligned}$$

(F) Ackley’s function

$$\begin{aligned} \begin{array}{l} \min f( x)=-20\mathrm{e}^{-0.2\left( {\sqrt{\frac{\sum \nolimits _{l=1}^n {x_l^n } }{n}} }\right) }-\mathrm{e}^{\frac{\sum \nolimits _{l=1}^n {\cos ( {2\pi x_l })} }{n}} \\ \quad + 20+e - 32\le x_l \le 32 \quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0}] \\ \end{array} \end{aligned}$$

(G) Griewank function:

$$\begin{aligned} \begin{array}{l} \min f( x)=\frac{\sum \nolimits _{l=1}^n {x_l^2 } }{4000}-\prod \limits _{l=1}^n {\cos \left( {\frac{x_l }{\sqrt{l} }}\right) } +1 \\ \quad - 600\le x_l \le 600 \quad \text{ for }\,\, \text{ all } \,\,l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0} ] \\ \end{array} \end{aligned}$$

(H) Rotate hyper-ellipsoid function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {\left( {\sum \limits _{ll=1}^l {x_{ll} } }\right) } ^2 \\ \quad -100\le x_l \le 100\quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {0\ldots 0} ] \\ \end{array} \end{aligned}$$

(I) Shift sphere function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {( {x_l -1})^n} \\ \quad -100\le x_l \le 100 \quad \text{ for }\,\, \text{ all } \,\,l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0 \,\,\text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {1\ldots 1}] \\ \end{array} \end{aligned}$$

(J) Shift Schwefel’s function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {\left( {\sum \limits _{ll=1}^l {( {x_{ll} -1})} }\right) } ^2 \\ \quad -100\le x_l \le 100\quad \text{ for }\,\, \text{ all } \,\,l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0 \,\,\text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {1\ldots 1}] \\ \end{array} \end{aligned}$$

(K) Shift Rosenbrock function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^{n-1} {( {100( {x_{l+1} -1-( {x_l -1})^2})^2+( {x_l -1})^2})} \\ \quad -100\le x_l \le 100\quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0\,\, \text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {1\ldots 1}] \\ \end{array} \end{aligned}$$

(L) Shift Rastrigin function

$$\begin{aligned} \begin{array}{l} \min f( x)=\sum \limits _{l=1}^n {( {( {x_l -1})^2-10\cos ( {2\pi ( {x_l -1})})+10})} \\ \quad -5\le x_l \le 5 \quad \text{ for }\,\, \text{ all }\,\, l{=1,2,}\ldots ,n \\ \text{ where } f( {x^*})=0 \,\,\text{ with }\,\, [ {x_1 \ldots x_n } ]=[ {1\ldots 1}] \\ \end{array} \end{aligned}$$