A novel chaotic Jaya algorithm for unconstrained numerical optimization

Abstract

Jaya algorithm is one of the recent algorithms developed to solve optimization problems. The basic concept of this algorithm consists in moving the obtained solution, for a given problem, toward the best solution and avoiding the worst one. However, it severely suffers from premature convergence problem and therefore can be easily trapped in local optimums. This study aimed to alleviate these drawbacks and improve the performance of the original Jaya algorithm. Here, three new mutation strategies were implemented in the original Jaya to improve both its global and local search abilities. Chaotic maps were proved to be able to boost the search capabilities of meta-heuristic algorithms. Therefore, after demonstrating its chaotic behavior through the sensitivity to initial conditions, topological transitivity and the density of periodic points, we proposed a new 2D cross chaotic map. The chaotic sequences provided by the proposed chaotic map were embedded into the original Jaya algorithm to generate the initial population and control the search equations. It is worth mentioning that the modifications incorporated in the original algorithm did not affect its two essential characteristics, i.e., simplicity and nonrequirement of additional control parameters. As case studies, sixteen benchmark functions were used to evaluate the performance of the proposed chaotic Jaya algorithm (C-Jaya) regarding solution accuracy and convergence speed. Comparisons with some other meta-heuristic algorithms for low-, middle- and high-dimensional benchmark functions show that the proposed C-Jaya algorithm enhances the performance of original Jaya significantly. Moreover, it offers the fastest global convergence, the highest solution quality and it is the most robust on almost all the test functions among all the algorithms. Nonparametric statistical procedures, i.e., Friedman test, Friedman aligned ranks test and Quade test, conducted to analyze the obtained results, show the superiority of the proposed algorithm.

Appendix A: The proof of chaos for G(x)

Definition 1

(Discrete chaos of Devaney) Consider a discrete dynamical system in the following form:

$$\begin{aligned} y_{i+1}=g(y_i),\quad g: J\rightarrow J,\quad y_0\in J \end{aligned}$$

(6)

g(x) is chaotic if the following conditions are satisfied [12].

• (1) Sensitive to initial conditions

  $$\begin{aligned}&\exists ~ \Omega>0,\quad \forall ~ y_0\in J,\quad \omega>0,\quad \exists ~ n\in N,\quad z_0\in J \nonumber \\&\left| y_0-z_0\right| <\omega \Rightarrow \left| g^n(y_0)-g^n(z_0)\right| >\Omega . \end{aligned}$$

  (7)

• (2) Topological transitivity

  $$\begin{aligned} \forall ~ J_1,\,\, J_2\in J,\quad \exists ~ y_0\in J_1,\quad n\in N,\quad g^n(y_0)\in J_2. \end{aligned}$$

  (8)

• (3) Density of periodic points in J Let $K=\{k\in J|\exists n\in N:\,\, g^n(k)=k\}$ be the set of periodic points of g. Therefore, K is dense in $J:\overline{K}=J$.

Definition 2

Let $f: I\rightarrow I$ and $g: J\rightarrow J$ be maps. f and g are topologically conjugated if there exists a homeomorphism $h: I\rightarrow J$ that makes $h\circ f=g\circ h$.

Theorem

Let $f:I\rightarrow I$ and $g: J\rightarrow J$ be conjugate via h. If f is chaotic on I, then g is chaotic on J.

Proof

• (1) (Sensitive to initial conditions) Let f have sensitivity constant $\alpha$. Let $I=\left[ \omega _0,\omega _1\right]$. Assume $\alpha <\omega _1-\omega _0$. Consider the function $|h(x+\alpha )-h(x)|$ where $x\in \left[ \omega _0,\omega _1-\alpha \right]$. This function has minimum value $\lambda$ as it is continuous and positive. So, h maps intervals of length $\alpha$ to intervals of length at least $\lambda$. We assume that g has sensitivity constant $\lambda$. Let $x_0\in J$ and B be an open interval about $x_0$. Therefore, $h^{-1}(B)$ is an open interval about $h^{-1}(x_0)$. By sensitive to initial conditions, there is $y_0\in h^{-1}(B)$ and $m>0$ which satisfy $\left| f^m\left( h^{-1}(x_0)\right) -f^m(y_0)\right| >\alpha$. Then,

  $$\begin{aligned}&\left| h\left( f^m\left( h^{-1}(x_0)\right) \right) -h\left( f^m(y_0)\right) \right| \\&\quad =\left| g^m(x_0)-g^m\left( h(y_0)\right) \right| >\lambda . \end{aligned}$$

• (2) (Topological transitivity) Let A and B be open subintervals of J. So, $h^{-1}(A)$ and $h^{-1}(B)$ are open subintervals of I (h is a continuous function). As f is topologically transitive, there is $x_0\in h^{-1}(A)$ that fulfills $f^n(x_0)\in h^{-1}(B)$ for some n. Therefore, $h(x_0)\in A$ and $g^n\left( h(x_0)\right) =h\left( f^n(x_0)\right) \in B$.

• (3) (Density of periodic points) Let A be an open subinterval of J and consider $h^{-1}(A)\subset I$. Since periodic points of f are dense in I, there is a periodic point $x\in h^{-1}(A)$ of period m. We have $g\circ h=h\circ f$, so $g^m\left( h(x)\right) =h\left( f^m(x)\right) =h(x)$. Therefore, h(x) is a periodic point of period m in A and periodic points are dense in J. According to the definition of chaos Devaney [12], g is chaotic on J.

$\square$

It is known that $\phi (\theta )=5\theta$ is chaotic [12] under the unit circle mapping $S^1\rightarrow S^1$, so $\phi$ is sensitive to initial value, topologically transitive and dense in $S^1$.

Considering $h(\theta )=\cos \theta$, we have $G\circ h=16\cos ^5\theta -20\cos ^3\theta +5\cos \theta =\cos (5\theta )=h\circ \phi$. So G is conjugated to $\phi$ in $y\in J=\left[ -1,1\right]$. Thus, as $\phi$ is chaotic on $S^1$, then G is chaotic on $J=\left[ -1,1\right]$.

1.1 The randomness proof of G(x)

Because function $G(x)=16x^5-20x^3+5x$ is a Chebyshev polynomial of degree 5 $(T_5(x)=\cos (5\theta ), x=\cos \theta )$, so its invariant density is

$$\begin{aligned} \rho _G(x)=\frac{1}{\pi \sqrt{1-x^2}},\quad -1<x<1 \end{aligned}$$

(9)

1.1.1 The auto-correlation proof

It is known that

$$\begin{aligned} \bar{x}&=\int _{-1}^{1}x\rho _G(x)\mathrm{d}x\\&\overset{x=\sin u}{=}\int _{\frac{-\pi }{2}}^{\frac{\pi }{2}}\sin u\frac{1}{\sqrt{1-\sin ^2 u}}\cos u \mathrm{d}u=0. \end{aligned}$$

Considering $x=\cos u$ and $\tau$ is the iterative times of G(x). We have $G^\tau (x)=\cos (5^\tau u)$. When $\tau \ne 0$,

$$\begin{aligned} \begin{aligned} C_G(\tau )&= \int _{-1}^{1}x\rho _G(x)G^\tau (x)\mathrm{d}x-\bar{x}^2 \\&= \int _{-1}^{1}x\frac{1}{\pi \sqrt{1-x^2}}G^\tau (x)\mathrm{d}x-\bar{x}^2=0. \end{aligned} \end{aligned}$$

When $\tau =0$,

$$\begin{aligned} \begin{aligned} C_G(0) =&\int _{-1}^{1}x\rho _G(x)G^0(x)\mathrm{d}x-\bar{x}^2 \\ =&\int _{-1}^{1}x^2\frac{1}{\pi \sqrt{1-x^2}}\mathrm{d}x-\bar{x}^2\\ \overset{x=\cos u}{=}&\int _{\pi }^{0}\cos ^2u\frac{1}{\pi \sqrt{1-\cos ^2u}}\\&\times (-\sin u) \mathrm{d}u\\ =&\frac{1}{\pi }\int _{0}^{\pi }\frac{1+\cos (2u)}{2}\mathrm{d}u=0.5. \end{aligned} \end{aligned}$$

Therefore, auto-correlation function of G(x) is:

$$\begin{aligned} C_G(\tau )={\left\{ \begin{array}{ll} 0.5 &{} \quad \text {if } \tau =0\\ 0 &{} \quad \text {if } \tau \ne 0\\ \end{array}\right. } \end{aligned}$$

(10)

The auto-correlation function of G(x) shows that when the iterative times $\tau \ne 0$, the time sequences generated by G(x) are independent.

1.1.2 The cross-correlation proof

Considering $x=\cos u$ and $\tau$ is the iterative times of G(x). We have $G^\tau (x)=\cos (5^\tau u)$.

Therefore, the cross-correlation function is given by

$$\begin{aligned} rr_G(\tau )=0 \end{aligned}$$

(11)

The result given in Eq. (11) shows that time sequences produced by G(x) with different initial values have no relation to each other at any time. According to the above characteristics, the average value of G(x) and the cross-correlation are 0, so the probability statistical characteristic is the same as the white noise and thus G(x) function can be used as an ideal chaotic sequence generator.

1.2 The proof of the equal probability of the pseudo-random sequence

When G(x) is iterated, a chaotic sequence is produced as follows: $g_0,g_1,\ldots ,g_p=G(x_{p-1})$ where p is an integer. The chaotic range $V=\left[ -1,1\right]$ is divided into M subdomains $v_i$, $i=0,1,2,\ldots ,M-1$. With $v_i=(t_i,t_{i+1})$ for $i=0,1,\ldots ,M-1$. Here, $t_i$ is defined by Eq. (12)

$$\begin{aligned} t_i=-\cos \left( \frac{i}{M}\pi \right) , \quad i=0,1,2,\ldots ,M-1 \end{aligned}$$

(12)

The initial conditions $(x_0,y_0)$ of the cross chaotic map are used to generate a value of chaotic sequence $\left\{ g_p\right\} _{p=1}^{\infty }$.

Definition 3

Mapping $S:V\rightarrow {0,1,\ldots ,M-1}$, $x_p\rightarrow i=s_k$, $x_p\in v_i$, $p=0,1,\ldots$, where $s_p$ is described as Eq. (13), so the N phase pseudo-random sequence $\left\{ s_p\right\} _{p=0}^{\infty }$ distributes in the N subdomains proportionally.

$$\begin{aligned} s_p={\left\{ \begin{array}{ll} \left\lfloor (1-\arccos (g_p)/\pi )\times M\right\rfloor , &{} \text {if} g_p\in \left[ -1,1\right] ,\\ M-1, &{} \text {if } g_p=1\\ \end{array}\right. } \end{aligned}$$

(13)

Proof

According to Eqs. (9) and (13), the probability of element i appearing in $\left\{ s_p\right\} _{p=0}^{\infty }$ is:

$$\begin{aligned} Prob(i)=\int _{t_i}^{t_{i+1}} \rho (x)dx=\frac{1}{M} \end{aligned}$$

$\square$

Obviously $\left\{ s_p\right\} _{p=0}^{\infty }$ obeys uniform distribution.