J Control Theory Appl 2008 6 (2) 141–145 DOI 10.1007/s11768-008-6067-5
A new adaptive mutative scale chaos optimization algorithm and its application Jiaqiang E 1,2 , Chunhua WANG 1 , Yaonan WANG 2 , Jinke GONG 1 (1.College of Mechanical and Automotive Engineering, Hunan University, Changsha Hunan 410082, China; 2.College of Electrical and Informational Engineering, Hunan University, Changsha Hunan 410082, China)
Abstract: Based on results of chaos characteristics comparing one-dimensional iterative chaotic self-map x = sin(2/x) with infinite collapses within the finite region[−1, 1] to some representative iterative chaotic maps with finite collapses (e.g., Logistic map, Tent map, and Chebyshev map), a new adaptive mutative scale chaos optimization algorithm (AMSCOA) is proposed by using the chaos model x = sin(2/x). In the optimization algorithm, in order to ensure its advantage of speed convergence and high precision in the seeking optimization process, some measures are taken: 1) the searching space of optimized variables is reduced continuously due to adaptive mutative scale method and the searching precision is enhanced accordingly; 2) the most circle time is regarded as its control guideline. The calculation examples about three testing functions reveal that the adaptive mutative scale chaos optimization algorithm has both high searching speed and precision. Keywords: Adaptive; Mutative scale; Chaos optimization algorithm; One-dimensional iterative chaotic self-map
1
Introduction
For optimization problems of some usual functions that are continuously differentiable, some traditional optimization algorithms, such as the steepest descent method, conjugate gradient algorithm, DFP method and BFGS method, can get their global optimal points with the advantage of speed convergence and high precision. However, these traditional optimization algorithms will easily trap into local optimization in solving optimization problems of some multi-modal functions. Despite its variational process with unorderly advantage, chaos variable is of an internal law. Therefore, using its stochastic properties, ergodicity and regularity can theoretically carry through optimization searching by using a chaos variable that had been mapped in optimization variable within the finite region. However, studies on chaos optimization algorithms (see, e.g., [1∼8]) reveal that there are problems of long calculating time and even no global optimal points in the process of optimization searching in solving optimization problems of large space and multi-variable references. In order to avoid the mentioned disadvantage of the chaos optimization algorithm, a new adaptive mutative scale chaos optimization algorithm (AMSCOA) will be proposed by using a chaos variable from the one-dimensional iterative chaotic self-map x = sin(2/x) with infinite collapses within the finite region[−1, 1], which will ensure its advantage of speed
convergence and high precision in the seeking optimization process.
2 Principle of chaos optimization algorithm 2.1 Choice of chaos model A one-dimensional chaotic self-map can be expressed as: x(n + 1) = f [x(n)].
(1)
There are some linear or nonlinear collapse phenomena in phase-space of the chaos process. In a sense, the Lyapunov exponent that can depict chaos characteristics of the model can be regarded as average collapse times of the chaos model. When the absolute value of the average slope of the chaos model is bigger than 1, the Lyapunov exponent is bigger than zero. Collapse times of some representative iterative chaotic maps such as Logistic map, Tent map, and Chebyshev map are finite. The one-dimensional self-map with infinite collapses within the finite region[−1, 1] is defined as: x(n + 1) = sin[2/x(n)], 0, 1, 2, · · · , n, (2) − 1 x(n) 1, and x = 0, If chaos variables are taken from the map, two facts below must be noticed: 1) Zero cannot be taken as the initial value of the selfmap.
Received 25 April 2006; revised 24 May 2007. This work was supported by Hunan Provincial Natural Science Foundation of China (No. 06JJ50103) and the National Natural Science Foundation of China (No. 60375001).
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2) None of the infinite roots of equation x(n + 1) = sin[2/x(n)] can be taken as the initial value of the self-map. The surpass number 2/(kπ)(k is an integer, k = 0) can not appear in the application process because it is always replaced by rational numbers with finite precision. Therefore, chaos phenomenon always appears when the initial value x(0) = 0 of the self-map whose defined region includes [−1, 1] expects x = 0 and x = 2/(kπ)(k is an integer, k = 0). The properties of stochastic sensitivity to initial value and ergodicity of the one-dimensional self-map x(n + 1) = sin[2/x(n)] is expressed as Fig.1 by iterating 1000 times. In Fig.1(a), it is obvious for its initial value x(0) = 0.250 that results of the system are of irregular. In Fig.1(b), it is shown that results of the two system, whose initial values are x(0) = 0.2501, y(0) = 0.2502 and vertical coordinate is x(n) − y(n), are totally different. In Fig.1(c), it is shown that results of the system, whose initial values are x(0) = 0.2501, y(0) = 0.1532 and whoes horizontal coordinate is x(n)(x(n + 1) = sin[2/x(n)]) and vertical coordinate is y(n)(y(n + 1) = sin[2/y(n)]), will spread all over root spaces after iterating sufficiently many times.
(a) x(0) = 0.25.
(c) x(0) = 0.2501, y(0) = 0.1532. Fig.1 Properties of stochastic, sensitivity to initial value and ergodicity of the one-dimensional self-map x(n + 1) = sin[2/x(n)] through iterating 1000 times.
An important index of chaos properties is always judged by Lyapunov exponent; therefore, Lyapunov exponent of the one-dimensional self-map xn+1 = sin(2/xn ) is expressed as: n−1 ln |f [x(i)]| LE = lim (1/n) n→∞
= lim (1/n) n→∞
i=0 n−1 i=0
ln |(2/x2 (i)) cos[2/x(i)]|. (3)
The result is obtained from the Lyapunov exponent expressed in formula (3) that chaos properties of the onedimensional self-map x(n + 1) = sin[2/x(n)] is more obvious than some representative iterative chaotic maps with finite collapses. The one-dimensional self-map x(n + 1) = sin[2/x(n)] of very obvious chaos properties can be proved by calculating formula (4) based on chaos searching from different chaos models. min(x) = min(4 − x2 )2 , (4) − 8 x 8, where f (x) has the minimum 0 when x = ±2. The Ergodic chaos optimization, only once searching, was carried through by chaos variable from chaos map x(n + 1) = sin[2/x(n)] and Logistic map in the whole region. The results of chaos optimization can be expressed as Table 1. The information is also obtained from Table 1 that chaos properties of the one-dimensional self-map x(n + 1) = sin[2/x(n)]is more obvious than some representative iterative chaotic maps with finite collapses. Table 1 Results of chaos characteristics comparing chaos map x(n + 1) = sin[2/x(n)] to Logistic map . CM x(n + 1) = sin[2/x(n)] Logistic map
(b) x(0) = 0.2501, y(0) = 0.2502.
CN/time AIN/time LIN/time 40 40
3564.3 2421.6
15473 5432
Note: CM is chaos map; CN is calculating numbers; AIN is average iterative numbers; LIN is longest iterative numbers.
J. E et al. / J Control Theory Appl 2008 6 (2) 141–145
2.2
A new adaptive mutative scale chaos optimization algorithm
Consider the following optimization problem about the minimum of functions continuously differentiable: min[x(1), x(2), · · · , x(n)], (5) a(i) x(i) b(i), where variable x(i) belongs to [ai ,bi ], i is variable number. Suppose K1 is rough iterative numbers, K2 is subtle iterative numbers, then the basic steps of the chaos optimization algorithm based chaos variable from chaos map x(n + 1) = sin[2/x(n)] are expressed as follows: Step 1 Initialization of the algorithm. Let K1 =1, K2 =1, two big positive integers N1 and N2 and there are i chaos variables xi,n+1 (i = 1, 2, · · · , n) from chaos map x(n + 1) = sin[2/x(n)] when random number x(0)(x(0) = 0) is taken as the initial value of the self-map. Step 2 Rough transformation of chaos variables in optimization design region of the variable. The region[−1, 1] of i-th chaos variable can be transformed into the optimization design region of the variable [ai , bi ] by using the following formula. xi,n+1 = ai + (bi − ai )xi,n+1 .
(6)
Step 3 Rough iterating searching of chaos variables. Let xi (K1 )=xi,n+1 , x∗i = xi (0), fi∗ = fi (0), and optimal solution of fi (K1 ) can be obtained, then 1) If fi (K1 ) fi∗ , then fi∗ = fi (K1 ), x∗i = xi (K1 ); 2) If fi (K1 ) fi∗ , then quit xi (K1 ) and begin next iterating search, K1 : =K1 + 1. When K1 N1 , otherwise, end rough iterating search. Step 4 Reduction of searching space about chaos variables. ai = x∗i − φ(bi − ai ), (7) bi = x∗i + φ(bi − ai ), where φ is contractive gene, φ ∈(0, 0.5). In order to ensure that the new searching space is not beyond the mark, disposing of searching space is taken as follows: If ai ai , then ai =ai . If bi bi , then bi =bi . Therefore x∗i can be replaced in the new region [ai , bi ] by the following vector yi∗ : x∗ − a yi∗ = i . (8) b − a Step 5 Subtle transformation of chaos variables in optimization design region of the variable. If fi∗ always stays constant after searching some times based on Step 3, then new chaos variables for the next optimization search are expressed by some linear combinations
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about yi∗ and xi,n+1 as follows: x∗i,n+1 = (1 − βi )yi∗ + βi x∗i,n+1 ,
(9)
where βi is adaptive control coefficient, 0 βi 1. The adaptive control coefficient βi is given as formula (10): K2 − 1 m βi = 1 − ( ) , (10) K2 where m is some integer, whose value is given from the optimization goal function formula (5). The main reason is that a bigger βi is needed in initial stages of subtle iterating search owing to a big change of chaos variables (x1 , x2 , · · · , xn ) and less βi is needed in order to search in a small range of chaos variables (x∗1 , x∗2 , · · · , x∗n ) when subtle iterating search is close to global optimal points of the optimization goal function. Step 6 Subtle iterating search of chaos variables. Let xi (K2 )= x∗i,n+1 and get global optimal points fi (K2 ) of the optimization goal function: 1) If fi (K2 ) fi∗ , then fi∗ = fi (K2 ), x∗i = xi (K2 ); 2) If fi (K2 ) fi∗ , then quit xi (K2 ) and begin next iterating search, K2 : =K2 +1. When K2 N2 , then end subtle iterating search. Although chaos is of ergodic characteristic in all states in the searching space, it will sometimes take a long time to get some global optimal points. Therefore, the problem can be solved by reducing the searching space about chaos variables gradually. After a rough iterating search space for some time, the approximate roots, which sometimes lie in the adjacent region, can be searched quickly. In the course of subtle iterating search, the subtle iterating search space will be reduced in the velocity 2φ and the adaptive control coefficient βi is also reduced gradually; therefore, in the course of the current optimization variable x∗i getting close to the real value of the optimization function gradually, the speed of iterating search is enhanced to a large extent so that the current optimization variable x∗i will soon arrive at the real value of the optimization function in the whole region when searching in the near region of the real value. It is very useful for arriving at the real value of the optimization function in the whole region soon to give suitable values to rough iterating maximum times N1 and subtle iterating maximum times N2 . When N1 and N2 are big enough, it will take a long time to search for the real value of the optimization function. Therefore, the change region of N1 is 150 ∼ 500 and the change region of N2 is 200∼700 in general, and whether the given values N1 and N2 change or not is decided by the obtained results of iterating search
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based on the given values N1 and N2 , which is an effective combination between optimization results due to chaos optimization theory and factual operation. 2.3
Astringency of adaptive mutative scale chaos optimization algorithm
It is well known that the roots of the optimization function will be searched finally if only the circle times of chaos searching are big enough owing to the ergodicity properties of chaos variables. Suppose R0 is the near region of the real value, the set that the series Xi brought by the adaptive mutative scale chaos optimization algorithm belong to R0 is Ai , then A1 ⊂ A2 ⊂ · · · ⊂ Ai ⊂ · · · and P (Ai ) 1. Thus lim P (Ai ) = P (
i→∞
∞
i=1
Ai ) = 1.
(11)
Let ε represent a positive number and 0 ε 1, then P (AN ) 1 − ε when i is more than positive number N , P (AN ) ε and P (Ai , i N ) εi−N . Therefore, the global optimal points in the whole region will be searched by means of the adaptive mutative scale chaos optimization algorithm with probability 1.
3
Simulations
In order to prove the advantage of the algorithm in the paper, we calculated the following three testing functions by using the algorithm in the paper, the algorithm in [6] and the algorithm in [9] based on the course of the same initial value shown in Table 2 and the comparison results shown in Table 3. min F1 = 100(x21 − x2 )2 + (1 − x1 )2 , − 2.048 xi 2.048,
(12)
2
min F2 = [1 + (x1 − x2 + 1) (19 − 14x1 +
3x21
− 14x2 + 6x1 x2 +
× [30 + (2x1 − 3x2 )2 (18 − 32x1 + 12x21 + 48x2 − 36x1 x2 + 27x22 )],
(13)
(14)
Table 2 The global optimal point and its global optimal solution of three testing functions. Testing function
Optimal point
Optimal solution
F1 F2 F3
(1.0000, 1.0000) (0.0000, −1.0000) (0.0000, 0.0000)
0.0000 3.0000 1.0000
Testing function In [6] F1 F2 F3
2905 1104 1092
In [9] In this paper 675 257 458
687 253 462
4 Conclusions A one-dimensional iterative chaotic self-map x = sin(2/x) with infinite collapses within the finite region[−1, 1] is of more obvious chaos characteristics than some representative iterative chaotic maps with finite collapses, such as Logistic map, Tent map, and Chebyshev map based on comparison results. In the optimization algorithm, the advantage of speed convergence and high precision is ensured by means of reducing the search space of optimized variables and controlling most circle times in the seeking optimization process. The optimization algorithm has many advantages, such as fast search, precise results, and convenience. The simulation results show that the performance of this method is better than that of other optimization algorithms. Therefore, it is useful for solving optimization problems of some multimodal functions and is also a quite promising optimization algorithm. References [1] L. Wang, D. Zheng, Q. Li. Survey on chaotic optimization methods[J]. Computing Technology and Automation, 2001, 20(1): 1 – 5. [2] Y. Pan, Q. Xu, H. Gao. The research of the fuzzy control algorithm optimization based on chaos[J]. Control Theory and Application, 2000, 17(5): 702 – 706(in Chinese). [3] F. Qian, C. Fei, B. Wan. A hybrid algorithm for finding global minimum[J]. Information and Control, 1998, 27(3): 232 – 235.
3x22 )]
− 2 xi 2, sin2 x21 + x22 min F3 = 0.5 − , [1 + 0.001(x21 + x22 )]2 − 4 xi 4.
Table 3 Results of comparing the iterative numbers of feasible solution before obtaining global optimal solution
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J. E et al. / J Control Theory Appl 2008 6 (2) 141–145 Jiaqiang E was born in 1972 in Hunan, China. Currently, he works at Hunan University as a vice-professor and M.S. candidate supervisor. He received his M.S. and Ph.D. degrees in heat energy engineering from Central South University in 2001 and 2004. His research interests include intelligent control, intelligent information fusion and intelligent fault diagnosis, etc. E-mail:
[email protected]. Chunhua WANG was born in 1987 in Jiangsu, China. He received the B.S. degree from Hunan University. He is now pursuing the M.S. degree in Hunan University. His research interests include chaos and fractal in heat energy engineering.
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Yaonan WANG was born in 1957 in Jiangxi, China. Currently, he works at Hunan University as a professor and Ph.D. candidate supervisor. He received his Ph.D. degree in control theory and control engineering from Hunan University in 1994. His research interests include intelligent control, intelligent information fusion etc. E-mail:
[email protected].
Jinke GONG was born in 1954 in Hunan, China. Currently, he works at Hunan University as a professor and Ph.D. candidate supervisor. He received his Ph.D. degree in vehicle engineering from Hunan University in 1997. His research interests include intelligent control in the work process in engine. E-mail:
[email protected].
