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JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 349 In response to the above deficiencies and combined with the solving mechanism of GA, we propose an improved multigroup parallel and adaptive Genetic Algorithm based on goodpoint setIMPGA-GPS. We use simulation technology to analyze the convergence of the algorithm, and good performance is proved under the strategy of optimal individual reservation, so that it can effectively avoid premature convergence. It is believed to be feasible for large-scale and high-precision optimization, and have broad application prospects in numerical optimization of complex systems. II. IMPGA-GPS DESIGN A. Strategy for Improvement GA is a simulation of biological evolution, which has been widely used because of its global search capability, strong adaptability and robustness. However, there are still many problems to be improved. Recent efforts to improve the performance of GA mainly focus on the implementation strategy, encoding, and design of the selection, crossover and mutation mechanism. Implementation strategy improvements include hybrid genetic algorithm [9-11] , messy genetic algorithm [12] , chaos genetic algorithm [13] , forking genetic algorithm [14] , symbiotic genetic algorithm [14] , nonlinear genetic algorithm [15] and parallel genetic algorithm [17, 18] ; Encoding improvements include string-coding, real-coding, structured coding and orderly coding; space searching strategies include space [19, 20] adaptive shrinkage and space shrinkage based on statistical theory [21] . On the basis of the above analysis, we apply the good-point set theory of number theory to uniformly design the initial population and the crossover operation, improving the overall performance of genetic algorithm. In the large-scale and highprecision optimizations, we can find the space compression factors according to the entropy strategy. Thus without the loss of optimal solutions, the scope of optimization can be narrowed, which is conducive to achieve more accurate satisfactory solution. B. Key Point of Algorithm Designed 1) Initialization of population The search speed of basic GA depends on crossover and mutation operators. Although it has a global search capability, it is dependent on the populations in a certain range. As a result, the initial population has a great influence on the convergence, search efficiency and stability in the algorithm [22] . In the absence of a predictable region of the optimal solution, the initial population must stands for the individuals of the whole space, represents all individual information to the maximum, and fully represents the characteristics of the solution space, thus ensuring the algorithm a fast approach to the optimal solution [23] .The initial population of basic GA is randomly selected (Monte Carlo method), and its coverage space is uncertain, which is prone to the phenomenon of premature convergence. In addition, maintaining the diversity of the population can improve the global convergence of the algorithm. So while ensuring a reasonable distribution of the initial individuals, a reasonable diversity should also be considered. Accordingly, in this paper, we set up initial population in a view of the good-point set. a) Good-point Set Theory l Assumptions (1) Let s G be an unit cube in an s -dimensional Euclidean space, then x Î Gs x = ( x1, x2, L, xs ) where 0 £ x i £ 1, i = 1,2, L, s . (2) Define a point set (with n points within) in the unit cube. (3) For any point r = ( r1 , r2 , L, rs ) in a given G s , we can define a N n ( r) = N n ( r1 , r2 , L, rs ) representing the number of the points satisfying the following inequality n in P n (k) . 0 £ x i ( k) £ r i i = 1,2, L, s . N ( r) rÎGs n | r | = r1 , r2 , L, rs .So we say the point set (k) f (n . If for any n , there isf ( n ) = O P n (k) has a coincident distribution with a deviation of f (n) n Definition 1 Deviation:Define f ( n) = sup | - | r || , where a deviation of ) P n has , we say Definition2 Good-point Set Suppose r Î Gs , and Pn ( k) = {{ r1 * k},{ r2 * k}, L ,{ rs * k}}, k = 1,2, L, n} -1+ e has a deviation of f (n) .If f( n) = C( r, e ) n where C( r, e ) is a constant only related to r , e (any small positive), (k) is called a good-point set and r a good point. P n Definition 3 Good-point set: Let 2pk r k = {2cos } ( 1 £ k £ s ) 1 p If p is the smallest prime satisfying ( p - s) / 2 ³ s , or r { k k = e } ( 1 £ k £ s ), then we say r is a good point. {a} Denotes the decimals of a . Theorem 1 If (k) P n ( 1 £ k £ n) has a deviation of (n) f Î B s s -dimensional limited function, then n 1 | ò f ( x) d x - å f ( Pn ( k)) | £ V ( f ) f( n) GS n k= 1 where V ( f ) is the total variation of f . f and 2 Theorem 2 If theorem 1 is valid for any f Î Bs , P n (k) ( 1 £ k £ n) is a point set with a deviation less than f (n) . This conclusion can be seen as the converse theorem of theorem 1. Theorem 3 Suppose f (x) satisfies: © 2009 ACADEMY PUBLISHER
350 JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 1 f | f | £ L,| | £ L £ i £ s x 2 f x x i 1 2 | | £ L, 1 £ i £ j £ s i 2 f x x j 3 | | £ L, 1 £ i £ s , i s Then if we use any weighed sum of any given n points’ function values to approximately calculate the integrals of the function on G s , we could never expect the error to be smaller -1 than O ( n ) . Note 1: This theorem is why the above set is called “Goodpoint Set” Note 2: Recalling theorem1, 2 and 3, we can know that using a good-point set for approximate integrals, the order of its error is related to n only rather than the dimension s , which provides a superior algorithm for higher order approximately calculation. Note 3: According all above, if we define n points and use the linear combination of their function values to approximate the corresponding integral, the good-point set can serve as the best solution. Theorem 4 If x 1, x2 L , xn is a uniform distribution on i . i. d. Dt and Pn = ( x1, x2 L, xn ) , the probability of its deviation written - 2 2 as D( n, P ) = O( n 1/ (loglog n) 1/ ) is 1. n Note 4: Theorem 4 implies that for an unknown distribution, we can randomly select n points whose -1/ 2 deviation is usually O ( n ) . If we use good-point set to -1 e find the n points, its deviation is usually O ( n + ) .For instance, if n = 10, 000, the deviation of the former -2 -4 is O (10 ) , while the latter is O (10 ) . Accordingly, the method of good-point set has a much smaller deviation than random selection. That’s the theoretical basis we implement good-point set to improve the crossover and initialization of GA [24,25] . ) The Establishment of Initial Population The establishment of the initial population is essentially an optimal design of how to use limited individuals to scientifically and globally represent the characteristics of the whole solution space. The initial population can neither be randomly generated nor traverse all the conditions, particularly in the multi-variable problems. Only if we set the most representative individuals who best reflect the inherent characteristics of the solution space as the initial population can we better characterize the space [24, 26] . Uniform design is a scientific and effective method to solve this problem, which has been used to determine the operating parameters of GA [27] .Therefore, researchers have put forward several methods for structuring uniform design tables. Good lattice point, Latin square and expansion of orthogonal designs have been used in population initializations, which have achieved good results [22] . But when the number of factors and levels increase, not only the experiment size increases, and the corresponding uniform design tables become very difficult to get, which bring difficulties to the uniform design methods. However, if we use a good-point set strategy whose accuracy has nothing to do with dimension to initialize the population, the mentioned disadvantages can be solved, and better diversity would be generated in the initial population. Suppose the size of the initial population is N , and the i i i i chromosome is A = a , a , L, a } .Firstly we define a N { 1 2 good-point set containing N points in a s -dimensional space H . The method is as follows: Establish a good-point set containing N points in a s - dimensional space H , P i) = {{ r * i},{ r * i}, L,{ r * }} With n ( 1 2 s i i = 1,2, L, n , 2 pk r k = {2cos } , 1 £ k £ s p If p is the smallest prime satisfying ( p - s) / 2 ³ s , or r { k k = e } ( 1 £ k £ s ), then r is a good point. {a} denotes the decimals of a . i (1) If a k ( k = 1,2, L, s i 1,2, L, N i single-digit binary, define ak [{ rk * i}] a
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