Simplex Crossover for Real-coded Genetic Algolithms

Technical Papers GA Simplex Crossover for Real-coded Genetic Algolithms 47 Takahide Higuchi Shigeyoshi Tsutsui Masayuki Yamamura Interdisciplinary Graduate school of Science and Engineering, Tokyo Institute of Technology higuchi@es.dis.titech.ac.jp Dept. of of Management and Information Science, Hannan University tsutsui@hannan-u.ac.jp Interdisciplinary Graduate school of Science and Engineering, Tokyo Institute of Technology my@dis.titech.ac.jp keywords: genetic algorithms, real-coded GA, simplex crossover, SPX, function optimization Summary In this paper, we perform theoretical analysis and experiments on the Simplex Crossover (SPX), which we have proposed. Real-coded GAs are expected to be a powerful function optimization technique for real-world applications where it is often hard to formulate the objective function. However, we believe there are two problems which will make such applications difficult; ) performance of real-coded GAs depends on the coordinate system used to express the objective function, and 2) it costs much labor to adjust parameters so that the GAs always find an optimum point efficiently. The result of our theoretical analysis and experiments shows that a performance of SPX is independent of linear coordinate transformation and that SPX always optimizes various test function efficiently when theoretical value for expansion rate, which is a parameter of SPX, is applied. We also show that BLX-α is equivalent to degenerate form of SPX. Experiments show that we have something misunderstood effect of epistasis on performance degradation of real-coded GAs...,, GA GA, GA. GA, [Salomon 96, Ono 97, 98b]. GA,, GA, GA,., [ 98, Kita 99] [ 98a],2 3 GA, (SPX) SPX 4. 5 SPX(SPX-n-m-ε), 6

48 6 Q 200 2. 2 GA, ( ), ( ),. GA, GA 2 2, GA. ( ),,, GA 2 3 GA / GA, [Davis 9] GA, GA BLX-α [Eshelman 93] UNDX[Ono 97] BLX-α p,p 2, c : c i = u(min(p i,p 2i ) αi,max(p i,p 2i )+αi) I = p i p 2i () c i,p i,p 2i c,p,p 2,u(x,y) [x,y] UNDX x c : n x c = x p + ξd + D η i e i (2) i= ξ N(0,α 2 ), η i N(0,( β i ) 2 ) x p,d 2, D 2 ( ) 3,e i,n,n(m,ρ 2 ) m, ρ BLX-α,, [Ono 97, Salomon 96]. UNDX[Ono 97],, (3 ) UNDX UNDX-m[ 98b],. 2 4 GA,. [ 98], GA, GA, GA. () GA,,

GA 49 (2),,, (), (2) GA,, GA.. () GA,,,,,, () ( )[ 98a], GA GA UNDX [Ono 97],, 2 5. SPX( )., GA. BLX-α α, (α =0.5). UNDX UNDX-m, BLX SPX, UNDX UNDX-m BLX SPX 3. GA. GA,,,,.,,, (A) GA, ( ), (B).,, GA ( A)., GA X A k???

50 6 Q 200 P2 P0 G P 2 SPX ( 3 ) X A k = A k X BLX-α UNDX,UNDX-m GA BLX-α UNDX (Simplex Crossover, SPX). [ 98a], SPX SPX (),. 4 2. 3 SPX R n n SPX [] (n +) P 0,..., P n. [2] G. n G = P i [3] i=0 x k = G + ε( P k G) (k =0,...,n) ( 3) 0 () C k = r k ( x k x k + C k ) (4) (k=,,n) x k, C k k =0,...,n. ε (Expansion Rate). r k [0,] u(0, ). r k =(u(0,)) k+ (k =0,...,n ) (5), k { 0 (k<0) r k = (6) (k n) [4] C. C = x n + C n (7) SPX, SPX. SPX SPX,SPX n+ n ε ( 2).. [ 89],. 3 2 SPX x = x. γ ij = (x i x i )(x j x j ). SPX 2 2( ) C P

GA 5 P k = P x k x k = 0. C k = r k C k C 0 = 0 C n = 0, C = C n + P n = P ( ) 3( ) m+, {γij C} {γij P } {γij} C = ( ) +ε 2 m {γ P m + m +2 ij} 2,. 3, m+ SPX { m + 2 (m =,2, ) ε = (8) (m=0) ε. 4., 3 2 ε. SPX,, 4 GA MGG [ 97] () p (2) (3) p 2. (4) (3) 2,,. 4 2,4 3 sphere-d f(x)= n (x i d) 2 ( 5.2 <x i < 5.2) i= 4 2 n, (d,...,d) 0. Rastrigin-d f(x)=0n + n {(x i d) 2 0cos(2π(x i d))} i= ( 5.2 <x i < 5.2) n, (d,...,d) 0. Rastrigin-d Rastrigin-d.,. Rosenbrock n f(x) = {00(x x 2 i ) 2 +(x i ) 2 } i=2 ( 2.048 <x i < 2.048) (,...,) 0. Scaled-Rosenbrock n f(x) = {00(x (ix i ) 2 ) 2 +(ix i ) 2 } i=2 ( 2.048/i < x i < 2.048/i), (, 2,, n ) 0. Scaled-Rosenbrock Rosenbrock. 3 2 ε,ε SPX n:0,20,30 : n 5, (Rastrigin ) n 90 :sphere-.0, Rastrigin-.0, Rosenbrock :n 0 :25 : (.0 0 7 ), ε = n +2( ) 2, ( ) (,...,)

52 6 Q 200 ε 0.9.0. sphere-.0 0 AVG 32404 5700 74780 SUC 25/25 25/25 25/25 20 AVG 705848 83452 02206 SUC 25/25 25/25 25/25 30 AVG 859672 226280 2902356 SUC 25/25 25/25 25/25 Rosenbrock 0 AVG 08577 275896 292576 SUC 25/25 25/25 25/25 20 AVG 396496 60624 SUC 0/25 25/25 25/25 30 AVG 379887 438356 SUC 0/25 25/25 25/25 Rastrigin-.0 0 AVG 4850 763828 2778348 SUC 20/25 25/25 25/25 20 AVG 7004557 959606 SUC 9/25 25/25 0/25 30 AVG 778353 2595923 SUC 6/25 24/25 0/25 2 ε 0.9,.0,..SUC, AVG, ε SPX ( ), ε, 2. sphere ε,,. Rastrigin ε., ε 3 2 4 3, (UNDX) :Rastrigin-0, Rastrigin-0, Rosenbrock, Scaled-Rosenbrock :Rosenbrock 300, Rastrigin 500 :20 : 00 00 SPX UNDX 0 e+06 2e+06 3e+06 4e+06 5e+06 6e+06 SPX UNDX 0 e+06 2e+06 3e+06 4e+06 5e+06 6e+06 3 Rastrigin ( ), Rastrigin ( ) SPX : ε = 22 ( ) UNDX :α =0.5,β =0.35( ) :200 :.0 0 7, 6.0 0 6 :25 3 4. UNDX SPX. SPX UNDX Rosenbrock,., SPX UNDX Rastrigin Rastrigin,. Rosenbrock Scaled-Rosenbrock,UNDX SPX SPX, 5. SPX BLX-α BLX-α SPX,, SPX BLX-α SPX, BLX-α. n (m )

GA 53 000 00 00 0 BLX-0.366 BLX-0.45 BLX-0.5 SPX UNDX 0. 0.00 e-05 0 e+06 2e+06 3e+06 4e+06 5e+06 6e+06 e-07 0 2e+06 4e+06 6e+06 8e+06 e+07 000 00 UNDX 00 0 0. 0.00 BLX-0.366 BLX-0.45 BLX-0.5 SPX e-05 0 e+06 2e+06 3e+06 4e+06 5e+06 6e+06 e-07 0 2e+06 4e+06 6e+06 8e+06 e+07 4 Rosenbrock( ),Scaled-Rosenbrock ( ) 5 Rastrigin ( ), Rastrigin ( ) m SPX. SPX-n-m-ε, [Tsutsui 99]. n R n,r n k R m., R n = R m... R }{{ m R } q k m ε k R m R q q SPX, SPX-n-m-ε. SPX-n-m-ε m =2 BLX-α. BLX-α α.,α = 2 ( ε),α α = 2 ( 3) 0.366, α =0.5. α 0.5 0.366 2 4 3. :Rastrigin-0, Rastrigin-0 :.0 0 7,.0 0 7 5, Rastrigin Rastrigin BLX-0.5(SPX- 20-2-2.0 ), BLX-0.366(SPX-20-2- 3, ) BLX-0.5, 4 2 SPX, 3, BLX-α,, GA,, 6., GA,,.,,[ 98a]. 5, BLX-α,,.

54 6 Q 200 [Davis 9] Davis, L.: The Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York (99). [Eshelman 93] Eshelman, L. J. and Schaffer, J. D.: Real Coded Genetic Algorithms and Interval-Schemata, in Foundations of Genetic Algorithms 2, pp. 87 202 (993). [Kita 99] Kita, H. and Yamamura, M.: A Functional Spacialization Hypothesis for Designing Genetic Algorithms, in IEEE International Conference on Systems, Man, and Cybernetics, p. 250 (999). [Ono 97] Ono, I. and Kobayashi, S.: A Real-coded Genetic Algorithm for Function Optimization Using Unimodal Normal Distribution Crossover, in Proc. 7th ICGA, pp. 246 253 (997). [Salomon 96] Salomon, R.: Performance Degradation of Genetic Algorithms under Coodinate Rotation, in Proc. of the Fifth Annual Conference on Evolutionary Programming, pp. 55 6 (996). [Tsutsui 99] Tsutsui, S., Yamamura, M., and Higuchi, T.: Multi-parent Recombination with Simplex Crossover in Real Coded Genetic Algorithms, in Proc. of the Genetic and Evolutionary Computation Conference, Vol., pp. 657 664 (999). [ 98a],, GA, 42, pp. 9 0 (998). [ 98b],, GA, SICE, pp. 57 62 (998). [ 97],,,, Vol. 2, No. 5, pp. 734 744 (997). [ 98], SICE, pp. 5 20 (998). [ 89], (989). 2000 3 23 GA R. GA n, g kn,p kn,c kn k n,, S R,X R R,, k f k, x f k (x) R N R,fk (x)., L g n f, L k A k. A k A k g kn ( )i kn g n i n A k i kn = A k i n,f k (i kn )= f(a k i kn). GA n,g kn g k(n+). 8>< >: p kn = P S R g kn c kn = (X R p kn ) g k(n+) = g kn p kn + N R,fk (p kn + c kn ) S R p kn = S R A k g n = A k S R g n (A.) X R,A k A k X R = X R A k c kn = X (X R A k S R g n)=a k X (XR S R g n) f k (i kn )=f(a k i kn) N R,fk (p kn + c kn ) = N R,fk (A k S R g n + A k ( P (X R S R g n))) = A k N R,f (S R g n + P (X R S R g n)) A k g k(n+) = + g n S R g n N R,f (S R g n + (X R S R g n)) = g (n+) k, A k A k, GA A k, A k X R. GA. ( ) SPX (m+), s k s k =(r m r k ( r k )) (A.2),s k P, S = m s k P k P k [ 89]. (7) C = ε S +( ε) G (A.3), C S., C ( ) 3 (m+), t k t k = ε(r m r k ( r k )) + ( ε) m + (A.4) (7) (3)(4)(6) mp C = mx t k P k (A.5) t k = 2, x C i x C i t k x P k i t k (x P k i x P i ) a b (x a i x i)(x b j x j) =0 x C i xc i = m P x Pi = m P, γ C ij = (xc i xp i )(xc j xp j ) = = = * m X * X m * X m t k (x P k i t 2 k (xp k i t 2 k + γ P ij mx + x P i ) t k (x P k j x P j ) + x P i )(xp k j x P j ) (A.6)

GA 55 r k = 8>< >: 0 (k<0) k+ (k =0,..., m ) k+2 (k m) 8 > < rk 2 = > : 0 (k<0) k+ (k =0,..., m ) k+3 (k m) (A.4) * X m + t 2 k = +ε 2 m m + m +2 {γij C } = +ε 2 m {γij P m + m +2 } (A.7) (A.8) (A.9). ( ) 999, 969.,( )., 987,.,,,.993 996 2000 4 200 3,,,IEEE. 982 987,.,, 996,..,,