Adaptive Simulated Annealing の基礎 - 知的システムデザイン研究室

92 2007 05 Adaptive Simulated Annealing 1 Simulated Annealing(SA) () 1) SA iSight SA(Adaptive Simulated AnnealingASA) 2) ASA .2 ASA n ASA ASA SA 3, 4) Fig. 1 Initial Parameters SettingGeneration2.1 α k α k+1 i α i k [A i B i ] (1) α i k+1 α i k+1 = α i k + y i (B i − A i ),y i ∈{−11} (1) y i (2) y i = sgn(u i − 1 2 )T i[(1 + 1 T i) |2ui −1| − 1], (2)u i ∈ U{01}T i ln e sgn (3) sgn(x)={ 1 if x > 00 if x =0−1 if x < 0(3)y i (4) Fig. 2 T i G Ti (y i )=G (y i Ti )D∏i=112(|y i | + T i )ln(1 + 1/T i )(4)NoAcceptance CriterionYesTransitionCoolingNo y iFig.2 ()NoReannealing CriterionYesReannealingFig.1NoTerminal CriterionEndASA () SA ASA 2.2 α k+1 E(α k+1 ) α k E(α k ) T cost (5) Metropolis (exp − E(α )k+1) − E(α k )>U, U∈ [01) (5)T cost1

2.3 (6) (7) ASA T i T cost 2 T 0i T 0cost k i k cost c i c cost (6) T i (k i )=T 0i exp(−c i k 1/Di) (6)T cost (k cost )=T 0cost exp(−c cost kcost) 1/D (7)2.4 ASA SA (Reannealing) T i SA () () (8) ∣ S i =∂L ∣∣∣∣∂α i (8) α i S i S max (8) S i S max (9) T i ′ T ′ i = T iS maxS i(9) (10) k i ′ k ′ i = ( ln(Ti /T ′ i )c i) D(10) (10) k i ′ (11) T i (k i )=T 0i exp(−c i k ′ 1/Di ) (11)3 ASA (1) 2 Rastrigin Griewank Ridge Rosenbrock ASA Fig. 3 Fig. 3 ASA 5) 44002200X2 0X2 0-2-200-4-400-4 -2 0 2 4-400 -200 0 200 400X1X1(a) Rastrigin (b) Griewank 60240120X2 0X2 0-20-1-40-60-2-60 -40 -20 0 20 40 60-2 -1 0 1 2X1X1(c) Ridge (d) Rosenbrock Fig.3 ( 5) ) (1) (2) y i y i [0, 1] u i T i (2) ASA 5) (2) (12) y i sgn y i = T i ( 1 ) |2ui−1| ,u i ∈ U{01} (12)T i (12) u i y i Table 1 Table1 u i y i ( 5) )u i y i u i∼ =1 y i∼ =1 u i∼ =1/2 y i∼ =T i ()u i∼ =0 y i∼ =1 Table 1 u i y i 1. u i 1 0 y i 2. u i 1/2 y i (2) u i y i 2 Rastrigin 2

u i y i Fig. 4 u i y i y i iuFig.4 u i y i ( 5) )Fig. 4 u i 0 1 y i (1)u i 1/2 y i 0 (12)Table 1 y i ASA (1) ASA u i y i 2 x 1 x 2 Fig. 5 x 1 x 2 X2u1u201/21X10 1/2 1(x1)(x2)(x2)(x1)(x1)(x2)Fig.5 u i ( 5) )Fig. 5 2 3 1. u 1 u 2 1/2 Fig. 4 y i 0 x 1 x 2 2. u 1 u 2 1/2 0 1 Fig. 4 u 1 1/2 x 2 y 2 x 1 y 1 0 x 2 x 1 u 2 1/2 x 2 x 1 3. u 1 u 2 0 1 Table 1Fig. 4 y i ASA 4 SA SA(ASA) ASA SA SA ASA ASA 2 ASA SA iSight 1) . . , Vol. 9, No. 6, pp. 875–880, 1997.2) L.Ingber. Adaptive simulated annealing (asa) :Lesson learned. Control and Cybernetics, 1995.http://www.ingber.com/.3) L.Ingber. Genetic algorithms and very fast simulatedreannealing. A ComparisonMathematicaland Computer Modeling, Vol. 16, No. 11, pp. 87–100, 1992.4) L.Ingber. Simulated annealing: Practice versus theory.Journal of Mathl. Comput. and Modelling,Vol. 18, No. 11, pp. 29–57, 1993.5) , , . . ,2004.3