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(a f (b f (c vv (d s (e s u (f s (g s ªªª ªªª } ªªª vv/z ««/ «z««s /zçèíéæí Éÿ ÊÊÊ yê~ ÊÊÊ y v Ò ÑÉy ÊÊvv ÑÑÒÕ ÑÑÒÕÌÊ Š Švv ÖÐÒÒ ÑÒ ÑÕ Ì ~Êvv ~{ Ê ~ÈÉvv v z ÊÉÌÊ vv/z s /zê Š f f C0«C«C w 5 30 0-5 -0 0 0 5 0 0 5 0 0 f'hxl fhxl 6 5 4 0 50 8 6 5 4 0 5 Š e } v e } v «temate codto«««ªªªªª 3 4 5 6 x 3 4 5 6 x ( f ' x ( + x = x ( f '' x ( + x x ε w f s vv«desg vaable«f«~ vv u ~ ˆ ~ ªªª s«f ~ ««x = { },, x x
w f s (obectve fucto f«s«f ( x = f f m f x x x w f s «costats«f«~ «vv «ˆ(sde costats (behavo costats «actve««actve«š s«vv s «ÿ f s «s x x x {,, k } g g g x = x x 0 w f s ÿvvš ÿ vv u Š {,, } f ( x fd x = x x mmze subect _ to x x x g x 0 w f s Š t vv u Š {,, } f( x fd x = x x mmze subect _ to x x x g x 0 w f s «temate codto«f«e } v «s«+ f ( x f ( x ε f x f ( x ε w f s vvv «feasble doma«f«vv «««vv«x={x, x } g (X=g (x, x <0 (=,,,k x vvv g g k x + N x ε 3 g g 3 g 4 vv x 3
w f s vv u ÿ v } f ( x f ( x f ( x = f ( x f ( x f ( x = f ( x f ( x Kaush-Kuh-ucke«95«vv u m f ( x λ g ( x = g ( x = 0 ( x 0, λ 0 + = 0 λ g g ƒ ««x } vv «vv (3-ba tuss L L vv L B x C D x x θ ( ( θ (3 A u P u ( (3 ˆ«x «f«( ˆ«x X = {x, x } «v «σ L < σ < σ U (=, ˆ««x L < x < x U (=, z f(x = M(x, x u (x ; ( =, ªˆ g = -x + x L < 0 g = x -x U < 0 g 3 = -x + x L < 0 g 4 = x -x U < 0 ª g 5 = -σ + σ L < 0 g 6 = σ - σ U < 0 g 7 = -σ + σ L < 0 g 8 = σ - σ U < 0 vvš«x = {x, x } f(x = M(x, x u Š zvvš «X = {x, x } z u Š M < M S, u < u S ( =, vvš Š x (adš ˆdŠ Š vv d vv d a x + a x + + a x b a x + a x + + a x b d f ( x = c x + c x + + c x a x + a x + + a x b ˆd m m m m vv ˆd g x 0 ( =,,, m (b~ Š Š x {,,, } ~ vv ~ = x x x ( x R l x x, x,, vv x ( x D vv ˆd f ( x ªÿ u vv ÁÁyÊÕÒÖÉ ÁÁvvŠÊ ÁÈÍÉ ÁÁu ÁÁ ÁŠÊ d Š vv d ~ Ñ ÔÖÒÐÑ Ï Ô Ï /~ vvêéìê ˆsÌÉÊ ÑÐ Ö Ð ~ ÁÕÒÖ ÁÁÁ vv u ÁŠ Ê ~ ŠÊ Á Á Á ÁÁÁÊv ÁÁÁv Êv Ê uêêvá ÊÔÖÐÖÕ ÖÓÑÒÊv ÌÉÊÔÖÐÖÕ ÁÁ vêííu Ê Êv Á Á ˆd ˆd ˆd ÊÈ ˆd ÊÈ/ˆd ÊÈ/ˆd ~ ~/ /~ ÓÕ Ò ÓÕ Ò ÿd h ~dv ~ v ÔÒÖÒÏ Patcle Swam ÑÕÕÖ ÒÏÒÎÓ Ö Ð ÎÖÐÖÑÕ ÁÿÊvvÊ 4
ªªªªª«Newto Method«f f f f ( x ( x + ( x ( x x + ( x x ( x ( x x f x = x f x x + l l l+ l x = x + α s f γ ªªªªª = f ªªªªª s x x s = f x + I f x ªªªªª«quas-Newto Method«l+ l x = x + α s H = H + H l + l l s = H f ( x l Davdo-Fletche-Powell l l l pp H yy H H = + σ τ Boyde-Fletche-Goldfab-Shao l l l σ + τ H yp + py H H = pp σ σ p x x l = py l l l f y = f x x σ = l τ = y H y } Hesses u «Steepest decet method«l + l x = x +α s s = f ( x «Gadet poecto method«~ Pq axf f axf Zx a f= Pq axf f axf Wq = m g a x f,, gq a x f g W q ªªª x 3 x (+ =x ( v Z x ( x ( Df g x P axf = I U axf [ U axf U axf] U axf q q q U axf= u,, u q q R g g u = U S,, V W q q x «Gadet poecto method««gadet poecto method«5
ÿd «Geealzed educed gadet method«mmze subect _ to {,,,,, m} f ( x fd x = x x s s mmze subect _ to x x x 0 s g x + s = 0 {,,,,, m} f ( x fd x = y y z z y y y z z z = g x 0 ÿd «Geealzed educed gadet method«d ( yz, ( yz, m f ( yz, f = y = df = dy + dz z y ( yz, y z ( yz, (, = + = 0 = f d + f dz dg y z Cdy Ddz g g y y C = gm gm y y (, (, (, g g z z m D = gm gm z zm df yz = f yz f yz D C dy G dy y z R ÿd «Geealzed educed gadet method«初期点を定める. ただし, zはdが特異にならないように定める 一次元探索問題 m f(y+αs を解きαを定める. ただし, 側面制約を破る場合には, 制約上になるようにα を定める h«feasble decto method«u ««sh Š u «停止条件を満たせば終了 縮約勾配を算出 独立変数を更新 y=y+αs 従属変数 z の値を求める fd maxmze β ( x β 0 ( x θβ s f + subect _ to s g + 0 s s h«feasble decto method«~dv«sequetal Lea Pogammg«d} dv «f ( x f ( x + f ( x ( x x g ( x g ( x + g ( x ( x x 0 x x x + 6
~«v«sequetal Quadatc Pogammg««} «} «g g + g 0 ( x ( x + ( x ( x x + ( x x ( x ( x x f f f f ( x ( x ( x ( x x ~«v«sequetal Quadatc Pogammg«fd mmze subect_to f + x ( x ( x x ( x x B ( x x g( x 0 ªªªª ªªªª v «Sequetal Ucostaed Mmzato echque«m f = f + x x f = f + g = g ( x = m ( x ( x max ( ( x, 0 Smulated Aealg«SA««aealg«d ªª ªª y u h u ~ h w ªªªªE f(x s ªªªªª p = + exp E / 0.8 0.6 0.4 0. -60-40 -0 0 40 60 Smulated Aealg«SA«ªªªªª«Geetc Algothm«~ «ªª ª ~w«t e } «~ ««d««ªªªª w vv x «ªªª ªª«ªªª u«h 7
ªªªªª«Geetc Algothm«Patcle Swam Optmzato ªªªªªªª «vv«f s «u vv ( k + ( k ( k x = x + x ( k+ ( k w x pesoal best ( k + c ( x x global best ( k + c x x x = J.Keedy, R.Ebehat, Patcle Swam Optmzato, Poc. IEEE o Neual Netwoks, 995 Patcle Swam Optmzato vvš u ªªª ªªªªªª u vv u «ªªªªªª s ÑÐ Ö Ð vv x ;( =,,, x = αx ;( =,,, g ( ;( =,,, k g( X = β g ( X ;( =,,, k Ê Ê f X f ( X = γ f ( X g f ;( =,,, k ;( =,,, x ;( =,,, g β g = α f γ g = ;( =,,, k ;( =,,, x α x ;( =,,, v Š vv x Ê ÈÍÊv [ˆs] [Ê] F Ì x Ê Ê K U = F ˆŠ Š Š K ÔÒÖÒÐÑ U ÔÐÒÖ F ÔÐÒÖ U F K = = = K F ; (,,, o U F K U = = K ; (,,, [Š] [Ê] (K λ M U = O ( =,,, l λ K M = U λ U ; ( =,,, l,( =,,, x x λ Ê ÉÉÈ U MU = U λ ÊÈÍÔÐÒÖ [ÔÐÒÖÊ] B=K λ M ÉÆÇ ÈËÉÊÇÊÍÉÈÉ Ê B ÔÒÖÐÑÊ Ê l- B ÊÔÒÖÐÑÎÊÍÆÊs U B b U B b B U + U + b U 0 U x = = = b b U 0 M U U MAU + U U ÇÇÊ B Ê(l- (l-êçèêôòöðñ b Ê (l- U Ma ÊÔÐÒÖ b ÊÑÏÖ ÉÉÈ B b a = b b B b B b B B A = b B b b b + b B b + b B ( =,,, l (=,,, 8
vv vv vv «vvv s ««ª ªªªŠ««Š «vv vv «Š ««Š ««f «ªª ªªª ªªª ª ªª ««vv vv vv vvv vv ÿ ˆ ŸŸ ˆ vv ~vv ªªªªªªªvv «ª ª«Š vv ª ªªªªªªªª «ª ª«y ªªªªª v ªª ªªª ªªª ªªª ªªªªvv h ˆŸŸˆ ª w ªªªªªªªªª ªª ªª ªª ªª ªªªªªªªªªªªª ª ªªªªª ªª ª w ªªªªª ªªªªª ªv ªu ªÿ ªªªªªªªªªªªª ªª ª w ªªªªª ªªªª x 0 vvš «vv«««g 4 g 3 p (x,x g x z X = {x, x } ««g (X 0 (=,,, k «vvš«f, f Ÿ g vv (x,x A-B C-D f f ªªªª ª ««Ÿ ªªªª«u B-C f f f A Q (f,f B C D f f = f (x, x f = f ( x, x F = { f, f } f ˆf ˆQ = ( ˆf, ˆf ˆf Q = ( f, f f zvv z Š««gŠ«««««f 0 f «(Wok f «(Salay f 3 «ªªª g, g, g 3,ªªª F (z F( Paetou f zu vvš u «««vv«x={x, x } g (X=g (x, x <0 (=,,,k «f (X=f (x, x f (X=f (x, x x g vvv g g 3 g k g 4 vv x f Optmum Soluto f f Paeto Soluto f /z Lpªªª š Geoffo-Dye-Febeg SW IFW ªªªªªªªªªª zªªªª ª 9
lpªªª ω l p f(x = ( w f p Σ / p (X p < = =Max w f (X p = PÀ f ÁÁÁ F = wf + wf+ + wf e e  dààâ ( F = wf + wf + + wf e e Â.m.s { w } F Max f [ÁÁ ÁÁÁ Á(chebyshevÁÁÁ] p= p= p= f maxφ w f w max w = C f max φ w f w f w maxφ w w f s «pefeece fucto«f«t ÿ s«v ( f( x ÿ lpªªª f φ max w = C Paetou f v«goal Pogammg«fd mmze k w d = ( x + ( + d + subect _ to f d + d = fˆ x ( ( x ˆ,0 ( ˆ ( x,0 d = Max f f + d = Max f f f «d + z~~«d- z~ zªªªª ª«Satsfcg ade-off Method«fd x, z mmze ( ( x a subect_ to g x 0 a a z w f f z w = f f «f a f zªªªª ª«Satsfcg ade-off Method««z f (x f (=,, x X «~ vva«b zu B A f C A B f f bette bette u DSS f = ( λ + α w f N + ( λ α w Impove bette Ÿ Paeto zpaetou u Ÿzªªªª ª 0
vv vv ˆ vv 3 vv ÿ~ ~ 4 t ªz w «wvv ( w vv ( u «f «ªvv (3 ªvv f ÿ ÿ s s ªªª ªªªªªªªªª ªªªªªªªªª t vv «z ( u w t ªvv s w(collaboatve Optmzato (~««««t vv t vv «Multdscplay Desg«u (3t u ÿ~ «t ª ªª ªª f v v