wolfe搜索下记忆梯度法的收敛性

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Vol.29No.120061ACTAMATHEMATICAEAPPLICATAESINICAJan.,2006∗Wolfe
(
276826)(E-mail:zjshi@qrnu.edu.cn)Wolfe Wolfe MR(2000)90C30;65K05;49M37O221.21nminf(

2、x),x∈R,(1)n1f:R→R
!"Æ
xk+1=xk+αkdk,k=1,2,···,(2)dk
f(x)xk#$αk
f(x)%$&'!∗∗fk=f(xk),gk=∇f(xk),f=f(x).)dk=−gk,!*Æ
"+#(#,"Cauchy),-+#!"#.$%#$/&'01%!$%(&'23)*(&+'()*4%+5)dk=−Hkgk,Hk!,f(x)xkHess

3、ian',+'#)-6!*Æ"
.Newton*7/#0#1-+#%223)*(&+''4%+5.+8#/3
!+5#9,*:1"+#*;2'23(&*)+'0Æ

−gk,)k=1;dk=(3)−gk+βkdk−1,)k≥2,[1−4]βk!*123βk'<4Æ,4556'<.+8#.367
=089>,:-78&'.+8#0##?-@%367;200294.9B<=200597.18B

4、?=∗@:/;<=>(10171054),0<D?ABEÆ=>(6765700)@0@BEÆ<=>12FA=1034GGC29'!=0>89B:-D78&',!Æ5#?-@367
*EDH@,.+8#'0#CD-@F!6(!'EÆ567#@
FGH-3H*HIIJ7IJJ@EÆ5[5,6].0#CD-@KLKKF88#88#9"+#*.+8#M62'23)*(&+'()4%N+5[7

5、]KK*)88#78&'#0#CD-@[8]KK*)L88#ArmijoH@&'#5OF0#CD-@"0#Q:@EM%αkL;#)%+MH
78&'*D78&'N!D78&'N!0#O@!WolfeN!PGoldsteinN!*ArmijoN!Wolfe>QkL;αkR?f−f(x+αd)≥−µαgTd,(4)kkkkkkk*g(x+αd)Td≥σgTd,(5)kkkkkk@0<µ<σ<1.
FST)*(&+'JJ-@E7I

6、+5!NU*)O88#HPFWolfeH@&'#0#CD-@*H@-+#:QRVOO!#92WSTVT3W5OCD-@4WHP-+#5WXUUJ*@A6WVK:QRV2BCDE!F)#YV(H):n167f(x)WZ[L0=x∈R

7、f(x)≤f(x0)X##(H2):8#g(x)=∇f(x)W]L0^>[BXLipschitz_(L>0R?g(x)−g(y)≤Lx−y,∀x,y∈B.(6)1

8、nGHA0<ρ<1,0<µ<2,σ∈(µ,1),x1∈R,k:=1.1Ygk=0!ZXY!I2;2xk+1=xk+αkdk(βk),
−gk,)k=1;dk(βk)=(7)−(1−βk)gk+βkdk−1,)k≥2,@βk∈[0,sk],)ρ/2,)k=1;sk=(8)ρg22+

9、gTd

10、,kgkkk−1)k≥2,"αk$Wolfe&'N!8Z3k:=k+1,I1.
[dk(βk)
dk.1
[J`Wolfe 11K1,k≥1,gTd≤−(1−ρ)g

11、2.kkkYk=1,!gTd=−g2≤−(1−ρ)g2.kkkkYk≥2,!$0≤βk≤sk5−gTd=(1−β)g2+βgTdkkkkkkk−1=g2−βg2−gTdkkkkk−1≥g2−βg2+

12、gTd

13、kkkkk−1≥g2−sg2+

14、gTd

15、kkkkk−1=(1−ρ)