A retrospective trust-region method for unconstrained optimization
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(31) x = x C ? R J*BNk R M3Me= J ( A ρ ∈ [0, η ) C Êxd8 Ì γ ks k ∆ = A ρ < 0 C min[γ ks k, max[γ , θ ]∆ ] R J\ J θ A=LJ*BJLABÊ|d8dÌ8 v G=@J C LJBJ Êxd8Ì f (x ) − f (x ) ρ˜ = m (x ) − m (x ) ;HBLXk R M3MV=@J A ρ˜ > η˜ C max[γ ks k, ∆ ] A ρ˜ ∈ [˜η , η˜ ) C Êxd8 s Ì ∆ ∆ = A C γ ks k ρ ˜ ∈ [0, η ˜ ) A ρ˜ < 0 C min[γ ks k, max[γ , θ˜ ]∆ ] R J\ J θ˜ A=LJ*BJLABÊ|d8 ,Ì8 - C¶ /> ½ - C¶ ?@! @ ! `¹ 8 f MeIK[O? J;K=@? J*[ ? R ;2?VÎ =@AO Yk*AhJBV? G5Y\ JrLOkJr=9? R JlIKMLJ*GÏ m ;>BL s = Ok R ? R ;2? x + s ∈ B 8 - C¶ ?B ½ @¶ % $!A ¶b¹ CD $ ¶ Cº ! V2¹ E @ f MeI}[O? J f (x + s ) ;>BLLJBJ Êxd8 ^|Ì f (x ) − f (x + s ) ρ = . m (x ) − m (x + s ) ρ > η C ? R JB LJ*BJ x = x + s ;>BL k*M>IK[O?gJ ∇ f (x ) F M>? R J\ A=@JLJBJ 8y Bk*\ JIKJ*BV? 435a;>BL7ieM}? MjV? J[ a>8. . 0. 1. 2. 0. 1. 2. 0. 1. 1. 0. 2. 1. 2. 1. 0. k. k. k. k−1. 1. k. k−1. k−1. 1. k−1. 0. k−1. k−1. 1. k−1. k−1. k−1. k. k. k−1. 2. k. k−1. k. k. k−1. k−1. k. 1. k−1. 2. k. 1. k. k−1. 1. k. 0. k. k−1. 2. 1. k. k. k. k. k. k. k. k. k. k. xk+1 = xk. k+1. 1. k. k. k. k. k. k. k. k. k. k. k. k. x. k+1.
(32) .
(33) ! " #$%&' )(*+ ,-(.)/.0. Ä"Á ;"6 Ã >"6Á5Ç36 Â B,6Ä2Ã 1 J BM2 AhB3peJ=@? Aie;2?gJ? R JFkMeBVpeJ*\gi>JBkJX[\ Me[J\@?gAhJr=}MHnMeO\Y;HGi>Me\ Ah? R I 8 j3AhBkJNAh?Yk;HB4JNk*M>B= ALJ*\gJL ;e= ;=@Ai>ptB;>A\ Ak;H;HBVBV?K?)MH= nAhIK? R AGJN;>\ 4Ah? ;eA=@JA=k7Ah? B \gO? =z? J*ux\gA\}J*ie[AhMe\ M3B M>IK=*8NJ? R B ML M>;H ?f MeMeBGhGBM2~J= ?;H9GxJ8xÊ|d>;t]>p>]>JX]SÌ ;2C ?@?gJ*JIKJ[?gJ[ L Jk?g?7MF= 4AI}JXAGJ;H\K[G\ AJrk*=@A?}OGM>?=YB ;H? BL J ;>=g=@OI}[? AM>B=;>BL7[\ Me[J\@?gAhJr=R C 4O?? MK\ J* J*\<?gM f R R ;H[?gJ*\ s MHC
(34) ? R A=R \gJ J\ JBkJl R JBJ*peJ*\<[MV= = A4GhJe8 R mnO\~;>=g=@OI}[? AM>B=;>\ JALJ*BV? Ak*;>G? M}? R Me= JO=@JrL7 Me\<? R J4;>= Akn? \gO=@?@u \ Ji>AM>BXI}J*? R ML8 U R k*M>BJ =@~?g;>JBe=g? = Aκ;>B M>=
(35) O? k R R J? MeR 4f;HÍz? JC k? MeAp>\~Jl;HGOG Bxk∈? AM> Z B ∇C f A=~OBA Me\ IKG574 M>OBLJL C A|8 J>8? R J\ JJ*A=@?g=~;K[ Me= Ah? Ap>J . n. xx. k∇xx f (x)k 6 κ .. . U R J0IKMLJ*G m A=S\g=@?@u M>\LJ\kM R J*\gJ*BV?Ah? R ? R J0 OBk?gAhMeB f ;2?9J;ek R A?gJ*\;2? AM>B x C A|8 J>8? R J*A\p2;>GhOJr= ;>BLi>\;>LAJ*BV?g=<;>\ JJVO;>G-;2? x Me\~;HGG k w ;>BL g = ∇ m (x ) = ∇ f (x ). m (x ) = f (x ) U = OR k J ~? J=g;H= ? A;>B MeM>\~
(36) ;H? GGR JIKM LJ*G ;H∇BLm Me\~;HA=0GG OBAh M>\gI}G574 M>OBLJL C A|8 J>8? R J\ JJ*A=@?g=~;Yk*M>B=@?g;>BV? κ
(37) > 1 R R C Z C k. k. k. def. k. >. x∈. k. n. k. xx. k. x. k. k. x. k. k. k. k∇xx mk (x)k 6 κ
(38) − 1.. B. U [ R M>J0ABVL? Jk*Ax\ 8 JeJr8 ;>= ? J<JM>\ BKJl? J*R 3J~A=@IK?g=~M3;KLJ*kG MemB=@?g;HABV=S? ;2?GJ;e=z?9;>=yI)Ok = R Ok ;>=S?; ;2\? ;>k? M>AM>\~BK;>MHGhG ? R ;2?Me4?g;>AhBJLY;2?? R J f ;>Ok R 5 R R R C kC F κ
(39) ∈ (0, 1) k. mk (xk ) − mk (xk + sk ) > κ . . kgk k , ∆k kgk k min βk. . Ah? R β = 1 + maxk∇ m (x)k 8 M>? J)? ;H?8 = [JrkAhJ=? JKBMH?gAhMeBMHÎ@= OkAJ*BV?l\ JrLOk?gAhMeBsÏ)O= JLAhB jV?gJ*[ 7M>M>O\;>GhieM>\gA? I AhGJ ? R JKk R MeR Ak*JMH m ABjV?gJ*[$R a)A=GAhIKAh? JrLF435N8d;HBL8 8 1J};>G= MBM>? J? R ;2? s 6= 0 R JBJ*peR J*\ gC 6=R 0 4Jrk*;>O=@JlM>8 8 def. k. x∈Bk. xx. k. k. . k. k. ! "$#% '&(*),+
(40) -.&0/213 4$ 56 +7&8+#%9;:=<>+? &(-. B ? R A==@Jrk?gAhMeB 9J [\gM2p>JX? R ;H?}? R JN\gJ? \gMe= [ Jk?gAhpeJ7?g\ O=z? ux\gJ*ieAhMeB ;HGi>Me\ Ah? R I A=YieGhMe4;HGG5$kM>B3peJ*\gi>J*BV?)? M [\\gM3=z? LuxOMe\gkLJrL7J\43k5Y\gAh? ? ACk*J;H;HG9G[ i>MeM>\ ABeAh? ?= IC AhB d8?aR =gJ ;2?g=@AJ=@B= JJ= ? R ;2?)Jp>J\ 5 GAhIKAh?}[ M>ABV? x M>~? R JX= JVOJ*BkJ7M>0Ah? J*\;2?gJ= (x ) R R k. ∗. Ah\g\gJ1= [JlJrk? \Ap>=@J?<ieMHAhpe? JR ;)J4 k R M>M>OABkLJMHM>BX? ?R JJ=@J?g;>\ \g\@M>?gAh\9B4i)J*[ ?zM>AJBVJ*? BXx? Jl;HB?gLX\ OAJBAhM>? 4fA;HÍzGJrk?g? \ AOp>=zJn? ux \gOJ*BiekAh? MeABM>B \g;eLA;>OB= L∆A?=8 kO\g\gJ*BV?I}MLJG R8 R f 2 ; ? ? J [ g \ * J 3 p A > M O 9 = h A ? * J \ 2 ; g ? J R m x ∇x f (x∗ ) = 0 0. k. k−1. 0.
(41) .
(42) ! " #$%&' )(*+ ,-(.)/.0. &¶S¹Nº¶:
(43). )*. j3O[[ Me= Jl? R ;H?~8a38 R MeGL-8SU R J*B 9J R ;tp>Jn? R ;H?. > . Ê8aEÌ. |f (xk ) − mk−1 (xk )| 6 κubh ∆2k−1. ;HBL C A Ah? J*\;2?gAhMeB. k−1. A=~= Ok*k*J=g=z OG C ? R ;2?. Ê8dÌ Ê8 ,Ì. |f (xk−1 ) − mk (xk−1 )| 6 κubh ∆2k−1. R J*\gJ. κ
(44) = max[κ7 , κ
(45) ]. def. >M¨Bº¤G¹ 5)¹ [C \ M2peJ U ÊRJ8 d,Ì4 8 M>:OBJrLJk*;>OÊ=@8J~aEÌ<? R LJnAh\gM>J4fkÍz? JrG57k? \gAp>J= J~O GO?B=<k ?g\gAhM>MeI B7;HU BR LKJ*M>? R\gJ*JnI IKs M38 L J*8aG ;HAB\gJ f CM>BB OJ*B?nk? ;HAM>Gx8 BÊx= dHC ]>9]eJ~],ÌIY8 ;t1 5}J;>[? R [OG5 = ? R JXI}Jr;HB p2;HGOJY? R J*Me\ JI MeB ? R J7Me4fÍzJk? Ap>JK OBk? AM>B f ;>BL M>B ? R JIKMLJ*G m C ;>BL M>4?;HAB \ MeI ? R ;H? x =x −s Ê|8 Ì f (x ) = f (x ) − hs , ∇ f (x )i + hs , ∇ f (ξ )s i Ê|8Ì m (x ) = m (x ) − hs , g i + hs , ∇ m (ζ )s i M>\~= M>IKJ ξ C ζ AB? R J= J*ieI}JBV? [x , x ] 8 :Jrk*;>O=@JMH8d C ? R J)Me4fÍzJk? Ap>J OBk?gAhMeB f ;HBL ? R J}IKMLJ*G m R ;tp>J? R JK= ;>I}Jpt;>GhOJ);HBLNi>\;>LAJ*BV? ;2? x 8SU R O= C = O4?g\g;ek? ABiÊ8 ,Ì \ MeI Ê8 Ì<;HBL7?g;ÉVABiY;H4=@MeGhO?gJlpt;>GhOJ=<53AhJGL=9? R ;H? . 2. k. k−1. k. k−1. k−1. k. k. k. k−1. k−1. k. k. k. x. k−1. k−1. 1 2. k. 1 2. k. k−1. k−1. xx. xx. k. k. k. k−1. k−1. k. k. k. 1 2 |hsk−1 , ∇xx f (ξk )sk−1 i − hsk−1 , ∇xx mk (ζk )sk−1 i| 1 2 [|hsk−1 , ∇xx f (ξk )sk−1 i| + |hsk−1 , ∇xx mk (ζk )sk−1 i|] 2 1 2 (κufh + κumh − 1)ksk−1 k 2 1 2 (κufh + κumh − 1)∆k−1 ,. |f (xk−1 ) − mk (xk−1 )| = 6. |Ê 8 s Ì R J*\gJlJ= Ok*k*J=g=@Ap>JGh5O=@JrL? R Jl? \gA;>Bi>GJlAhBJVO;HGAh?z5 C ? R J f ;HOk R 5Vujk R <;H\¾~ABJVO;>GhAh?z5 C ? R JABLOk*JL IKÊ;H8d? Ì9\gAYk*GhJrB;HMe\g\ GhI5 [M>\gGM>L[ =*J*8 \ ? AJ= C 8a C 8 C ;HBL? R J0;>k?? R ;2? x ∈ B AhIK[GAJ=? R ;H? ks k 6 ∆ 8Sj3M R 2 U R O=9? R J;HB;HGM>iM>
(46) U R JM>\gJ*I s 8 8a0M> f MeBB7J*?;>G|8´ÊxdH]>]e],Ì R M>GL=9ABXMeO\k*;>= J C R J\ J9Jn\gJ*[G;>k*J~? R J M>\g90;>=y\gLXMeOL\SAhP BJJ\ JB\k;2J ? AfM (x O= )J=y−? mJ\g(xJLOk? )AM>43B}57AB Ah?g=<\gJ?gAh\ BMV=z=@?g[ J;>JL}k? M>Ap>J? p2J;H\g\ AJr;>LBeO? kf? (xAM>BKAhB ) − m (x9J;H\g)J<8 AhBV? J\ Jr=z?gJL AhBN;}4 M>OBLMeB7? R J*A\0Lρ˜AhP J\ JBkJ R C R Ak R A=[\gM2p3AmLJrL435? R A=BJ3R ?\gJ= OG?r8 m C 6. 6. k. k+1. k. V¶:
(47).
(48) ! > . k−1. C. k−1. k+1. k−1. k. . k−1. k. k. k−1. k−1. k−1. j3O[[MV=@J? R ;H?)8a8 R M>GL8 U R J*B 9J R ;tp>JK? R ;2? C Me\Jp>J*\g5= Ok*k*J=g=z OGSA?gJ*\;2?gAhMeB Ê8 ^|Ì | [m (x )−m (x )] − [m (x ) − m (x )] | 6 2κ
(49) ∆ . k−1. k−1. k−1. k. k. k−1. k. 2 k−1. . k. ¨º¤¹ ¹ C o =@ABiN? R JIKMLJ*G<LAhP J\ JBe?gA;>4AGhAh?z5 C JX;H[[G5 ? R JXI}Jr;HB p2;HGOJY? R J*Me\ JI MeB ? R JXI}MLJG C ;HBLXJlMe4?g;>AhBX? R ;H? m Ê|8 `,Ì ,∇ m (ψ )s i m (x ) = m (x ) + hs ,g i + hs M>\~= M>IKJ ψ AhB? R J=@Ji>IKJ*BV? [x , x ] 8 Z J*IKJ*I)4J\<? R ;H? Ê8 ,ÌABX? R Jl[\gJ*p3AM>O=9[\gM3MH
(50) ieAhpeJ=? R ;2? Ê|8 b,Ì , ∇ m (ζ )s i m (x ) = m (x ) − hs , g i + hs . k−1. k−1. k. k−1. k−1. k−1. k−1. k−1. k. k−1. k. 1 2. k−1. k−1. xx. k−1. k−1. k. k. k−1. k. 1 2. k−1. xx. k. k. k−1. k−1.
(51) s.
(52) ! " #$%&' )(*+ ,-(.)/.0. M>\n= M>IKJ ζ AhBF? R J}=@Ji>IKJ*BV? [x , x ] 8njO4=@? Ah? O?gAhBiJÊ8 `SÌ;HBL Ê8 bSÌ<AhB=@ALJ? R JGJ,? u R ;HBLF= ALJMH Ê8 ^|Ì ;>BLO=@ABiN8 ? JK?g\ A;HBieGhJKABJVO;>GhAh?z5 C ? R J f ;>Ok R 5euzjk R <;H\¾AhBJVO;HGAh?z5 C ;>BL ? R JYABLOk*JL IK;H? \gCAXBM>\gI [\gM>[ J*\ ? C AJ=<R 53AJ*GL? R ;2? k. k−1. k. | [mk−1 (xk−1 ) − mk−1 (xk )] − [mk (xk−1 ) − mk (xk )] |
(53)
(54) =
(55) − hsk−1 , gk−1 − gk i − 1 (hsk−1 , ∇xx mk−1 (ψk−1 )sk−1 i + hsk−1 , ∇xx mk (ζk )sk−1 i)
(56) 2. 6 ksk−1 k · kgk−1 − gk k + κ
(57) ksk−1 k2 .. M2 Me4= J*\gp>Jn? ;H? 4 Jk;HO= JlMH8 d }I Jr;HBp2;HGOJl? R JR M>\gJ*C I MeB ∇ f ;HBLXCMekg4?g;>AhBX? −R ;2g? k = k∇ f (x k−1. k. x. k−1. Ê 8har],Ì 8 1Jl? R JBN;H[[G5Y? R J ) − ∇ f (x )k x. k. x. ∇x f (xk ) = ∇x f (xk−1 ) +. Z. Ê8haeaEÌ. 1. ∇xx f (xk−1 + αsk−1 )sk−1 dα.. U R O=9? R J f >; Ok R 5euzjk R <;H\¾~AhBJVO;HGAh?z5 C ;>BLX8halieAhpeJn? R ;2? kgk−1 − gk k 6. Z. 0. 1. k∇xx f (xk−1 + αsk−1 )k · ksk−1 k dα 6. j3O4=@? Ah? O?gAhBiY? R A=<4 M>OBLABÊ8a]SÌ C 9JlM>4?;HAB7? R ;2? 0. Z. 1 0. κ7 ksk−1 k dα = κ ksk−1 k.. Ê8hatdÌ. | [mk−1 (xk−1 ) − mk−1 (xk )] − [mk (xk−1 ) − mk (xk )] | 6 (κ + κ
(58) )ksk−1 k = 2κ
(59) ∆2k−1 2. R J*\gJlJB;>GhG57O=@JÊ8 SÌ C ;>BLX? R Jl;ek?<? R ;H? x ∈ B 8 2 1 J}kM>BkGOLJ \gM>I ? A=n\ Jr=@OGh?~? ;2?n? JLJ*BMeI}AB;H? M>\=ABN? JJ[\gJ=g= AhMeBM> ;>BL LAP J*\n435 ;7VO;>Be?gA?z5 R Ak R A=nMHS? R R J}=g;HIKJ)M>\R LJ*\;>R =0? R J}J*\g\ Me\04 J?z9J*JBNR ? R J}IKMLJ*Gy;HBL ? R ρ˜J}M>4ÍzJkρ?gAhpeJ OBk? AM>B 8 \go ;e= LAhBAOiY=0? AR =A=~= O Mek4= J*AJ*\gBVpt;H? G? 5FAM>=@B IYC ;>GhJGSk*;>M>\ IKJB[M2;H\gJLNk*;>? [MX;H4? R GhJJKM>i> \;>= RLM2AhJBeA? BC iK;HB? R L ;2?~;HG?=@RMXJ? RAh? ;2J?\g;H? ? R AM>J}B ? \gIOO=@?@=@u ?0\ J4 i>JAM>= BFOk*\k*;>JL=gAh=zO O= G
(60) R A;e =n? R ? MJ AhBk\gJ;>= JAhB? R A=~k;>= J>8 k. k−1. k. &¶S¹Nº¶:
(61). )*. > >. U R J*B Ah? J\g;H? AM>B. j3O[[ Me= Jl? R ;H?~8a38 R MeGL-8j3O[[ Me= Jl O\ ? R J\ IKM>\gJn? R ;H? g ∆k−1. k−1. k. 6= 0. k−1. ;>BLX? R ;H?. (1 − η˜2 ) κ
(62) 6 min 1 − η1 , kgk−1 k. (3 − 2˜ η2 ) κ
(63) . A=~= OkkJ=g=@ OG-;HBL. . Ê|8a,Ì Ê|8a Ì. ∆k > ∆k−1 .. @=¨Oº¤k¹ kJr¹ = C =@ OG;>1 BJL}? R\g=@O?0=;H? [R [;2Gh? 5 xU R =J*Mex\ JI +s 8 s8 d)MH 6=f xMeBBNJ8S?nmn;>4G|8 = J*Ê|\gdHp>]eJ]>]SBÌSM2?gM7? LR Jr;2L?O? kR JJ0? k*R M>;HB?0=@?gA?g;>J*BV\?g;2= ?gηAh˜MeB ;HkBL −κ1
(64) A= GhAJAhB? R JAhBV?gJ*\gpt;>G (0, 1) C R Ak R AI}[GhAJ=<? R ;2? ;HBL? R O= κ (1 − η˜ ) < 1. Ê8hatÌ 1 (1 − η˜ ) < <1 (3 − 2˜ η ) 2 (3 − 2˜ η ) U R Jnk*M>BLAh? AM>B=:Ê8har,Ì C Ê|8arÌ C ;>BL'Ê8 SÌ C kMeI4ABJLKAh? R ? R JLJBA?gAhMeBMH β AhBX8 AI}[Gh5}? R ;2? Ê8ha s Ì kg k kg k ∆ < < . . k. k−1. k−1. k−1. 2. 2. 2. mdc. 2. 2. k−1. k−1. k−1. κ
(65) . k−1. βk−1.
(66) ^.
(67) ! " #$%&' )(*+ ,-(.)/.0. 0=~;KkMeB= JVOJ*BkJ C 8 AIKI}JrLA;2? JGh5Xi>Ap>Jr=? R ;2?. Ê8ha2^|Ì mnB7? R JMH? R J*\ R ;>BL C 9JlIY;t57;H[[Gh57T J*IKIY;K8d);>BLO= Jn? R Jl? \gA;HBieGhJlABJVO;>GhAh?z5Y? MKMe4?g;>AhBX? R ;H? mk−1 (xk−1 ) − mk−1 (xk ) > κ
(68) . . kgk−1 k kgk−1 k min , ∆k−1 = κ
(69) kgk−1 k∆k−1 . βk−1.
(70)
(71)
(72)
(73)
(74) mk−1 (xk−1 ) − mk−1 (xk )
(75) −
(76) mk (xk−1 ) − mk (xk )
(77)
(78)
(79) 6
(80) mk−1 (xk−1 ) − mk−1 (xk ) − mk (xk−1 ) − mk (xk )
(81) 6 2κ ∆2k−1. ;HBL7? R J\ J* M>\gJ C Ah? R Ê8ha2^|Ì C ? R ;H?. |mk (xk−1 ) − mk (xk )|. > |mk−1 (xk−1 ) − mk−1 (xk )| − 2κ
(82) ∆2k−1 > κ
(83) kgk−1 k∆k−1 − 2κ
(84) ∆2k−1 .. M2UÊ|8aSÌAIK[GAhJr=<? R ;H? (3 − 2˜η )κ
(85) ∆ . 2. k−1. 6 (1 − η˜2 )κ
(86) kgk−1 k. ;HBLX? R O=9? R ;2?. (1 − η˜2 )(κ
(87) kgk−1 k − 2κ
(88) ∆k−1 ) > κ ∆k−1 > 0.. JlB;HGGh5IY;t57;>[[G57U R JM>\gJ*I 8ha;HBLXLJrLOk*Jl \ MeI 8d C Ê|8dÌ C Ê8a`SÌ9;HBLbÊ8harb,Ì? R ;H? 1.
(89)
(90)
(91) f (xk−1 ) − mk (xk−1 )
(92) κ
(93) ∆k−1
(94)
(95) 6 |˜ ρk − 1| =
(96) 6 1 − η˜2 .
(97) mk (xk−1 ) − mk (xk ) κ kgk−1 k − 2κ ∆k−1. Ê8har`,Ì Ê8harb,Ì Ê8 d>],Ì. U R J*\gJ Me\ J C ρ˜ > η˜ ;HBLbÊ|d8 s Ì? R JBJB= O\ Jr=? R ;H? Ê8a Ì R M>GL=8 2 kO\g\g J*?BVA?9=Ah?? RJ\gJ*;H\g? JJ MeA\ =<J}Bi>MHO?~;>B\g;>JBV;H\? J*k*Jr\ LNAh? ?Ak*R ;>;2Gh?lG5>? 8yR U JKR ? A\g=OA=z=<? uxJ*\gBJ*= ieOAhMe\gJBFLX\4V;>L57AO? R =lJk*;>BBJ3B?<M>? ?R J4M>J\gJ*LI Jrk8 \gJ;e=@JrLFABLJBA?gJ*G5Ah? R J )* &¶S¹Nº:¶
(98) > B j3O[[ Me= J? R ;H?08a38 R MeGL-8yjO[[ Me= Jl O\@? R J*\gIKM>\gJ0? R ;2?<? R J*\gJJA=z?=;)k*M>B=@?g;>Be? = Ok R ? R ;2? kg k > κ Me\~;HGG k 8SU R JBX? R J*\gJlA=0;}k*M>B=@?g;>BV? κ ? = Ok R ? R ;H? κ Ê|8daEÌ ∆ > κ ? M>\~;>GhG k 8 k. . 2. k. . k. ¨º¤¹ ¹ C U R Jl[\gM3MHA=<? R J=g;HIKJ;>=9 M>\~U R J*M>\gJ*I s 8 8 }AB f eM BB J?0;>G|8ÊxdH]>]e],ÌJkJ[?? R ;2? . κ . (1 − η˜2 ) γ1 κ
(99) κ = min 1 − η1 , . (3 − 2˜ η2 ) κ
(100) . i>GM>46;>\gGVM>k*I M>B3R p>J*J\g\ JieJ*M>BB k*C JS? MHR J? [R \ JM3;HM>G
(101) i>Me Me\ \Ah? R? R I J)?g4M0;>= A\gk=@?@? u \gM>O\=zL?nJ*\\ Jki>\gAAhM>? BAk*;H;HGV[[ [M>GAAJBe=?= Ah? R R J*MeB)O?A?yk ieR J*;HBBJie\g;HJ>? 8J1=
(102) M>JBGh5l\=@?nBLA?gJJ*LGO5kIKJl;>? B3R 5J =@Ok*kJr= =@ OGAh? J\g;H? AM>B=8 j3O[[ Me= J? ;2?8a8 M>GL8j3O[[ Me= J O\@? J*\gIKM>\gJ? ;H?0? J*\gJ);>\ JMeBG5XBA?gJ*G5 IY)* &;HB3¶S5¹Nº=@:¶O
(103) kkJr>= =@ OGVA?gJ*\;2? AM>B=*R 8U R J*B x = xR M>\ ;HGGV= O kAJ*BV? G5R G;H\gi>J k ;HBR L x R A=
(104) \=z? uxMe\gLJ*\ k*\ Ah? Ak*;>G|8 2. . k. ∗. ∗.
(105) `.
(106) ! " #$%&' )(*+ ,-(.)/.0. ¨º¤¹ ¹ C U R J=g;HIKJ;H\gi>OIKJBe?0;e=ABFU R J*Me\ JI s 8 8 AB M>BB J?;HGx8 Ê|d>]>]e],ÌIY;t574 J);H[[GhAJL = AhBkJ ? R J<\;>LAO=
(107) O[ L;2? J<A=ALJBV? Ak*;HGV?gMn? R ;2? MH? R J<4;e=@AkS?g\ O=z? \ fJi>AM>BIKJ*? R ML M>\ OB= Ok*k*J=g=z OGVAh? J\g;H? AM>B=8 2 6
(108) AB;>GhG5 ? J~BJ3?y?zM\gJ= OGh?g= JB=@O\ J<? Ji>GM>4;>GkMeBVpeJ*\gi>JBkJ9MH ? J~;>GhieM>\gA? I ?gMl\g=@?@u M>\LJ\ k*\ Ah? Ak*;>G [MeAhBV?= C 435C = R R M2ABi7AB;7\=z?=@? J*[ ? R ;2?;2R ?lGhJr;>=@?MeBJK;>k*k*OIOG;H? AM>BR [ M>ABV?MHSR ? R JKA?gJ*\;2? Jr=l=@JreOJ*Bk*J)A= \=z? uxMe\gLJ\9k*\ Ah? Ak*;>G|8 )* &¶S¹Nº:¶
(109) > j3O[[ Me= Jl? R ;H?~8a38 R MeGL-8SU R J*B M>BJ R ;>=9? R ;2? Ê8 dedÌ lim inf k∇ f (x )k = 0. . . x. k→∞. k. ¨º¤¹ ¹ C j3JJU R J*Me\ JI s 8 8AB f M>BBJ?0;>G|8ÊxdH]e]>],Ì8 2 \=z? 0uxMe=)\gL M>J\)\9k*? R\ AhJ? Ak*4;>;eG|=@8 AkY?g\ O=@?@u \gJ*i>AM>B IKJ*? R ML C ? R A=)k;HB$4 JXJ*3? J*BLJL$? M = R M2 ? R ;2?Y;HGGGAI}Ah?K[MeAhBV?g=K;>\ J )* &¶S¹Nº:¶
(110) > j3O[[ Me= Jl? R ;H?~8a38 R MeGL-8SU R J*B M>BJ R ;>=9? R ;2? Ê8 d>,Ì lim k∇ f (x )k = 0. . . k→∞. x. k. ¨º¤¹ ¹ C j3JJU R J*Me\ JI s 8 8 s A B f M>BBJ?0;>G|8ÊxdH]e]>],Ì8 . . 2. ! "$#% '&( "#% 4/.1 4 +
(111) &8+#%9;:=<>+ &8-. JYBM2 ABVpeJ=@? Aie;H? J}? JY[ Me=g= Ah4AGA?z5N?gMNJ[GM>Ah?=@JrkMeBL3u M>\LJ*\AhB M>\gIK;H? AM>BM>B? JYM>4fÍzJrk? Ap>J} OBk? AM>B A? R ? R J;>AhI M>JB= O\ RABiKkMeBVpeJ*\gi>JBkJ0? M= JkMeBL3u M>\LJ*\<k\gAh? Ak*;HG[ M>ABV?g= C A|8 J>8y[ M>ABVR ?g= x = Ok R ? R ;2? C ;HBL ∇ f (x ) A=[ Me= A?gAhpeJ=@JI}ALJ*BAh? J . ∇ f (x ) = 0 m0 k*M>O\= J 9JBJ*JrLX? M7k*G;>\ Ah 5 ;H?0J[\ JrkA=@JGh5IKJr;HBN435 Î =@JrkMeBL3u M>\LJ*\<AB M>\gIY;2? AM>BfÏ8 1J? R J*\gJ Me\ J AhBV?g\ MLOk*JC ? R Jl MeGhGM2AhBi};eLLAh? AM>BR ;>G-;>=g=@OIK[?gAhMeB=w U R JI}MLJGA=S;>= 53I}[? MH?gAk;HGGh5)=@JrkM>BL3uxMe\gLJ*\ykM R J*\gJ*BV?yA? R ? R J~Me4fÍzJk? Ap>J9 OBk? AM>BYBJr;H\ \=@?@u M>\LJ*\ k*\ Ah? Ak*;>G[MeAhBV?g= C Ax8 Je8 R J*BJ*p>J\ lim kg k = 0. lim k∇ f (x ) − ∇ m (x )k = 0 U R J 0J=g=@A;HBMH<J*peJ*\g5NIKM3LJ*G A=T Ah[=gk R Ah?«¾YkMeBV? ABVOM>O= C ? R ;2?A= C ? R J*\gJYJA=z?=; kM>B=z?;HBV? κ m = Ok R ? R ;H? C Me\~;HGG k C k∇ m (x) − ∇ m (y)k 6 κ kx − yk Me\~;HGG x, y ∈ B 8 ? R J=@IY;HGGJ=@?<JAhieJ*B3p2;HGOJ MH? R J 0J=g=@A;HBMH
(112) ? R JIKMLJ*G m ;2? x A=<BJie;2?gAhpeJ C ? R J*B τ 1. ∗. x. xx. ∗. ∗. . k→∞. xx. k. xx. k. k. k→∞. . k. xx. k. xx. k. k. . k. k. mk (xk ) − mk (xk + sk ) > κ 7. Me\~=@MeI}Jk*M>B=@?g;>Be? κ 7 ∈ (0, ) 8 . 1 2. k. |τk | min(τk2 , ∆2k ). k.
(113) b.
(114) ! " #$%&' )(*+ ,-(.)/.0. U R J= J;>=g=@OI}[? AM>B=;>\ JlALJ*BV? Ak*;>G ? MK? R Me= JlO=@JrLAhBFj3Jrk? AM>B= s 8);>BL s 8 s M> MeBB J?~;HGx8 ÊxdH]>]e],ÌS M>\<? R J 4 ;>= AkY? \gO=@?@u \ Ji>AM>B IKJ? R ML8 B ;ek? C ? R J =@JrkMeBL3u M>\LJ*\}kM>B3peJ*\gi>J*BkJY[\gM>[ J*f \ ? AJ=)MH? R JX\gJ?g\ MV=@[ Jk? Ap>J ? ;H\gBOLX=@?@? u \ JJ*i>A\~AM>[BK\gM3IKMHJ=~? R kM3;HL7B;HJG=g=@=@MlJBe?gO?gA\ ;>B7GhG57M>O4 ?J*? JMB 4 4J0MeJ*\ \gM2;ek?gJrGhLY5) ? \gR M>JI = ;>? IKAJ0=0;>k=S;>= ? J R MV=@JnAh? MH ? ? R JJnJ4;>k= JAk<[? ? A\gM>OB =@?@M>u \ JT i>J*AM>IKBKIYIK; J? R 8M38 L C8 R R C R R s 1 J? R J\ J* M>\gJBJ*JL?gMY[\gJ= J*BV?0;K[\gMVM>M>
(115) ? R ;2?n[;H\ ? AkOG;H\\gJ= OGh? M>\? R JBJ IKJ*? R ML8<0=0JABLAk*;H? J 4JGhM2 BFC M>;HOG\G MH;H? BR ;>J*Gh\9MeiY\ JrMH=@SOGhT-?g=JIKi>JIKB;J*\s;HG8A©¾8 J~ 9Ah? JR Me;eO= = ?9Ok IKR J;>B? i>R Jn;2?0;HB? RLYJ9IKJ0MMeLBJG5}G
(116) \gIKJLJOBe?gkAh? MeAM>BYBN? R A=~J*I Jp>J M>Be\?gO? ;HR GJlG5=g;|= É>AhieJ0BMHAh k*k;>G;HBe\g?nAh?zAh5>B 8 ? R JX=@JB=@J? R ;H?A?}A=);2?GJ;e=z?)MH<? R C JX= ;>I}J7M>\LJ*\;e=? R JJ*\g\ Me\4J*?zJJ*B ? R J7IKMLJ*G9;HBL? R J7Me4fÍzJk? Ap>J OBk? AM>B-8 1Jl? R JB= R M2#? R ;2?? R J? \gO=@?@u \ Ji>AM>B7\;>LAhO=94 Jk*M>IKJ=;e=@53IK[? M>? Ak*;>GhG5YAh\g\gJ*GJ*p2;HBV?<A
(117) ? R J=@? J[= ? JBLX? M¾*J\ M8 V:¶
(118)
(119) ! > j3O[[MV=@Jl? R ;2?8a38 C ;HBL8 R M>GL89j3O[[MV=@J;>G= M}? R ;H?~? R J*\gJJA=@?g=;Y= JVOJBkJ ;>BL ;KkM>B=z?;HBV? κ 7 > 0 = Ok R ? R ;2? (k ) Ê8 d Ì m (x ) − m (x + s ) > κ 7 ks k > 0 M>\~;>GhG i =@O k*AhJBe?gGh57G;>\ ieJ>8y6AhB;>GhG5 C = O[[ Me= Jl? R ;2? . . . i. ki. ki. U R J*B Ah? J\g;H? AM>B k A=~= Ok*k*J=g=z OG-;>BL. ki. ki. ki. lim kski k = 0.. i→∞. i. M>\ i =@Ok*AhJBe?gGh57G;>\ ieJ>8. 2. . ki. ρ˜ki +1 > η˜2. ;HBL. Ê|8d>Ì. ∆ki +1 > ∆ki. @=¨Oº¤k¹ kJr¹ = C =@ OG1 Me\Jni\g= =@O?<k;>[A[J*BVG5? G57T-JGI};H\gIYi>Je; 8 s 8M28 C M>k* M>f BM>= ABLBXJ*\ J?~k ;HGxM>8´BÊ|Jld>]>= O]>]Sk RÌ ?gAhM}? J*L\;2Jr?gLAhOMek*B-Jn8? U R R ;2J?<J*JrpeVJ*O\g;25K?gAhAMe?gBJ*\=2;2Ê? AM>8 B Ì<k;HBAL = Ê8ÌAIK[Gh57? R ;H? M>\~= M>IKJ ξ ;HBL ζ AhB? R J=@Ji>IKJ*BV? [x , x ] C . i. i. ki +1. ki +1. ki. ki +1.
(120)
(121)
(122)
(123) f (xki ) − mki +1 (xki )
(124) |˜ ρki +1 − 1| =
(125)
(126) mki +1 (xki ) − mki +1 (xki +1 )
(127)
(128)
(129)
(130) hsk , ∇xx f (ξki +1 )ski i − hski , ∇xx mki +1 (ζki +1 )ski i
(131)
(132) =
(133)
(134) i − hski , gki +1 i + 12 hski , ∇xx mki +1 (ζki +1 )ski i
(135). Ê8 d s Ì R J*\gJ)9JK;HG=@MO= JLN? R J f ;HOk R 5Vuzjk R 9;>\«¾ABJVO;>GhAh?z5>8}:95F=@O4=z?gA?gO? ABi g = ∇ f (x ) Ê4J*u \ k*J;HO=@\ JXAh?@?gMHJ*nB ;>8= d,ÌAh? R A?=}J[\gJ=g=@AM>B$ABXÊ|8a>a¤Ì C ? R J LJ*BMeI}AB;H? M>\ D M>~? R JXG;2? ? J*\} \;>k? AM>B k;HB 4 J ks k2 · k∇xx f (ξki +1 ) − ∇xx mki +1 (ζki +1 )k
(136) ki
(137)
(138) − hski , gki +1 i + 1 hski , ∇xx mki +1 (ζki +1 )ski i
(139). 6. 2. ki +1.
(140) Z
(141) D =
(142)
(143) − ski , gki +. U R J*B C \ J[G;ekABi D. 1. . ∇xx f (xki + αski )ski dα +. 1 2. x. ki +1.
(144)
(145) hski , ∇xx mki +1 (ζki +1 )ski i
(146)
(147) .. 435Ah?g=J*[\ Jr= = AM>B7ABÊ8 `SÌ C 9JMe4?g;>AhB. 0. − hski , gki i
(148)
(149) =
(150)
(151) mki (xki ) − mki (xki +1 ) +. 1 2. hski , ∇xx mki (ψki )ski i. Z + 21 hski , ∇xx mki +1 (ζki +1 )ski i − ski ,. 1 0.
(152)
(153) ∇xx f (xki + αski )ski dα
(154)
(155).
(156) a].
(157) ! " #$%&' )(*+ ,-(.)/.0. M>\Y=@MeIKJ ψ AhB ? R JN= J*ieI}JBV? [x , x ] 8 U R J7?g\ A;HBi>GJXABJVO;>GhAh?z5 C [\gM>[ J*\ ? AJ=}MH~? R J AhBV?gJ*i>\;HG C Ê8d Ì C ;>BL f ;>Ok R 5Vujk R <;H\¾~ABJVO;>GhAh?z5i>Ap>J? R J*\gJ Me\ J? R Jl M>GGhM2ABi}GhM29J*\<4 M>OBLMeB D w ki. D. ki. ki +1. > |mki (xki ) − mki (xki +1 )|
(158) Z 1
(159) 1
(160) [∇xx f (xki + αski ) − ∇xx mki (ψki )]ski dα − 2
(161) s ki , 0
(162) Z 1
(163) + s ki , [∇xx f (xki + αski ) − ∇xx mki +1 (ζki +1 )]ski dα
(164)
(165) 0. > κ. 7 . 2. kski k − 21 kski k −. > kski k2 (κ 7 − 12 i ). R J*\gJ def. i =. 1 2 kski k. Z. 1. Z. k∇xx f (xki + αski ) − ∇xx mki (ψki )k · kski k dα. 0. Z. 1. k∇xx f (xki + αski ) − ∇xx mki +1 (ζki +1 )k · kski k dα. 0. 1. k∇xx f (xki +αski )−∇xx mki (ψki )k dα+. U R Jl? \gA;>Bi>GJAhBJrVO;HGA?z57BM2 AIK[GhAJ=? R ;H? 0. 1. Z. k∇xx f (xki +αski )−∇xx mki +1 (ζki +1 )k dα.. 0. k∇xx f (xki + αski ) − ∇xx mki (ψki )k 6 k∇xx f (xki + αski ) − ∇xx f (xki )k. ;HBL C =@AIKAhG;H\gGh5 C ? R ;H?. Ê8 dV^|Ì. + k∇xx f (xki ) − ∇xx mki (xki )k + k∇xx mki (xki ) − ∇xx mki (ψki )k. k∇xx f (xki + αski ) − ∇xx mki +1 (ζki +1 )k 6 k∇xx f (xki + αski ) − ∇xx f (xki +1 )k. j3AhBkJ9JBM2 M>4=@J\ peJ0? R ;2?. + k∇xx f (xki +1 ) − ∇xx mki +1 (xki +1 )k + k∇xx mki +1 (xki +1 ) − ∇xx mki +1 (ζki +1 )k.. k(xki + αski ) − xki k 6 kski k, k(xki + αski ) − xki +1 k 6 kski k,. Ê8 d>`,Ì Ê8 d>b,Ì. kψki − xki k 6 kski k, kζki +1 − xki +1 k 6 kski k,. JlIY;t57LJrLOkJl? R ;2?~4 MH? R. H; BL k∇ f (x + αs ) − ∇ m (ζ )k kM>B3peJ*\gi>JS?gM2¾*J\ MA? R ks k 4Jrk*;>O=@J9? R J<\=z?;>BL? R A\gL)? J\ IY= M>? R J\ Ai R ? u R ;>BL= ALJMHÊ8dH`SÌ;HBL Ê8dHbSÌy?gJ*BLX?gM ¾J*\gM4357k*M>BV? AB3OA?z5YM> ? R JM>4fÍzJrk?gAhpeJ0 OBk? AM>B =;>BL7IKMLJG| = ~Jr= = A;>B= ;>BLX4Jrk*;HO=@J ?R 8JY7IK;>BALLFLU GJY? JJM>\ \gI J*I AB8 ? ^3R 8lJ70\ A=li ;7R ?@k*u RM>B;>B= JLVO= AJLBJYkJ MH? R J= JYABJV O;>GhJ*AhB ? AJ=A=l;>G= = O M kkM>AB3J*peBVJ*? \gG5Ni>JG=n;>\ ?gieM5J ¾J*;H\gC BM LN4? Jrk*J;H\ OJ*=@ JKM>\gMHJ kM>I)4ABAhBivR Ê|8d s Ì;>BLbÊ8de^Ì C ;>BL7O=@ABiK? R C Jl ? \g6A;>Bκi> G7Jn ABJrR eO;HGiA?z5 C JlM>4?g;HAB C R C k∇xx f (xki + αski ) − ∇xx mki (ψki )k. xx. ki. ki. xx. ki +1. ki +1. ki. i. |˜ ρki +1 − 1| 6 6. . 2 k∇xx f (ξki +1 ) − ∇xx mki +1 (ζki +1 )k κ 7 2 k∇xx f (ξki +1 ) − ∇xx f (xki +1 )k κ 7 + k∇xx f (xki +1 ) − ∇xx mki +1 (xki +1 )k + k∇xx mki +1 (xki +1 ) − ∇xx mki +1 (ζki +1 )k. . Ê8 e],Ì.
(166) a>a.
(167) ! " #$%&' )(*+ ,-(.)/.0. : 5 ? R JX=g;HIKJ\gJ;>= M>BAhBiN;e= M>\Ê8dH`SÌx´Ê8dHbSÌ C ? R JX\ Ai R ?@u R ;>BL =@ALJM> Ê8 >]SÌn? JBL=? M'¾J*\gMN R J*B ieMVJr=9? MABBA?z5 ;HBL ? J\ J* M>\gJl? JBL=?gMFa>8 ?nA=<? R O=0G;>\ ieJ*\<? R ;HB η˜ < 1 M>\ i =@Ok*AhJBe?gGh5 i G;>\ ieJ;>BLJÊ8d>,ÌyC MeGhGM2~ρ˜=8 R 2 0=AB$T-JIKIK; 88 M> M>BB J*?);HGx8VÊ|d>]>]e],Ì JYIY;t5 ;H[[Gh5? A=\gJ= OGh?? M ? JJ*BV?gAh\gJ7= JVOJ*BkJYMH A?gJ*\;2?gJ=y;HBL}LJLOk*Js ? R ;2?;HGGf Ah? J\g;H? AM>B=y;>\ JJp>J*BVC ?gO;HGGh5)=@OkkJr= =@ OG;>R BL)? R J?g\ O=z? ux\gRJ*ieAhMeB\;>LAO= 4 M>OBLJL ;t9;t65}\gM> \gI M>I0J*¾*\gJ*J\gMM>8 B ? J? JM>\g5XAB M>BB J*?;HGx8 ÊxdH]>]e],Ì9ieJ*BJ\g;>GhA¾*Jr=<A? M>O?=@Ai>BAhk;HBV?0k ;HBi>J 5VAJ*GLABi R R C f ? R Jl MeGhGM2R ABi)\gJ= OC Gh?gR=8 R ki +1. 2. j3O[[ Me= J? ;2?8ha38 M>GLN;HBL? ;2? A=0;= O4= JVOJBkJMH ? J)A?gJ*\;2?gJ=ieJ*Bu *J)*\ &;2?g¶SJ¹NL º¶:43
(168) 5 0> Gh ieM>\gA? R I Z U Z R k*M>B3p>J\ ieAhBi7R ?gM; \=z? uxMeR \gLJ*\x k\gAh? Ak*;HG[MeAhBV? x R J\ JR ? R J 0J=g=@A;HB M> ? R J}M>4ÍzJk?gAhpeJ OBk? AM>B ∇ f (x ) A=0[ Me= A?gAhpeJ)LJ*BAh? Je8j3O[[MV=@J O\@? R J*\gIKM>\gJ? R ;2? s 6= 0 Me\n;HGG =@AO Yk*AhJBV? G5XG;H\gi>Je8U R JB ? R Jk*M>IK[GJ?gJ=@JrVOJ*Bk*JMHyAh? J*\;2?gJ=~k*M>B3p>J\ ieJ=? M x C ;HGG Ah? J\g;H? AM>B=0;H\gJ kJ*peJ*BV? O;>GhG5peJ*\g5=@OkkJr= =@ OG ;HBLX? Jl? \gO=z? ux\gJ*ieAhMeB7\;>LAO= A=4 M>OBLJL;t<;t5K \ MeI¾*J\ M8 C R ∆ ki. ∗. xx. k. ∗. ∗. k. ¨º¤¹ ¹ C j3JJU R J*Me\ JI s 88AB f M>BBJ?0;>G|8ÊxdH]e]>],Ì8 2 GhJr;>=@1 ?M>JlBBJM2= #Jk[Me\gBM3L3MHu M>?\R L;2J*?<\<Ak \gAh? ? R Ak*Jl;H=@G-Jr;eVk*Ok*J*OBIk*JnOGM>;2-?gAAh?gMeJ*BX\;2[ ?gM>JA=BV\ ?JAIK=;>ieAhOB;H=\A;HBXBV?g;}J*JkL-MeI}8 [;>k?<=@J*? C ? R J*BX? R JJA=z?gJ*Bk*J0MH;2? . j3O[[MV=@JF? ;H?X8a8 ^ M>GL#;HBL ? ;2? ;HGG0Ah? J\g;H? J=\ JIK;>AhB AB = M>IKJFkMeI}[;>k? @= J*&?8¶S¹NU ºR ¶:J
(169) BF? > R J*\g. JJ*3A=@?g=l;2?GhJrR ;>=@?nM>BJ)GAhIKAh?R [ M>ABV? x MHyR ? R JY=@JreOJ*Bk*JMHSAh? J\g;H? Jr= x [\gMLOk*JLN435 ~Gi>Me\ Ah? R I Z U Z C R Ak R A== JkMeBL3u M>\LJ*\<k\gAh? Ak*;HGx8 )*. k. ∗. ¨º¤¹ ¹ C j3JJU R J*Me\ JI s 8 s 8AB f M>BBJ?0;>G|8ÊxdH]e]>],Ì8 2 :5 ÍzO=@?0=z?g\ JBiH? R J*BABi}? R J\;>LAhO=<O[ L;H? J\gOGJ435X\ JreOAh\gAhBi? R ;2? A ρ˜ > η˜ ;>BL ∆ 6 ∆ , ? R J*B ∆ ∈ [γ ∆ , γ ∆ ] Ê8 aEÌ GhAM>IK\A=@?~Me[ I}M>J ABVγ?~MH>
(170) ? γ J>= JV1O;>JBBkLNJl= MHM> IKAh? J J*∆\;2?gJ=<ie>J*B0JC\g;H9? JJL7IK43M>57\gJ*~M2peGi>J*Me\\ M>Ah? 4?gI ;HABNU? R J)8 = Jk*M>BL3u M>\LJ\0k\gA?gAk;HGA?z5MH;>B35 R R Z Z . k. 4. k. 2. 3. max. k+1. k. 3. 4. k. max. j3O[[MV=@J)? ;2?8a38^ ;HBLÊ8 aEÌ M>GL;HBLFGJ? 4 J};>B35NGhAIKA?[ M>ABe?lMH? R J =@Jr&VO¶SJ*¹NBºk*:¶ Jl
(171) MH > A?g2J*\ ;2? Jr=*8yU R JB Rx A=;Y=@JrkMeBC L3u M>\LJ*\<k\gA?gAkR ;HG[ M>ABV?8 x. )*. ∗. ∗. ¨º¤¹ ¹ C j3J*JU R JM>\gJ*I s 8 s 8 `)AB f MeBBJ?0;HGx8 ÊxdH]>]e],Ì8 2 U O=y? J0\gJ? \gMe= [ Jk?gAhpeJ<? \gO=@?@u \ Ji>AM>BY;HGi>Me\ Ah? I = ;H\gJ=S;>GhG? J Ê ABe?gJ*\gJ=@? ABiSÌSkM>B3peJ*\gi>J*BkJ<[\gM>[ J*\ ? AJ= MH? R JR 4;>= AR kn?g\ O=@?@u \gJ*i>AM>BIKJ? R MLXOBLJ*\0? R J=g;HR IKJ;eR = = OIK[? AM>BR =*81JkMeBkGOLJl? R A=? R J*M>\g57= Jk? AM>B 435 BÊ [ MHMe?gAh=gB=@Ai)4G? 5 R ;2A??gJ*? \R ;2J? AM>;H4 B M2Lp>J*Jl[ kJ*MeBB3Lp>J*J*BV\g?i>Ì)JBOkBJAh \gM>J\g= IKOGGh?5 =Jr;HV\gOJAh=@p2? ;>AGhGJG-Bep2?};HGBALM>\gAhI MeB? JJ*\ \gJJ*[43G;>5 kJr;H=GGhM2? RJ ABv iFO[kG\gAM>L4JrGh;HJB I BMe=gk*\ ;>I GhAB43i5;H;HBB3L5 C R [\gJkMeBLAh? AM>BABi8 .
(172) ard.
(173) ! " #$%&' )(*+ ,-(.)/.0. Ã 3È È3Á,?2Ã Á Æ C 6 ÃÈ3Ç3? AN6S8:6 ÃÈ6Á5Ç36 1 JBM2k*M>B= ALJ*\ ? R J~B3OIKJ*\gAk;HG4 J R ;tp3AM>O\yM>? R J~BJ;>GhieM>\gA? R I C AhBk*M>IK[;>\ A=@MeBAh? R ? R J4;>= Ak9? \gO=z? u ?\ RJi>J7AM>a B s ;>OGhieBM>k\gMeA? BR =zI?g\g;>:<AhBU JLZ [Ê\g= M>J*4JGJ*[IY;>i>=JM>a>a? Rs J MH f Me BB
(174) Sk*J*M>?KGGhJr;>kG|? 8AM>ÊxdHB ]>]eÊ|=@],JÌ«JÌ8 M>1OJXGL ?gC Jmn=@?}\ 44 ;HMHBN? R ;HB;>LNGhieU
(175) M>\gM>AA? BVR ? IYC =dH]>Me]eB ,Ì;>8GhG<6MeMH\ ? R J[\ Me4GhJIY= R Me= JLAI}JB= AhMeBIY;t5" 4 J+3 k R>;> + B i>JrL C 9Jk R Me= J;K\ Jr;>= M>B;>4Gh5= IY;HGG-p2;HGOJABMe\gLJ*\BM>?? M M2p>J\ GMe;eL? R J
(176) 9AhBV?gJ*\ ;>kJ}Ah? R 8KU R JY=z?;H\ ? ABi[MeAhBV?g=;>\ J)? R JK=@?g;>BL;>\gLM>BJr=l[\ M2p3ALJL 435Y? R J
(177)
(178) GAh4\;H\g5>8 6M>\7? R J 4;>= AkF;HGi>Me\ Ah? R I ? R JF? \gO=@?@u \ Ji>AM>B \;>LAO=O[ L;H? JF<;>=7AhIK[GJ*IKJBe?gJL#435O= AhBi$? R J \ OGJ [\gM>[ Me= JLAB f M>BBJ?0;>G|8 Ê|dH]e]>C ]SÌ C [-8 ^H`>w A ρ > η C max[γ ks k, ∆ ] A ρ ∈ [η , η ) C ∆ ∆ = A ρ ∈ [0, η ) C γ ks k A ρ < 0 C min[γ ks k, max[γ , θ ]∆ ] R J*\gJ γ A=JL ;H?]8 ] s de C γ ;H? 0.25 C γ ;H? 2.5 C η ;H? 0.05 ;>BL η ;2? 0.9 ;HBL R J\ J θ A=i>Ap>J*B 435Ê|d8 d,Ì8SU M};tp>MeALY4A;e=@ABi? JlkM>IK[;>\ A= M>B J ;tp>JLJk*ALJLY?gM)IY;É>Jn;e= J* ;eL;H[?;2?gAhMeB=;e=[ Me=g= Ah4GJ ? OMB= ? OR k*;2k*?YJ=g\ =zO OGhJ G AAxB8 J>8MeO\K\ J*? \gMe= [RJrk?gAhpeJXp2;H\gA;HBV?vC ÊA|8 J>8 R jV?gJ*[#d AhB#~Gi>Me\ Ah? R I d8aEÌ8 U R O= C AhnAh? J\g;H? AM>B k A= ;>BL kMeB=@JrVOJ*BV? G5 9J;HG= MKLJk\gJ;e=@J? R J? \gO=@?@u \ Ji>AM>BXO=@ABiK? R J ;H4 M2p>J)\ OGJ>8C C M>Bρ? R <JKkηMeBe?g\g;>\ 5 C A?gJ*\;2? AM>B k xA=l=@=OkxkJr= =@ OC G C Ax8 J>8 ρ > η C ? R J)? \gO=@?@u \ Ji>AM>BFA=O[ L;2?gJL dlMH-;H~BGLi>Me\ Ah? R I d8ha~ R J*8
(179) \gU JJ0J0IKk R MMVLMVJ*=@GJ<9? ;eR =Jnk =g;HMVIK=@JJ~B p2;>AGhBYOJr4=M>?;e=S;Hk;>4 = M2Jp>= J OM>= \ AhBγiC? RγJ~[;>\ BML kJrγLOC \g;>J0BL}LJr?g= ;k*É>\ JA4η˜JrLK=AhB7ηj3? =J*[0.05 ? ? MNJ4 ? JK\gO? R=z? J7ux\gJJ*ie;>AhMekB?U
(180) <;t;>53=GM>k*\rM> =lIK= [J*\gOAh?gJrJ=L ? \gJOA? Bk*J*;H\)? JrJL;>?gkMF? G=@5Jrη˜kOMe= =BAhBL3iηu M>? \=LJJ*0.9 \ ENC ;HM>B\gLquzR j3? RM>\gJJ*BIK=@AhJBB AhIK;HAG¾*i>J*Me\\ R AhMH? 9I ? R CJYÊ= J*IKJ7MEFLJ*M>R G\gAhqB=@;HABLL J C j3R M>\gJ*B=@JB C abe`>SÌ0C Me\;H[[\ MtAIK;H? JGh5NOR = ABi? R JXj3? J*A R ;>OiHuRU MeAhBV?;HGi>Me\ Ah? R I Ê= J*JjV?gJ*A RR ;HOi C arb>`> C U
(181) M>ABe? C abe`aEÌ8K B ? A=k*;e=@J ? Jk*M>BÍzOie;2?gJKi>\;>LAJ*BV?A?gJ*\;2?gAhMeB=;H\gJ}=@? Me[[ JLA<? R JK? \gO=@?@u \ Ji>AM>B4 M>OBL;H\g5FA= IKJ?~M>\;e==@M3R M>B ;>=9? R C JR IKMLJ*G=*i>\;>LAJ*BV?= ;H? A=zJr=9? R Jk*M>BLA?gAhMeB 6A %C. 2. k+1. 1. k. k. 0. 1. k. k. 0. 2. k. 1. 2. 1. 1. 1. 2. 1. k. 2. k+1. k. 0. 2. k. 1. k. 1. k. k. k. 1. k. k. k. 2. 1. 2. @= IYJ<;>kGhGMeJ*B\= ?ALR J;H\ B JrLl10? R ;2?8F? R6J9;>AhAG?gOJ*\g\J;2?g<Ahpe;>J=[L\gJrMkkGJ;H=g\g= Jk*LM>B3Ahp>J? R\ ieJJLl;HGi>M>R \gJAh?BR ? I R J Lv AL OkBGAMHL?}J;Hk*B)M>B3Bp>M>J\g\ I ieJ)MH? Ah?R R J9AhBi>\;>? R LJ7AJ*BVIY?
(182) ;H4 3JAkI;HOIKI J B3OI4 J*\M>y>]]e]>])A?gJ*\;2?gAhMeB=*8 M>\B31 OIJ4 k R J*\MV=@MHJS?g MOBkkM>?gIKAhMe[B;>\ J*Jnp2;>? RGhOJ;H? BVAOM>IB=4 8lJ*\~ BM>L
(183) JAhJ? L J*\C ;2? ?gR AhMeJ}Bk*=9Me? =@M?n[ ;eJ*k R\lAAhJ*? p>J*Jl\;2k?gMeAhMeB3BFp>J*A\g=~i>J? RBJYkJn= Ah;>BI}=z?gJJ;> M>L\M>4
(184) M>? ? R R J ;Hf Gi>QM>o \gAh? ? R AIKIYJ = ;HBL)? R J*5}4M>? R Jp2;HGO;2?gJ<? R J~Me4fÍzJk? Ap>J< OBk? A" M>BY+3 MeeB+ kJ [ J*\A?gJ*\;2?gAhMeBK;>BLYkMeIK[O? J0M>BJie\g;eLAhJBV?y;H?J*peJ*\g5 =@Ok*kJr= =@ OGAh? J\g;H? AM>B-8yENMe\ JM2p>J*\ C ? AI}ABiV=AhB ;H\gJlMH,? JBNLA k*OG??gMYAhBV?gJ*\g[\ J*?8 ~GG \gOB=9J*\gJl[J\@ Me\ IKJLXAB E;2? G;H4 p 8V^8hae8 ]8har`>ËÊ Z a Ìj3J*\gp3Ak*J Q ;>kÉ7KMeB;Y8 dK R ¾ BV?gJ*G = AhBi>GJu [kMe\g\ MHJG[J\ MMH k*J? =gR =@J0Me\BVOk*M>IIK4 [J*\O?gM>J* \ Ah? JA\g? ;HR ? AdM>Bl=S: MH M>? R Z J0?z9E$M8H6
(185) ;HGAi>i>MeO\ \ AhJ ? R IK8a=\g8 J*Q [J\ Jr\@=@MeJ\ BeIY?=-;H? BR kJJ<k[Me\gI}MH[G;HJ\g=yA= ieM>AhpeBJ 43C 5 M>;n\S[Jp>J\@J*\gMe5 \ IYσ ;>>Bk1J C ? R J[\gM>[ M>\ ? AM>B p(σ) MH3?gJ=@?
(186) [\gM>4GJ*IY= M>B R Ak R J;ek R kMeB=@ALJ\ JrL;HGi>M>\gAh? R IKAk pt;>\ A;HBV? R ;>= ;~[J\@ Me\ IY;HBkJ ?R AA? =XR AB i>O;\ J;eC k?gM>J \ R σ;tp>MHJM>? RBJG5 4É>JrJ*=z[??7Ê|=@? JR JFJ W0[Me\gM>G;>4B GhJIK;>B=7L M>EN\M> \gq R C Ak dHR ]e]e4 d MHC ? R M>\7;HG; i>MeIK\ AhMe? R \ JNIYk=XMekIKM>[B3GhpeJ*J*? \gJFi>JLL A=gk?gOM =g=@? AR M>J BAÌ=g8 ;HIK B J Gh%!M#%k;HG3 = 8(M>9 GO% ?gAhMe B'%'Ê 9 4 J~ J kGO'L5Jr% L ;H B <L C 3= !,#" ($ %'&($ C Ì)8 * + ,? %-.JF =@OC 4/ 0[1 \ Me2 4)G* J* I 3 C A0=Y4=@Me* Ghpe+ J5L " ;H5C [6 [1 \g" Mt1 'AhIY7 C ;H%!? J*4%G5 ) 3 48(M>9.? : C ;HGi>Me) \ Ah? R IY=C ;H1 AGJL M>B?;C > 2 + 0(6 1 2 A@ )C 4 " % $ C 1 ) + 6!5% 1 C R ) 9' 0 "@5B C ) 9' 0 " : B};HBL? ) 9' 0 "'7 B8 EFM>\gC J*M2p>J\R C U ;HAGhJrLKM>B 3 7 Ak <;>=y= M>Gp>JL}435K:9U 83mnB}? J0MH? J*\ ;HBL A? J0= O4[\gM>4GhJI A== M>Gp>JL JZ ;>Z k? G5 C 4 MH? R ;H/ Gi>0(M>1 \g2 Ah? ) R IY=
(187) C ;HAGhR JrLR MeB / 01 2 ) 3'7;HBL}:<Z U Z ;>AhGJR L}MeB R ) R !5% 1 C C R R Ak R 9;e=y=@MeGhpeJL4V5 Z U Z 8 M>? J7;>G= M ? R J7BVOI4 J*\M>A?gJ*\;2? AM>B=BJ*JrLJL?gMF\gJ;>k R kMeB3p>J*\gi>JBkJKAh? R ? R J Z U Z ;HGi>Me\ Ah? R I M>B? R J RkMeAi I}R [G5GhJ*B? JlMeBB3kOMeIKBVpeJ*\gJA k;H* G9 \g6 J>'= }OG;H?=BL;H\g0 Jn(ieAh*pe+J*B5 Ah")BN[0\ [Me[4JGhJBIYLAh=A=<I8 Ok RXR Ai R J*\9? R ;>B7 Me\<? R J:<U Z ;HGi>M>\gAh? R I 8yU R J k∇x mk (xk + s)k 6 min[0.1, k∇x mk (xk )k1/2 ]k∇x mk (xk )k.. 1. −5.
(188) a.
(189) ! " #$%&' )(*+ ,-(.)/.0 RTR vs BTR 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2. MS−BTR MS−RTR TCG−BTR TCG−RTR. 0.1 0. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 6AhieO\ J 8a>w Q J*\ M>\gIY;HBk*JS[\gMHGJk*M>IK[;H\gABi? R JB3OI)4J\
(190) M>A?gJ*\;2? AM>B=-MH? R J Z U Z ;HBL:9U Z ;HGi>M>\gAh? R IY= mnO\l\ Jr=@OGh?g== R M2 ? R ;2?l? R JK\ J*? \gMe= [Jrk? Ap>J};HGi>Me\ Ah? R I [ J*\ M>\gIY=l;>=JGhGS;e=n? R JYkG;>=g= Ak;HG M>BJK;HBLA= ÍzO=@?9;>=\ JGhA;H4GhJ~Ah? R J~? \gO=@?@u \ Ji>AM>BY= O4[\gM>4GJ*I A== M>Gp>JrL};>[[\gMtAhIY;2?gJ*G5>8~M29J*p>J\ C A? R J0[\ Me4GhJI = A¾*J M>\9=z?g\ Ok? O\gJ0;>GhGM2~=S;HBYJ*;>k?9=@MeGhO?gAhMeB C ? R Jn\ J*? \gMe= [Jrk? Ap>J~;>GhieM>\gA? R I A=S? R JB7=@Ai>BAk;HBV? G5}I}Me\ J0<J kAJ*BV? Ê,? J~AI}[\ M2peJ*IKJ*BV?SA=y?z53[Ak*;HGG5MHMeBGh5}; J* Ah? J*\;2?gAhMeB= 4O?9A=Sp>J*\g5)k*M>B= A=z?gJ*BV?Ì ;>BL9ÍzO=@?9;>=S\ JGhA;H4GJ>8 LJR ?;HAGhJrL);>B;HG5=@A=MH M>O\y\gJ= OGh?g= = R M2~=? R ;H? Z U Z A= AhB}i>C JBJ*\;HG= GhAi R ? G5IKM>\gJ<kM>B=@J\ p2;2?gAhpeJ? R ;HBK:<U Z AhB ?BR J*;HiV?y;2?gA?Ahpe? JJB<L;t=5>8-?gM0 B}?;|[É>;HJ9\ ? IYAk;HO\gGi>;>A\ B?;HGGJ5lG=M>R BMei>\@J*?g\ J*\=@? =zJ*?g[J*=
(191) [M>=8:<0UM2Jp>M>J,? \ JC B? R \gAJ=y= OLGM3? JrA=
(192) B}B;0MH?SG;>=@\ JieJ*J*I \[?g\ MlMe;>[GMe?g\@J*?g\ AhMe[ BJ*\ MH M>\gOIKB= ;>OBk*kk*JJAh=@BK O; G A?gJ*\;2?gAhMeB=Ê ? R A=<IY;t5K4JlLJLOR kJrL \gM>I ? R J\gJ= OGh??;HZ 4GJl=@ABkJn? R JnB3OI)4J\9M> OB= OkkJ=g=@ OG A?gJ*\;2? AM>B=A= i>1 Ap>JXJBX;HG=@43M 5B? RM>? JJYL? AhRP ;2J*\g?J*?BR k*J7Jk4R J*Me?zAk*JJKJ*BMH~? ;HR B JBV;>OkkIO4 \;2J*\~?gJKM>I}AhA? BJA\gIK;H? A©A¾rM>;2B? A=M>;>B BM>LX ? R JJ*<BV?gOM>IB-4 =J*\~IKM>MLieJ*\gG;eAhLB AhJ? BVR ?<JYJ?gp2\ ;HOG=@O?;2?g\gAhJ*Mei>BAM>=ÌB 8 ;HG=@MN;>[[Jr;H\=? MF4 JkMeB= ALJ\g;>4G5FIKM>\gJY<J kAJ*BV?? R ;HB$;HB ;>[[\gMt3AIY;2?gJ}MeBJ ;2?GJ;e=z?AB ? J\ IY=M>? R J B3kMeOBI=@4 JrVJ*O\ J*MHB k*AJ?gJ*\? ;2?gJ~AhMe\gBJ? =\gBMeJ*= Jr[ LJJkL?gAhpe M>J\yp2k;>Me\ B3A;Hp>BVJ*?y\gi>A=yJBk*kGhJrJ ;HC \gAhGh\g5)\ Jr;2=@?y[ AJ?k=y? A4 p>JJ9=@?SMH ? R J*J~B}k R? MeJ~Ak*kJMe4 =@J?S?zMH9C J*J*Jp2B);HG:<OU ;H? Z AB;>iB? L J~Z MeU 4fZ ÍzJ8Hk0? A=yp>;J AhBOLBAkk? ;2AM>?gJ~BY? ;HB;2L}?<C ie4\gR M>;e? LAhJ;>BVGh?yieM>L\gM>AIK? AhIYB;2=? ;HJr\g=J~? RJr= ;H= ?J*MHBV? A? ;HR GGJ05)M2Op>BJ*\L;HA=@GG3? AAB?gJ*ie\O;2A?g= AhMe;>B-4R8
(193) GhJnLLR AJ*?gBAhMeBVO;HG;>?g= AJu =@?SJB<MH? ?SMe\gBJ*[ ;HM>[\ [? R \gJrMtL R AhIYJ*\g;HJ~? A;HM>GB=@M = R R R R R Ê|j Z aNM>\Y:<6
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