Matematika | Felsőoktatás » Pólik Imre - Lineáris optimalizálás belsőpontos módszereinek újszerű vizsgálata


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„ˆ›4Ÿgz|” w!Ð4¡ Î ž x∈X . Y . M ” “•{utgv|ž u ∈ R , v ∈ R Ö4{žo“•›o„Z,؎ ›u4Ò m AµõnMöõnMùGXI AµõnM õ@M I AµõnMöõnMöøI A õ@M õ@M HI l f 0 (x̄) + g 0 (x̄)u + h0 (x̄)v = 0 uT g(x̄) + v T h(x̄) = 0 AµõnMöõnMösI AµõnMöõnMöEI A õnMöõnMKCFhXI [¨•¾¾´«µ·`¸u§¿¨k´•¾´u§¿´k¬d·»´k±†¾´«¯ ´•¾¾áoÆ*Á °¬o¤]¤f5À5¬ ´kº µÀ§[´k§´•¾¼4´ku¬d≥À§k®‹0.¤ ¨«°¬g¼o¤ ¬/°†¾¾@*¤†¨–´k·`¸k´«¾·`à ± à ¾¿¾ µ´’Ã’5Ç`৐§¿¨k´•ìíá ±± ¸u§¿´«Æ`ìí´«Â¾º µº§¤FÁ ¸«± ´«¾¿¾ ë µ¤ Æg¾À5Ægċ§¤ ¼ ¼ÅÁ °

¾´«± ²Å§¿¨«´k·`¸k´«¾´k§3¾º µ±²¤†º†§·`¦¹¶;È CFõ õnMùô Mo P (κ)>?'(,!"'1!<19+LK+L$&1'Y)*+ ∗ b™†9W1i(''( 3#(.6+ 3  W+(' S517'1, ~+ ]6+ ('|$&1'+ ]6+ 1'(µ `W@u#6+u,@S5^,j pW i(A HdzvI™]("(A H1ltI™$&(')*+ 5.#J+L5+ #(:„,!"'(!"(X+L5+ 5€$&1'Y)+t'¡,!<'6> !<19+L+ $&1'Y)*+ó'(5# j!<1'W~8,†‚`6+ ,(#13<$&!<ZQ) .*+ :U!"(& Ú −P v − Au + v̄ AT v − Qu + ū u, v, ū, v̄ u ū = 0, v v̄ = −b = c , ≥ 0 = 0 A LCP I QP  .' P = BB ]( Q = C C #(K+ }K`<5#((!") ô* +ó!<+K=n,jM ú('†'

S5¡RAIHdzwIv](%AIHdldI $&('Y)*+ ,@Z:'p9W(r+8A LCP I$&('Y)+ƒ!<+ 5K= + M MNP,9,¡# M !<+ 5K=R",†F`1+,(#(3 'Y,90R'(‹# Ú " # P A AµõnMöõnMKCC‚I M= , −A Q .' P ]1 Q #(K+ }`y5#(1!<) ô* + 1,j&M ‰O(#1(57,RZJ8,†‚`6+ ,1#(3uS5('†']11,1g+ Ú ]( s := (v̄, ū) M*NP,@,™# (LCP ) $&1'Y)+ó(W@#1(k ˜( q := ( ) , x := (v, u) T T QP T −b c QP − Mx + s = q x, s ≥ 0 x·s=0 'Y,90R'(# j.' # M !<+ 5K=y1m(Y'#1(5,(#(1+u˜a 0aM Ÿgv|sutgvxwhwm{ “¿„†v|ž+;sdw!Ð4“„†ví} M ¢ ûpêûaê ð "ê !„$ì # & %('X’
 ê (A ]#(!<!U1+ 5,9^O!<+ 5K=I^) t M = ¡ Ÿgv|sutgvxwhwm{ “¿~ ∈R + N, (8 ; „ v|ž Ñsqw!Ð4“¿„ ví} ؎ Î P ”s Q y ›gtgvx“ xÖ@suto{

whv¿ƒ#{ –¡ vx“ów!Ð4“„†ví}š›ž . P A −AT Q N ×N Be (S51W(#1#(7, a. WBy)^*'€$&(')+Z:' ) ^'`<^WX(#6+uy,!"'(!"(X+L5+ 5p$&('> )*+ +ƒ,‚ +^,~`' M  ø!U mU ù P∗(κ)` a ü^ ^]Y,ÿdXa Y,bqiab‹^füÿ@úÿdY üa^fb1XEa¹i e@û o¡#w('3#(3ƒ#(,5#(,9Z*fZ1!8^n+L+5+ ^,j ‚.W%ƒ'*u,5'Y+#(+5+ 'a`Wu,X`)n>  + ,9^óm(]('K$&7`](9W1™5!<#5É$&1'Y)+ ,"$&1'}.*+:, Mx + q = s x, s ≥ 0 x·s=0 Cô AœzwI õnMùô Mo P (κ)>?'(,!"'1!<19+LK+L$&1'Y)*+ ∗ 'Y,9Z*a .' # M !<+ 5K=ct'

5|m1]('K$&7`]19Wc(6+ ]1U$&() ](U#1!"!<6+ 5,9^1 ,X`)n>  + ,9^ƒ(51+Z(VZ #1!"!"1+,9^²!<+ 5K=MjeB5+™(#1+¡<!"@)1''K+ƒ'K+L'}K+&S5^,jMñ{1,99+r> 7,„#%'ZZÉ) ô*}m1:q+ pö4õ orÚ ûpê ð ê·éj"ê !„$ì # & %('X’
 ê A P (κ) !<+ 5K=I ) t M ∈ R w!Ð4“„†ví} P (κ) w!Ð4“„†ví}N,Ž Î whvx¡
ƒ#{ ¡ u∈R N ×N ∗ N Ö4{žo“€›‹„†„ Î (1 + 4κ) X ui [M u]i + i∈I+ Î Ž ›oz X i∈I− I+ = {i : ui [M u]i > 0} , ∗ ui [M u]i ≥ 0, I− = {i : ui [M u]i < 0} . ‰O(W@7,˜]1#(5 |.W  #y )6=n.*'!<#(, $&7(,˜# u `(,X+ 5+ :']18# M !<+ 5K=@+ :' (M|ú1'†'TS5 P #1ò!<+ 5K=n, .*'!<#(+

!"('KW(,V`'Y!"'KW1 κ>gB,91']1}+,kQ$&(X+  )ô* }m(:+ n`Wn [ AµõnMùô MCFI P = P (κ). ∗ ∗ ∗ ƒW@'K` `': @. W~ !y+ = ,"#1+ 'Wf)ô* }m1:85#(19+ó!<1(W(#(,yf#(K+ }` P∗ (0) #(1!<)ô +!<+K=n,f5#1+L'WXF`' M@NP'3#1†#1+!f^ +L+&S5^,8!<1 @.W"(#tc!<+K=n5#6> κ≥0 +L'KWy+ 5+ '!<#(#(8#u('3#13",]6+ƒ$&gS5(#11+ Z1B+LWX'K+ó!y+  =n,+(M ûpê ð êûp"ê 1 ¿(%Y ý ï ê ) Ô¹{ „kƒ#” ¡sutgvxwhwm{ “¿„†v|ž+;sŠ”s Î Ÿgv|sutgvxwhwm{ “¿„†v|ž+;smw!Ð4“¿„ ví}š›ž!y ›gtgvx“(8xÖskto{ ~ ; -, . ë
í c<@% ý ïd| 1W( M $&(5)](B#1!"!<6+ ,9^ó!y+  =M*NP,9, x M

x = ¡x M x¢ = wv¿ƒ#{ –¡&vx“•{ž óÖ Î 4Òov|s P (0) w!Г¿„†ví}š›ž ∗ T `W@w!" )( x (51+]( x M x = 0 9}W"e›#1+}`f5#((!")ô* +M e5](#6+¡'(W1 Z#(!<!"6+ K,9^ó!<+ 5K= . ' P ]( Q #(K+ }K`y#(1!< > )ô* +!<+ 5K=n,jM9ñ€1,@9+7,<# x M x #(#+ +](ZX+ 5^,U$&('# x>g6+# M !y+  = #(1,1#(1+ ]((,V!"($&('1'3@1 (x , x ) 'Y,9Z*aMaNP,9, x M x = x P x + x Qx ≥ 0 !<K`(' P ]( Q #1+ }K`y#1(!") ô*K+ (#1#(1'{#%''} +L5+óZ1'Y+5+ ^,jM o4,†F`1+ ,1#(3"](')*8#1+ó!f^ + +·S @. W~ P !y+  = #6+L'KW"Z3‚`(ZZQc$&1)]1Q5#(!U> !<6+ ,9^²]1™8Z5#(!<!"6+

K,@^ ó!<+ 5K=n,9*'µM ûpê ð ê ð fê e #î ¿€ ê Q {žovx¡ “xs Õў Î t " # T xT M T x = −xT M x ¡ P A¢ M = −A T Q 1 T T T T 2 ∗ 1 −3 0 1 M= C(G T 1 1 T 2 2 õ@MùGMo P (κ) !y+  =n,y'9`6+ 38+^'Y‚S5)  X ∗ , w Ð4“¿„†ví}š›o“ žžÑ›o„@{4Òo„ˆ”sutg“ Î t u = (1, 1) Ö4{žo“€›o„†„ Î z uT M u = −1 ó“€{ŽÉÐ4“ M 1!®#(K+ }` #(1!<)ô + w!Ðos „ˆ”sutg“FŸÐ4„†wm{ zÓÒ {s{ “€” ¡ Î 5 ÖoÐ4z Î sutg“Ðoss Î z .T , (1 + 4κ) u = (x, y) X ui [M u]i + I+ X I− κ = 16 2 2 ui [M u]i ≥ (1 + 4κ)y + x − 3xy = Ö Î 4Òov|s M Î P ( 5 ) ›sutg“ñÐ4zÓÒ4Ÿ Î “ Î „†“•›gtgv|ž ∗ 16 . µ 3 y−x 2 ¶2 ≥ 0, ñ{1,@9+.1+7,R P (κ) !<+K=n,9*'a'+ '  Z ZR#1+ 'W9+²Ú ûpê ð íê €"ê !„$ì

# & %('X’
 ê A P !y+ =*If) t M w!Ð4“¿„ ví} P “g#z Î ‘ƒ#›‹¡šsgНSh,Ž Î whvx¡
ƒ#{ ¡fÔ¹‡owhvx¡Ë›o„’‘ Î ∗ g8 . 0 ¡Ë{ wh¡Ë{ñ Î “ xÖ 0 o Z6`(#11+ 3Z(Ò!<u1!<'}+1+5+ 7,j p.Wi P (κ) !<+ 5K=n,9 RQZ('3X+ %('TS , ® '!"'K,jM¹jg!"(5+(,d'WX%]('), P !y+  =n,9,'µ F!"1'W(,¡`@5#(X+ N P >?+('TS5(5(, A·' 3M pùõ orYI6M ∗ 0 ji ø!U U ù P∗(κ)  b1YXalk‚ûqe ÿqüÿGmC?^`Y@n Y&E üÿRoú û@Ši[bqý?ÿqa ON Z Z1˜~#,5#(Z*k$&''TS5^ ,Q†5#1"#(,+u#U''}K+ ,+( É!<1'W(,1+%~,]13Z ZPZK> #(XWn}K+L5,9Z*R$&('É$&^ ,~.*5#('µM ûpíê  ê·éjêxüOì  ê ŒdÎ Î t

M w!Ð4“¿„ ví} P (κ) “gšz Î ‘ƒ#›o¡ÉsgНS$, Î žžÑ›o„ Î t ∗ M0 = (8 " −M I S X A õnM GMKC‚I #  w Ð4“¿„†ví}mwhvx¡
ƒ#{ ¡ S ”s X y ›gtgvx“ xÖƒ4v Î #›o¡
Ð4zÓv|sdw!Ð4“¿„†ví}‚wm{ zxz|{ “¿“¡Ë{ w?sutgvx¡# #zÝÐ4„ v|s . ë
í c<@% ý ïdxp(W1 1'35#(† M P (κ) !<+K= x ]1 s #1+}`y`(,X+ ,jMñ{1W@7,R$&1' .W M #1^'Y1 (,9,ƒ']6+ (#1, ζ = (ξ, η) 6= 0 W M ζ = 0 `Wn A õ@MùGMöõI sξ +xη =0 AµõnMùG MöôXI η = M ξ. jg+5+ 9}WUyAµõnM GMùGXIO''}K+ P!<++x+L''.*+:u'WX i )l= @. W ξ 6= 0 ]( ξ η ≥ 0 M ξ 6= 0 e5](#6+t`@#(9+ ξ η = −s ξ /x < 0 !" 1''19+!<)*1MjNO#1#(1'|Z1'Y+5+ ^,j j.W M (!"#1^

'Y51M C ∗ 0 i i 0 i i i i i 2 i i i i i 0 õ@MùGMo P (κ) !y+  =n,y'9`6+ 38+^'Y‚S5)  X ∗ ûpêí êûpêØ9äì;"  ì;" < í înï ê J { zÝÐ4“’Ž Î “€-, Ž ›u4Ò Î t M w!Ð4“¿„ ví}Fy ›o¡ “€›s Î ¡ Î žžÑ›o„–¡Ë{ w?sutgvx¡#šzÝÐ4„†v|sl, Ž Î ( Sh, Z, 0 g8 p Yrqs .@T Î Î Î Î ¡ { wh¡Ë{ Î “ xÖ t Î tg“v|s M P0 “ #z ‘ƒ#›o¡ÉsgН óÖ 4ҋv|s ؎ M whvx¡
ƒ#{ ¡@Ô¹‡owhvx¡Ë›o„x‘ Ë w #“ Î “ ‘ Î
Ž ›u4Ò*whvx¡
ƒ#{ ¡ P (κ) w!Ð4“¿„ ví} P w!Ð4“¿„†ví}1v|s={4Ò4Ÿ { ¡ #ƒ ›oz #›gt Î “%“•›oÖoÐџŸgv„ˆ”suto”gŸ { ¡ ∗ 0 { „ „ˆ{ Î t{ „ˆ{uƒ4wm” ¡ ҋ„ˆ{d¡Ë{ w z|{sut1sutgÕ;žs”4Õ#¡Éž m , .) . o÷,†‚`6+ ,1#(3y'1!<!<f P (κ) !<+K=n,"(,X`@`'1xS51''1!"#(](5]1+ƒ)‚S M

ûpíê  ê ð êxüOì  ê ) Î zÝÐџŸgvžÑ” “»Ð4zxz98x“Ð‹s³{žoÖvxÖ Î z|{ ¡És 2 κ ≥ 06 V ∗ X . M ∈ R P (κ) w!Ð4“„†ví} Y . J Ð4„†wm{ zÓÒ D ∈ R y ›gtgvx“g8xÖƒ4v Î #›o¡
Ð4zÓv|sdw!Ð4“¿„ ví}Š”s h ∈ R Ö4{žo“•›o„1wm{ zÜz|{ ““ Î n×n ∗ n×n n A õnM GM G9I −M x + s = 0 −1 t, D x + Ds = h „ˆ{ ¡
ƒosuto{ „d“•{ “xsuto‡oz|{#{s (x, s) wm{#›oz݃ÑÐosgÐ4„ Î “•{ z ‘¹{s Õ#z óŽ ›u4Ò A õ@MùGM5I ë
í c<@% ý ïdƒW@'K`X`': .Wft!y5@),%''}K+ | h `1,9+O ](',@7 'x!"($&X'!"#l> .*+ É: Ú ]!"('KW x, s ∈ R `(,X+ n j.' M x = s + 1' S5(57' W ° ° A õ@MùGMùøXI x s ≥ −κ inf °D x + Ds° . NO'3#1†²#1+²'Y+&S5^,<Z *.W X ° ° A õ@MùGMöHI inf

°D x + Ds° = 4 xs, xT s ≥ −κkhk2 . n T D 2 −1 D 2 −1 i i  .' I = {i : x s > 0} MNO (#V'(W1 D = diag d ' d ∈ R {0} #(K+ }K` )Y*'K!y+  =MNP,@, ¶ µ ¶ X µx X ° ° A õnM GMùsXI x inf °D x + Ds° = inf +ds = inf +ds + D i∈I+ n i i ‰t#1X' S5^,‰# n 2 −1 d>0 2 i i i di i=1 n 2 i i=1 di >0 i i di + ,‚+MFÙ¡ x s < 0 ,9, d = q (51+]( α(x , s ) = 0 M[Ù¡ x s = 0 ,9, x = 0 A`W s = 0I6 p}W d 0 − A d ∞I16+ ]1 α(x , s ) 0 Mo¡# x s > 0 (51+ Z1 #%'ZZÉZ(m1'](ƒX4# Ú ¶ µ ¶ µ A õ@MùGMùEXI x x +ds = 4x s + −ds ≥ 4x s . xi si i α(xi , si ) = inf di >0 i i ³ xi + d i si di i i i i i i i i i i 2 i di ´2 i i 2 i i i Cø di i i i i i i õ@MùGMo P (κ) !y+  =n,y'9`6+

38+^'Y‚S5)  X ∗ oÞZ1m('](5Z(iX+ k,@,d'' $&(V1W( '35]( É. d = q `Wn α(x , s ) = MƒNP#(1,1+ †5#1) `;):@),¹ A õnM GMöHI„''} +L5+MóNPZZ3'%,‚@S5^,j ƒ.W-‰'(!"!< 4x s !y5@),y''} +L c(,X`@`'( ™ X AµõnM GMChXI x M x + 4κ x [M x] ≥ 0, ∀x ∈ R xi si i i i i i T i n i i∈I+  ''}K+ 5 'µ p!<x](1Ò P (κ) !<+K=n,)ô* }m(:S M NP#1#(('v#"1,X`nK`'(m1YR!<),]6+  9WX+óZ('++^,jM -dWX,9R$&S5^,y.*#('  U,†F`1+,(#(3"+ ^'S5)  +¹Ú ûpíê  íê €"ê 1 ¿(%Y ý ï ê Œ1ÎIÎ t M w!Ð4“¿„ ví} P (κ) “gšz Î ‘ƒ#›o¡ÉsgНS$, Î žžÑ›o„ŸÐ4„†wm{

zÓÒ x 6= 0 Ö4{žo“€›o„ ∗ N, {s{ “•” ¡ z|” “€{utgv|ž ›ozÓÒ Î ¡ i vx¡
ƒ#{k} ؎ ›u4Ò ∗ A õnM GMKCC‚I ë
í c<@% ý ïd€o¡#%''} +L+^'Y‚S5) ,](1#6+ƒ!")‚S8,9 .W~Z*5!<1'W x > 0 11+]( # I )l= .'!<#™m(,y,9,'11+75(( * I # (#¡`@5#(X++ 5`@Y'w,†‚`6+ ,(#l> !<]19W% P (κ) +^'Y‚S5)  ,jM ûpêí ê ' êØ9äì;"  ;ì " < í înï ê T t Î “gšz Î ‘ƒ#›o¡ÉsgНhv|s@{žoÖvxÖ Î z|{ ¡És Î P w!Ð4“¿„ ví}“g#z Î ‘ƒ#›o¡ÉsgН Î z . pvu s ñ{F`ZZ p+^'Y‚S5)  ,y]1²$&X+ óZ(m1']11,„+L'Y'.*+ :Ì, pùõ rK>?Z1aM xi 6= 0, xi [M x]i ≥ 0. x + x − ∗ 0 CFH w LON àUßyàâå[àÒÙ P à;ÚRyx Ü ÙQÜ R Ý

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7'5:Z'TS,c(' > ](  9.W<J,!< '(!"(X+LK+L$&('K+ ]6+ ('jO+ 1' S5(57' S5†aM9o WX,'Y+ Z*y1#tJ)^'K+LD> ](v!"!"'K#(' +ÉS5('(X+ µ 9!"Z3'!<O'Y+#1,j X.WU1#((,"J!":@)5#((5(?, D8'('YZZ Cs ô MCM*NO'!<]1'6+ É.++](  #<16+ (,iWk]1#1](Z(ÏDk (! ,]1(5(,˜X+ %!<1')*r+J)  M NP#y#(@Z˜m(, 'Y+#(:'YƒS51'19+²5Z'](!<+¹Ún1W@](5#1+¡8WX,'+ É','!<#5,9Z*„'(+†ZZ5#(†ó1']1 (Wk†#(}K+ 6++%. Z*,'+ VZ('7 'O!"(')r+d+L''µ

É!<5](#6+fy |g!<‚S5)(! 9+ } !<1')Z:'X1W%'!"'K ,1(,9}K+ ](5 1' S r+{.*5#('`™1'3™'(.6+x''} +LX™X+   + !f^!"m+ p C‚H r M|NP1, (''1](5„Z1'39+f!":@)#1(5(,i$&DS5'3@)](5]1+8]18WX,5'Y+ ('+(S5()]1]1+™5,8`@5#`1+1++c# *.W„*W„X+ 5 0B5#!"}+ 5+ó](9W1'+(,jM  UT@U ~ ü Zfü ^`Y[ab€b1YY[Z`X m 5‚ƒ„‚ ƒ„‚ † ‡ˆ ‰vŠŒ‹Ž„^‘Œ7Š>’lˆ ‰j‘Œ“”‘•ˆ ‰ Be 5+ 9+:'P#U z-$&('Y)*+d.1'W6++J~,9,'O'K+L'ZZ P (κ)>?'(Kt,!<'(!"(n> +LK+L$&1'Y)++

'$&'Y',#1^,jMOo™!<9+"#6+yZ1!f^ +L+5+ ^,Ò#Q('3#(3k$&DS5(#11+Z(a v1# !<@)1''*+L5+ '!<#1#cc'(w](óc,9`) + ,@^ óm1]('K$&7`]19W1² + !y'K#('jZ']l> !y+ó1Mñ€(,9X+ 7 ,„+(.*+™#%'YZZ É$&1'Y)+ +gÚ ∗ AµônMCMKC‚I Mx + q = s x ≥ 0, s ≥ 0, s·x=0 .' M P (κ) !<+K= q ≥ 0 Ml¡ ô*Y' S5^,~8,†‚`6+ ,1#(3y.*'!<#(,‚+¹Ú eB1(()6++™!<1'),
Ú F = (x, s) ∈ R : M x + q = sª ª n#(0R!<11() 1++ƒ!"(')*, Ú F = (x, s) ∈ R : M x + q = s ª +dn+ !<'K!"(')*, Ú F = (x, s) ∈ R : M x + q = s, x · s = 0 ∗ 2n ⊕ + 2n + 2n ⊕ ∗ o

5#(0Ò!<1( () 1++U!"(')*,‚+fWX,9 Ÿ { z5s‡ y ›o¡ “ ,j Ónf#„ +!<'K  !<1'),‚+()  žÑ›ow*y
z|{ wm{ ¡ “ñÐ4„†v|s !"1'),"$&S5^,<6`(#(  M
@#(7,9]17,„'1# #%'ZZ$&('K+ ]6+ ('‹ Ú ð ê·é*ê·éj"ê !„$ì # & %('X’
 ê (A ]1'39+R$&('+]1+('µf jrz œóI ) “€›oÖoÐџŸgv Î ž4Ÿ Î ¡Ô¹{ zӓ€{ssutgÕ;ž-,³Ž ›u4Ò Î Ÿ{ z5s¹‡/y ›o¡ “•›ž F Ž Î zÓw Î t Î ¡Ë{ w + l, Õ#„ˆ{s Ö Î 4Òov|sqz|” “€{utgv|ž Ÿ { z5s‡/y ›o¡ “ . o Z('53c9+']6+ 1#(](5](1,y,]1)]1u'YX`1+3J$&X+ 5 0ÉMno-+‚`ZZY,@ZyZ('Y+·S5^, UZ('3U9+ƒ']1+(#1](]1(,„$&9+

 ZZR,†‚`6+ ,(#1!<]19W(+gÚ ¢ ß ¨´«5¬ ´kÁ ´«¨«¸u§’¾
¤†¨3À5¬o¶° Æ ° ÿ¤o®¯ ° ± ²¤ó§¿¨kÀ±° µ½o¤†¬·`´«±´«¬ ±´k¶´«¾¿¾ ·`´«±° ¶ º §° Ɩº†Â¾¤ 5º ¼o¤†¬F¼4´«§Ç ë ° ¬¾ ÿ¤†À ¤»·`´k± ´«¬o± ´k¶ ´•¾¿¾[·»´k± ° ¶ º§¿°Æ`¯o¤ ·f¤†¨uº†¬o¤ ÆÈ CE ô MCM*NO'!<]1'6+ É.++]( ð ê·é*êûpê >JîX ì  ê ŒdÎ M P (κ) w!Ð4“¿„†ví}Š”sqz|” “€{utgv|žmŸ { z5s‡y›o¡ “–, Î žžÑ›o„ Î t A·ô MCM õI ∗ F w = {(x, s) ∈ F : x · s ≤ w} Z, sutgvx¡ “’Ž Î zÓw Î tqwhvx¡
ƒ#{ ¡ w ∈ Rn {s{ “•” ¡ ¡Ë{ whÕ#„{s ؞ћow*y Î žo“ØŽ Î zÓw Î t ⊕ . ë
í c<@% ý ïdxp(W1 (x , s ) Z1'53<9+™]( w = x · s > 0 Mjo÷+]1+ 1'+²+ †Z Z5](5#(Z( Z#19W@}+&S5^,jM

—G˜"™@šr›•œ@˜"hžp(W1 (x, s) ∈ F w(,9, x · s ≤ w MFÙ¡5#(*' S5^,Ò+ F`Z Zò c := ]( c := x s = e w S5('†'](r+M|o P (κ) !y+ = ,˜) ô* }m1:Sk'@S x s = e w }.+·S5^,j  W 0 0 0 0 0 w 0T T 0 T 0 T (x − x0 )T (s − s0 ) ≥ −4κ X i∈I+ ∗ (xi − x0i )(si − s0i ) ≥ −4κ i T 0 i 0T 0 T AµônMCMöôXI 0 i i T i∈I+ (xi si + x0i s0i ) ≥ ≥ −4κ(x s + x s ) ≥ −4κ(c + c0 ), .' I = {i : (x − x )(s − s ) > 0} MU/gW
Ú + X T s0 x + x0 s = xT s + x0 s0 − (x − x0 )T (s − s0 ) ≤ c + c0 + 4κ(c + c0 ) = = (1 + 4κ)(c + c0 ), `W@ AµônMCMùG9I AµônMCM5I n o T T F w ⊂ (x, s) ≥ 0 : s0 x + x0 s ≤ (1 + 4κ)(c + c0 ) , !<É,5'Y+ó.*'!<#M Ÿ@›f™rœ„ž»‰O1#(17,;ZV# f : R R $&7`]19W9+ !<1'W@5 f (x, s) =

x · s M +r> .*+ : .W~# f $&7 `]19Wy$&'KW9+ c+ 1' S51](5+('!<1#(](5É+ 5+!<9WXaM NP,9,²# F #(9+.*'!<#d#%'ZZa',9ZQ''} + .*+ :f('&3 Ú ¡ ¢ AµônMCMùøXI F =F ∩f w−R , .' w − R = w − w : w ∈ R ª f 1)f+ ('TS5(ó9`1#u,]11+xS5('†' Mno™# f $&7> `](9Wc$&'W9+1  w −R .*'!y#v#5+ }Wc#3,]1"x# + 1.*+O F .'!<#v#r+  X !<++ F ó#r+M G¡£¢¤¦™@¡£hž{ñ€(W@7,V$&1'P) 1,X+%!":@)a a. WV']6+ 1#(, ŵ > 0 !"('KW@ F = ∅ Mpo™# .'!<#™1!®71( !"`1'+ 5+ '!<#(#(c# (x , s ) X+ +]1²f!<vZ#1XWn}K+ +5+L, F

!<++,!"*,X+²(Mñ€1,@9+7,R8,†‚`6+ ,(#13y.*'!y#6+¹Ú ¯ n o Aµô MKCMöHI ¯ G = w ≥ 0 ¯ ∃ (x, s) ∈ F : x · s = w o G .*'!y#1:'É#%'YZ Za!") .*+ :ÉÚ õh 2n n w w n ⊕ w 0 n ⊕ 0 −1 n ⊕ n ⊕ −1 ŵ w0 0 0 ô MCM*NO'!<]1'6+ É.++]( ð ê·é*ê ð êxüOì  ê ) G Ž Î zÓw Î t@¡Ë{ whÕ#„ˆ{s³”stÐ4„†“ . ë
í c <@% ý ï dao Z('3fX+']6+ 1#(]1t!<++ w ∈ G @`Wn G (!q75((Mno #5+ .# #1+u,1''|X#('µ .W Z*5!<1'W G >gZ1'|9+ 5#+d'!U(5#(X+·S~ G >?Z(V`aMp l> W(Q+(.*+ w ,9`11™#(+ !"('W@5 w w̄ Mo™#1+ƒ,(''pZ1'Y+ µ .W~']6+ (#1,

É!"('KWn5 w̄ = x̄ · s̄ MÉñ€(,99+ 57,(W'WX w̃ ∈ R 1'1!<6+ É!"1'W@J+(' > (x̄, s̄) ∈ F S5(7 ' p. W ∀n w̃ > w ]1 w̃ > ŵ Mpo G *'!<#f) ô}m1:S~!<++d w 5#+.# ']1+(#1,c'WX (x , s ) 5#+( X.W w = x · s 9 ' (x , s ) ∈ F MXo™# F '!<# ,!<,X+ }W~# (x , s ) 5#+ *,<`Q,X`(5(ƒ]1#15#+  j!"('KW@(,y.+Lr> ](5+](,]6+uS5('†' S57 , (x̄, s̄)>g*' Mxo™# (x, s) 7 x · s '1,]((#1]("$&'W9+ XV!"Y+5+8(,9, `W@ w̄ ∈ G M*NP#(#1('€U'(!"!y+Z1'Y++ ^ ,M w̄ = x̄ · s̄ ¤ p(W1 w ∈ R 'KWX„`(,X+( *.W w > w ]1 w > ŵ ](ó+ 1,@9+7,„#

ª Aµô MKCMùsXI A= w∈R :e w≤e w (!f75(( *,!",X+².*'!<#6+&M l™ ô'TS5^,y(W f : A ∩ G R $&7`]19W9¹+ Ú  0, .* w ≤ ŵ , Aµô MKCMùEXI f (w) = 9 ,  7  '  † @   Z 1  a  M w − ŵ o[) ô*}m(:²!"Y+5+p# f $&7`](XWt$&'W9+ 1 }Wt k.k $&'W9+ X!"Y+5+p kf (w)k $&7`]19W¹~$&'W9+  (MóeB`1'd,!"*,X+R.*'!y##r+Q5](#1.*'!y#Ò,!",X+( ó}W; Ê 115+ 5• D9+]1+ 1'w!<++u# $&7`](XWk$&('K`(5#(vR!<!c^!<+%# A ∩ G .*'!y#(a f `W@ó']6+ (#1, ρ .W ° ° A·ô MKCMKCFhXI ρ = kf (w )k = min kf (w)k ≤ °f (w )° , 0 n n n ⊕ n n n n 0 n n n n n n w̃ w̃ n n 0 0 n ⊕ 0 T T 0 n ⊕ i i

i i i ∞ ∗ ∞ 0 ∞ ∞ w∈A∩G ∞  .'É#t^ +'5:f(W1'3+ '1]1U#(]15+ó# @!<15+ w ∈ A M&ٙf!"5+ ρ = 0 '1 ,9, ']1+(#18'WX w ∈ A ∩ G É!<1'W@ w ≤ ŵ )f(,9, w ∈ G !"++ ∃ x , s É.W (#ó()d(''(X+ !")f*,j .W F = ∅ `Wn ρ > 0 Mñ€(,9X+ 7 , x ·s = w ≤ ŵ !<r+ƒ Aµô MKCMKCC‚I T (w , ŵ) = {w : w ≤ w ≤ ŵ} .*'!<#6+ ‚!<1'Wd+^'Y‚S5),](181W n>?)!"1#(:A&(6+ '(t1'$·‚S5^':XIÉ+ ]1'Y+(r+4M œ]1'^, # .W8 w X+ Z:'* ŵ 9"+ | $&('O] }('']19`ó'WX w X+ +x+L''TS5^,J¡+ ]('Y+(5+r> Z1a

!"('KW@(,¡!"(S57,t# x ·s = w 1'3X''} +L + ((,9,| kf (w )k < kf (w )k õ@C 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ŵ ∗ ∗ ∗ + + + + + ∞ ∗ ∞ ô MCM*NO'!<]1'6+ É.++]( ( ''(X+ !")*J!<++J#~)(,X+J$&('K+ 1`]1f(! X# `W@ F (! 7 (1Mxo­,†F`1+5> ,(#(3„1() !<]19W@f!<]1~. `+ ,#1€$&^ ,By,](53ZZ(,9Z(a a}WB'K+L'Yƒ'(!"!y,](X+ $&X'!<#1#(^,<!<1M ð ê·é*íê €êxüOì  ê M {4ÒÑ{ ¡ M ∈ R P (κ) w!Ð4“¿„†ví}N, q ∈ R Ö4{žo“•›o„Z,d“€›oÖoÐџŸÐ w < ŵ -, n×n n ›ozÓÒ Î ¡õÖ4{žo“•›o„ˆ›ž *Ž ›u4ÒÏz|” “€{utgv|ž w 2 ∈ R+ M x1 + q = s 1 Œ1Î . M {4Ò;{ ¡ ,Î ∗ (x1 , s1 ) ∈ R2n + , n Î

wm{ zÓÒÑ{žo„{ ª T (w1 , w2 ) = w : w1 ≤ w ≤ w2 . 1 w1 = x1 · s1 ”s , A·ô MKCMKC‚õI Î int T (w 1 , w2 ) 6= ∅ žžÑ›o„z|” “•{utgv|ž α ∈ (0, 1) ”s (4x, 4s) ∈ R2n fŽ ›u4Ò t x+ = Î x1 + α4x > 0 s+ = s1 + α4s > 0 w + = x+ · s+ Ö4{žo“•›o„ˆ›žž z w+ ∈ int T (w 1 , w2 ) , , ë
í c<@% ý ïd{ñ€1,@9+7,R# . Aµô MKCMKCFôXI A·ô MKCMC(G9I s 4x + x 4s = w − w ( )#1(5+(M€eQ`1' M P (κ) !y+  = }WVAµõnMùG MC‚I²'1!<!<"!"Y+5+™(1, y( )#1( (, ']1+(#1,y(W](5+('!h˜ ( (4x, 4s) !"(')* M oä+ F`ZZ Y,9ZR'WX α ∈ (0, 1)>g6+ƒ,('' +L''µ !"('KWn5 Aµô MKCMKC I x = x + α4x > 0 Aµô MKCMKCFøXI s = s + α4s > 0 Aµô MKCMKC‚HI w < w < w ∀i,

.'|y*!<),„$&1'+ ]6+ 1'|Z #1+}K+·S j.W  w X+t`':Z*+ ]('Y<Z('5DS5]1Z(V`aM o¡#ƒ1'3%,]6+w$&('K+ ]1+('K+O,@) *+L,]19+j'Y,9dZ9+5`Ju,†F`1+ ,(#(3J$&('+]1+('1,1+w,nS5^ , # α #(!" ÉÚ n o A·ô MKCMCsXI α < min − =: α o n A·ô MKCMKCFEXI α < min − =: α , .'É# α ]1 α #!<,y# (x, s) 5#(0~#(+`@+ J!"Y+5+v#(K+ }K`,jM@o4*5!y), $&('+]1+('a('3U$&1'u#u'YZZa'Y,9Z*R}.+ :&Ú M 4x = 1 4s 1 2 1 ∗ + 1 + 1 1 i + i 2 i + i:4xi <0 i:4si <0 1 x1i 4xi s1i 4si 1 2 2 + 1 1 1 1 1 1 wi1 ≤ wi+ = x+ i si = (xi + α4xi )(si + α4si ) = xi si + α(4xi si + 4si xi )+ + α2 4xi 4si = wi1 + α(wi2 − wi1 ) +

α2 4xi 4si , õõ A·ô MKCMöõhXI ô MCM*NO'!<]1'6+ É.++]( `W@ Aµô MKCMöõnC‚I (wi2 − wi1 ) + α4xi 4si > 0. ON #<R$&1'+ ]6+ 1'Om1 ,k#(,9 B i )l= 1,9dS5('(9+%!"(,†+ ](r+ €!"('KW(,9 4x 4s < 0 !<]1()"8,†‚`6+ ,(#13<$&!<Z*3Ú i i A·ô MKCMöõõI ½ ¾ wi2 − wi1 α < min − =: α3 , i:4xi 4si <0 4xi 4si .' α !"]1+²#(K+ }`y5#!QMo÷!y,"$&1'+]1+ 1'++ 1,99+5`4Ú 3 + 1 1 1 1 1 1 wi2 > wi+ = x+ i si = (xi + α4xi )(si + α4si ) = xi si + α(4xi si + 4si xi )+ + α2 4xi 4si = wi1 + α(wi2 − wi1 ) + α2 4xi 4si , `W@ A·ô MKCMöõôXI ·A ô MKCMöõG9I eB`1' w − w > 0 91#(](r+(#ƒm(,",9,aS5('(X+v!"(,†+ ]15+ 9. 4x 4s > 0 M@NPZZ(~#

(6+ Z(V#%(W1'6+¡,9(Z Z,~W†,J1X+}` *}W„#%(W('3+ '(5](y#1,9 <"#(+}` 5#!",9 f+ 1' S517'µ *!"('KW(,9‹ Ú α 0 < (1 − α)(wi2 − wi1 ) − α2 4xi 4si . 2 i 1 i i i ·A ô MKCMöõI !< 4x 4s > 0 !<++ó#(K+ }`Mo÷,++²$&('K+ ]6+ 1'(,6+ƒ†5#(1`6+`c,@S5^,j j.W„Zr> !<1'W Aµô MKCMöõøXI α < min {α , α , α , α , 1} #! ,9('](}+ PB'(!"!<R,9}`*'!y +3M jg'W1ò()  # α #!<,V#(K+ `@K+L R!"Y+5+ ']1+(#1,jM eBr+ó!<v,†XW(RZ1$&gS5(#1.1+&S57,„J+]1+ 1'ÉZ#(XW@}+L +( @5#(1a * int T (w , ŵ) ¤6= 9,9,w#²('3ZZUZ('Y+

+5+O'1!<!<t!"++x']1+(#1,c'KWX w = x · s X.W ŵ < w < ∅ j`W@ kf (w )k < kf (w )k !< (''19+ !" )*(
M ٙ int T (w , ŵ) = ∅ ,9, w $&'W9+ a!"($&X+ '5,<!<++$&1'`1.1+7,R1W w X+ +™ w !"($&1'1'3@1R,9 ,†XW(#(6+ ](Z(a @!<1'W@1,c!"(S57,c x · s 1'3X''} +L +|](O5 dt9+ t!<)S5^,J1'* Z#19W@}+ 5+(Mo¹Z#(9W@}+ P#(]15+P$&d³! ˜,†@)µ !<15+x('](ut'1*WZZ8,@)+LZ ,†#(1'(Z ZB,(57'   ŵ 9+ . #M NP#(#1('€8+]1+ 1'p!<)1W@,y''} +L +Z('++^,jM α< i wi1 − wi2 + p (wi2 − wi1 )2 +

44xi 4si =: α4 , 24xi 4si i 1 2 3 4 i ∗ + ∗ + ∞ ∗ + + + ∗ ∞ 0 0 0 õô ∗ ô MCM*NO'!<]1'6+ É.++]( ð ê·é*ê ' ê ‚@ ì ÿ ì@í ˜în=< ê Œ1Î z|” “•{utgv|ž Ÿ { z5s‡hy ›o¡ “–, Î žžÑ›o„ Î t̛y
“vxw!Ð4zÓv|s 2 žÑ›owy
z|{ wm{ ¡ “Ð4~ „ v|s 6 wm{#›oz݃ÑÐos›ž A·ô MKCM õHI ª F∗ = (x, s) ∈ R2n : −M x + q = s, x, s ≥ 0, x · s = 0 Ž Î zÓw Î t Î žÑ›ow*y Î žo“ . ë
í c<@% ý ïdBp(W( (w ) ⊂ R 'KWX25#+ u.W w > 0 w M*NP,@,ƒ('3#(38+]1+ 1'nS5('†']1(K`('É}.+·S5^,j  W lim w =0 i i∞ n i i+1 < wi ]1 i 0 F∗ = F = ∞ i=1 i A·ô MKCMöõsXI 0 Fw ⊂ Fw , oÞ+]1+('x5#(1X+% F .*'!<#1,R,!",X+L,Q](d(! 711, p}W# F .'!<#J#5+

](ƒ5](5#(J# F .*'!y#(,j !"a,!<,9+( `W@² F .*'!y#tó,!",9+(M o¡!"X+t#1+t!y™(!"'}K+ 6++7,R, ZZ*É a~Z('39+ t!<:@)5#(1(,Bm( , ~,!<'6> !<19+L+ O$&('K+ ]6+ ('K+f5('Y‚=' S,j €](U+†ZZv$&1'+]1+ 1'!"(+ 5+  !"(''6++J+†1,()1,  n w.W/# x · s `(,X+ F |?,†#(1'™'1W(r }˜@^''`1,9+ #MvNPZZ3'¡˜, m(1m1:Z:' ):@),~#%'YZ Z = § ›o¡ ¡Ë{ Ö4{ ¡
ƒ¨`Ò$o„4Ò +3–' p ôXH@ õos rp#!<#(:U) ô}m1&: Ú ð ê·é*ê Gp"ê !„$ì # & %('X’
 ê A œ(X+ 'ó0 +LRI ) t wi ∗ w0 ∗ A·ô MKCM õEXI Mx + q = s x ≥ 0, s ≥ 0 s

· x = µe „{ ¡
ƒosuto{ „ (x, s, µ) wm{#›oz݃ÑÐos Î vx“–™ Î Ž ›oz e = (1, . , 1) ™ m((X+ 'ó0 + ¡ Î ž!¡Ë{ Ö4{uttgÕў . ­¬ 8 o÷)ô}m1:¡S5X5:'a5#(:' #u'YZZa''}K+ Ú ð ê·é*ê J "ê 1 ¿(%Y ý ï ê Œ1Î M P (κ) w!Ð4“¿„ ví} ”s Ϊ2« . X X 6 „ˆ{ ¡
ƒosuto{ „†¡Ë{ž‚z|” “€{utgv|žŠŸ { z5s‡?y ›o¡ “ ‘ Î , Î žžÑ›o„whvx¡
ƒ#{ ¡ ∗ Z, r Î wm{ zÓÒžov{ z|” x“¿v Î µ > 0 ” „†“•”žŽ {utŠ{4Òє „ “€{ zÓw { ¡z|” “€{utgv|ž (x, s) Ö4{žo“€›o„ †{ ¡ “¿„kÐ4zÓv|s #“qƒ#{ –¡  v•‘Ð4“ ÅÖ Î 4Òov|s Î { ¡ “¿„uÐ4zÓv|s #“?{4ҏy Î „ Î wm” “€{ „{s³ o„Ÿ { Î y
„†vxw!Ð4zÓ~«ƒ ÉÐ4z ÖoÐz“•›gtož“€{ „ˆ”gŸ{ ¡ ®S ; U8 . ;, ®S NO ](' 135(ZZ''}

+LóX#M ð ê·é*ê ê >JîX ì  ê ) t Î zÝÐџŸgv–Ð4zxz98x“Ðos›ž‚{žoÖvxÖ Î z|{ ¡És{žV õG  ô MCM*NO'!<]1'6+ É.++]( X . )¯2« X X 6 „{ ¡ƒ‹skto{ „ ¡Ë{žhz|” “€{utgv|ž Ÿ { z5s‡/y ›o¡ “ ‘ Î Y .° vx¡
ƒ#{ ¡ µ > 0 sutÐ4w?Ž ›gt!z|” “•{utgv|ž {4Òє „ “€{ zÓw­¬ { ¡ ›ozÓÒ Î ¡ (x, s) ∈ F Ö4{žo“€›o„Z,fŽ ›u4Ò x · s = µe . « .° vx¡
ƒ#{ ¡ w ∈ R Ö4{žo“•›o„ˆŽ ›gtŠz|” “•{utgv|ž7{4Òє „†“•{ zÓw­¬ { ¡ ›ozÓÒ Î ¡ (x, s) ∈ F Ö4{žo“€›o„Z, Ž ›u4Ò x · s = w . ë
í c<@% ý ï²d A·b™S5!< pùõorYI]ƒW@'K`B1'](y#yArC‚I AµôIó9W9+™Z1'Y+ !<K`('€U+ †ZZ + K`n'K(M ±²¦³"´£™rœ@¡=šr¢µ&+´£²jž ñ{1W@7,R$&1' )1,9+ƒ!":@)a *.

W„`'Y!"('KW w > 0>?#%']1+(#( (, + 2n + + `(,X+ ,j !"('KW(,y,91']1}+,<U.*!<),<''}K+ r+MNP,9,Ú A·ô MKCMöôhXI M (x − x ) = s − s Aµô MKCMùô C‚I x ·s =x ·s >0 eB`1' x − x 6= 0 x(#1](r+y‰A õ@MùGM G9IJ'1!<!< '@Sò+ 1,9X+ .6+ 7,ò'KWX i ) 6=n1+( !<1'W@ x 6= x ]1 0 ≤ (x − x )(s − s ) ËM ƒW@'K`B$&('+(.6+·S57,j ÉW ' M x > x )t1,9,óc,‚+5+(W1'3+ '(5](f!"Y+5+ s ≥ s n1#¡`@#(9+(''19+ !" )y*,j n.W MjNP#(#1('€#u1W]15+ 1'³! ˜5](6+™Z1'Y+5+ ^,jM x x =s s >0 o¹']1+(#1](5ƒ,]6+wZ#19W@}+ 5+O)

^,jMXo¡#²1'53%1WU*'+,9^x(#(5#1+(m1YZ#(9W@}+ ¤( U!<,<(W„, 5+ 5^,X+ }K`ZZB5#(1''1³! ˜a ƒ ó+# D@5()5#((5J](7 '3UZ#19W@}K+L1M ¶Ž´Uœ@¡£·h´=ª¸l¹•º7»c¼–·h˜"½³Ž¾¿œr›¦À-ž p(W( w = x · s ∈ R p.' (x , s ) ∈ F Z('3 9+( *'KW(Q8Z('3X+ ó$&('+]1+('a!<++']1+ 1#(,jMñ€1,@9+7,„!"5+ƒ# A·ô MKCMöôõI x · s = (1 − t)w + tw, t ∈ [0, 1], (x, s) ∈ F (W1'6+ (,1+Mp(W1 >g (, ∀t ∈ [0, t̂)>?d']6+ 1#(,<!<1')*  ª A·ô MKCMöôôXI t = sup t̂ : (3.132) NO,9,%`òB!"1'),9*, (Wk'WX (x , s , t ) #(+Ln .W lim t = t M NOZ Z1U#(+ Z* A·ô MKCMùôG9I

x s = e (x · s ) = (1 − t )e w + t e w ≤ e w + e w, !<Z3'j~Aµô MKCMöõIO''}K+ P$&('.*#1' `'j,@S5^,j n.W< (x , s )ª '!<#²,'+ 1 `W@O$&('K+ (. 1+ 3 @WU+ 5+v1W (x , s ) ∈ F X+ #šM —*'W9+5 !<1$&X+ '5, õ4 (x1 , s1 ) 6= (x2 , s2 ) > 0 1 1 1 1 1 2 2 i 1 i 2 i 1 i 2 2 2 1 i 1 2 i i 2 2 i 1 i 1 i 2 i 2 i 1 2 i i 1 1 1 n ++ 1 1 1 + + ∗ kT k T k k k T k k k 1 k T k∞ T 1 k ∗ ∗ + T k k ∗ ô MCM*NO'!<]1'6+ É.++]( !<++(5%U9+ UX# .W A·ô MKCMùôÑI o;S5ZZQ')'Q'':f`1,X+ ƒ#(K+ }` `W@ƒ".*+ ]15+](,9,](X+t,‚+5+ (x , s , t ) 9+

²,91']1}+pBAµônMCMùôXõI1)5#(15+M%Ù¡U!"5+ t = 1 j,9,ƒ,]1#(1`W@^ ,M%ٙ t < 1 ,9,ƒU!<Z#19W@}K+ +5+²^ m1+ ó!"Y+5+ó# A·ô MCMöôøI u:R F , u(x, s) = x · s '(,]1(#1](ƒ','.!"(!<k ô*#1!8^  (x , s )>gZÉMNO,9,²#(@ZQRAµô MKCMùôXõIw( )#1(r> (,f!")( t>g. (#ó(']1u,†#((' t >? ¡`"!<1')* !" (''19+!< ) t )  ô }m(:S,jM x∗ · s∗ = (1 − t∗ )w1 + t∗ w. ∗ ∗ ∗ n×n ∗ ∗ + ∗ ∗ ∗ ∗ ¶Ž´Uœ@¡£·h´=Á¸Â=º&»”¼–·h˜"½³Ž¾¿œh›fÀÞ%ٙ#1*' S5^ ,„BAµô MKCMöõIv+ ]6+ 1' S5('†']11+(MNO,9,™# F .*'K> ¤ !y#J!<)( w ∈ R

11+](k,!<,X+ (!f75(t.*'!y#Mñ€(W@7,B$&('|)(,X+( .W ']1+(#1, ŵ > 0 p!"('KW(! ''|1'3 x̂ · ŝ ',9Z*ÉMañ€^)S5^,j . WV# F *'!y#c(! 71( O}Wò(#Qm1 ,Ò0 Wò$&)^ '.*+8('3 x.* ∃ i : x̂ ŝ < ŵ O!"Z3'v,†‚`6+ ,1#(,j w.W M x̂ ŝ < e ŵ ñ€(,9X+ 57,˜# f : F R , f (x, s) = x s $&'W9+  u$&7`](XW9+M€o F .'!<# ,!<,X+  XU!"++']6+ 1#(, (x̄, s̄) ∈ F j!"('KW@ Aµô MKCMùôXHI x̄ s̄ = max x s. ñ{1,@9+7,k!"r+J T (ŵ, w̄) + ]1'Y+ p.' w̄ := x̄ · s̄ ,M ٙ T (ŵ, w̄)  (! 75(( €,9,J Aµô MKCM G9I'1!<!<c!"Y+5+ó']1+(#(, (4x, 4s) ]( α ∈ (0, 1) .W

A·ô MKCMùôsXI (x , s ) ∈ int(ŵ, w̄) ∩ F , .' x = x̄+α4x s = s̄+α4s MFoÒ+ F`Z ZY,9Z*%'WX α ]15+](,6+{`'#6+ ^ , F.W ]( w ∈ F +('TS5(7 ' S5†a !" (''(X+ !") (x̄, s̄)  + !y' +L *,Mjo (x ) s > x̄ s̄ ) ô* }m(:,y!"Y+5¹+ Ú w n + ŵ i i T i T T ⊕ ŵ ŵ T T F ŵ + + + T + + + + T + ŵ w+ = (x̄ + α4x)(s̄ + α4s) = x̄ · s̄ + α(s̄4x + x̄4s) + α2 4x · 4s = = w̄ + α(ŵ − w̄) + α2 4x4s, `W@ (x+ )T s+ = eT w+ = eT w̄ + αeT (ŵ − w̄) + α2 4xT 4s ≥ ≥ eT w̄ + αeT (ŵ − w̄) − 4κα2 õø X i:4xi 4si >0 4xi 4si , Aµô MKCMùôEXI A·ô MKCM GXhXI ô MCM*NO'!<]1'6+ É.++](  .'t#V^n+ ':Ò1W( '3+'( ](]('t,9*5#('+^,â P (κ) + ^

'Y‚S5)+M`+t'WX α> + ,(17 , !"('KW@ X Aµô MKCM GC‚I e (ŵ − w̄) − 4κα 4x 4s > 0, ∗ T i .B+( )(#1](™^ + Q,nS5^,„# 0<α< 4κ i i:4xi 4si >0 P A·ô MKCMùGXõI eT (ŵ − w̄) i:4xi 4si >0 4xi 4si $&('+]1+('+(MÉjg'W1 α # e (ŵ − w̄) #(K+ K`nK+L f!"Y++']6+ (#1,jM Û ¤ NP#(#1('€8+]1+ 1'+²Z1'Y+5+ ^,jM o÷m(19+ 'ó0n+²+ F`ZZp*'+,9^+ ^'S5) XK`'$&'',#1,Ìp ôÑ rOCFø M $&gS5(#11+M ð ê·é*ê C fê e #î ¿€ ê Q {žovx¡ “xs ÕўGS†‘†„ ÎhÎ t M = ( ) w!Ð4“¿„†ví}š›o“ Î q = ( ) ‘¹›4ŸŸ ›oz݃ Î zÉwm{ zxz|{ “¿“ . T T S 1 −3 0 1 žžÑ›o„ Î { ¡ “¿„uÐ4zÓv|s š“ µ ” „†“•”žŽ {ut@“ Î

„†“•›gto (x, s) y ›o¡ “ ‘Ð4„ Î V 1 0 x1 − 3x2 + 1 = s1 x2 = s 2 x1 , x 2 , s 1 , s 2 > 0 xi s i = µ T tg“–ž$o¡ ¡ ÒÑ{ ¡ wm{#›oz݃oŽ Î “ ‘Äў-,»”sdž Î y ‘Äў-,؎ ›u4Ò Î { ¡ “¿„kÐ4zÓv|sÅSš“ µ ~ŽUgt@“ Î „†“€›gtody ›o¡ “ ‘ Î xµ1 = xµ2 = sµ1 = sµ2 = √ 3 µ−1+ √ µ √ √ 13µ−6 µ+1 √2 √ √ −7 µ+5+ 13µ−6 µ+1 √2 √ √ 3 µ−1+ 13µ−6 µ+1 2 ) ž Î y ›o““f{ „ˆ{uƒ4wm” ¡ ҋ“ Î:2« . X 6 Ðџg„uÐ4¡ zÝÐ4“xŽ Î “ ‘Äў Be K`('n¡m1(X+  '{0 + (x, s) 9+&S x·s = µe ]1P#1'3Z ZcZ#19W@}K+ +5+€+ ]6+ ('@#(19+ Z*!"('KW µ ]15+](,9.1#„']1+ 1#(,k!"($&('('3 (x, s) 9+( €(#(]15+ µ 0 16+ ]([B!"($&('1'3 ŽÆÈÇ ÉÊ Þ ß ¨ ÈÂæÈ ;5´«·`·f¤E¤†¤ ë ÿº†¬FÆ °

¬ ƹµ¸•¾É¼4´u㫧¿5¸k§¿¾&À§&¤¶¯o¤¾¬oº†¬ Ɩ¤ ¨ ¸kµ¿¾¸«Æ¹µ´ ®ˆ¤ ¨«° ¬¹¼o¤†¬´kµµ´%·`°§¿¾&¬ À5¬o㫧 α §¨«áoƹ§¸«±á ¬ ÆÈ õH ô MCM*NO'!<]1'6+ É.++]( ô MCMjZ  M 8 Ë ð ê·é*ê CcÌ@Pî# ý0Í  ï íXì@þ‚ì ¿ ì & € F' ì  þ‚ý µ
·ï­Îp• ý €ÿ þ î@ï í  ì ì o[XW@',¡óm1†,9,13 µ ](r+ ](,1,d9WZ*!f^ +L+,jM(oÒ](W¡`'K+ #1:ó,†#(7 'Xm(,t.*5!y+ Z #('K+ ^,j !"`1' x = s M 2 2  9+5#+<.*+ ]15+ ]1,B'KWX (x , s ) 9+ O!"('KWn5 x · s = 0 P`W@  + !y'KƒA&,!"'1!<19+ LI{!"(')*(MÑn#(1!<']('1+ 11<(#™#1+aS5('(X+  .WUum(19+

' 0 + m| `]1.*') `+ }Df`W@w(Wn5™,95(ZZ µ ](r+ ](,1,9.(#t,95#!"}K+`u# (x, s) `1,9+5+ D~ +!<'K!"(')*.#¡+Lr+ ^ ,M ð ê·é*ê·ré Ïpfê e #î  ê ) t/{ z|‡4ŸŸgvÑy ” z݃ÑÐ4“Ëwm{ñ4ÖvÝt†suÑÐ4zÓÖ Î zÝÐ4“ ‘Äў-,%Ž ›u4Ò Î { ¡ “„kÐ4zÓv|sÐS#“zÓvxwm{sut«y ›o¡ “¿~ (xµ , sµ ) ∗ ∗ ∗ ‘Є Î x∗1 = 0 x∗2 = 0 s∗1 = 3 , Ö Î 4Òov|s x∗ = (0, 0) s∗ = (3, 0) @. T žžÑ›‹„ s∗2 = 0, 0 M x∗ + q = ( 10 −3 1 )( 0) + ( 1 0) = ³ ´ 3 0 = s∗ x∗ · s∗ = (0, 0) · (3, 0) = (0, 0), Ö Î 4Òov|s1Ö Î z|4Ÿ Î ¡Iwm{#›oz݃ÑЋs ÎmÎ Ô¹{ z Î ƒ Î “¿¡ Î ž . õs ∗ ô MCM*NO'!<]1'6+ É.++]( oÞWX,5'Y+Z* (#81! ,9`@+('(#(. 1+ 3 É!"`('{1(, ~(]()

$&1'Y)+ *, ~Z9W> '^'K+  X"!"(1W(#1,B#f11)1+{$&('Y)*+]1`'µ%M œ]('^,Q,ZZV# .W<m(19+ '™0 +5> .#J`'!"' W(B](5+('(!fZ( ,†#1('|!< )S5^,jMo m(19+ '™0 ++:' `':<+ F`' +™(W $&7`]19Wò!<]1µ |!<1'W@1, ,,9]1+<',(S:'v,](53ZZ‰'(5#„#1:MPo™#Q'K+r> δ(x, s, µ) !8^ ,<(W~1'35d)+5+ Ÿ { z5s‡ X+ Z:'É)^ '*,j @`W@$&('K+ (5#17,j j.W~']1+(#(,<'KW( 9+(Mxo¡#R1'3ZZ['Y++ ^ , |. Wò1ZZ(â#„(6+ Z1â']1+ 1#(,˜,!< '1!<19+Lc!<1')*(

]1')^'jum1(X+  'P0 +w.*+ X+·Sd'KW(aMNO#ƒd$&('K+ ]1+('#²(W@,c,9+,9^P9+&S%uZ(' > 3X+ ó!":@)#1(5(,91,Mp]6+ (#1(,~^WXQ!":@)5#((5(,j j!"1'W(,"(W~'WXiA&*WZZjI $&('Y)*+ Z~WX#(#, yZ '](!<+( !<1'W@1,B!<¡']1+(#1,QZ('3„X+·S )f(#1(,(! ','!<#(.*+ :,U+ 1' S51ƒ'+ ' ZaM 5‚ƒ„‚jÑ‚ † ÒӈÔՑŒ„×Ö+ØÚÙÛRŠrܔˆÝbބ‰j‘Œß„‰ގ‡Rß7 oæ'(f,!"'1!"(9+ K+ 'v`':k,‚m(5'Y+<)â'Y+U5  x.W[','!<#(#1^, R(!"'t

!"#(d1#(,†#((KZ+ Kc( a]1')^'OŠ ƒ ó+n>g!":@)5#((r+ p(!"' (W1'6+ (,~!<1')*   M p(W1 F : R R )È à(5(m1Y'.*+:U$&7`](XW É!<1'W@1, J(x) úm1Z >g!<+ = !<)( x (51+ ]1(!"#1^ 'Y51Mo ƒ ó+ # D@!":@)#1(™# F (x) = 0 1W( '6+ 1+™')S !<1J+( + }`f,†#1('}+ 3%!":@)#1(51' 9!")1<'](](5Z(~1W".(5},9,'j,†#(('}+5`¡u$&7> `](9W9+(
M —*!<' x = x + 4x .' 4x = −J(x ) F (x ) âM á™o™#u1' S,n> `(' .*t™,(#1)3X+w™W†,9.1'WJ!<1$&('('3™,†59W1#(6+ ](Z(U`a4M

ö™WX(#ó¡!":@)#1( .*#1'.*+ :"!"!"'K#('a5Z'](!<,y!<1')  8g Ú.*< g : R R $&7`](9W9+ ,(''!"!"'K#('· ,@,w ∇g(x) = 0 (W('1+ 6+v')S5^,J!<1MXNP1,8!<1')* t5#( > 0B!"!c^!Q .* g ك(56>?!y+  = <#(+}`„)  ôK+ `Wn™]1')^ ' .* g #10 ,9`l=ÉM 2P#11`(.1+&S57, Wy³  ƒ ó+ n>g!":@)#1( (!q,(#1('É#¡('3S51'!"1,†+ ]11,1+( 1#(]15+ (#(1,1+!<O!":@)U,1''*!"(S51'( }+ 1#ƒ'K+!f^ ZÉMXNP5ƒ#1'',c#@Î ž Î ƒÑÐ4zÓҋ~ Ԇ՝4Ö4” ¡ Ò

1,Mñ{1,99+7,~+(.*+ƒU,†F`1+ ,1#(3y) ô}m(:¹+ Ú n n k+1 k k k k −1 k n [㒧¿·`¸«¾ ľñ¤†5Ä ¬ ÆF¤†µµ¤o®u¯ °± ²¤E¬ ´«·`5À¬ ´uº†µÀ5§ ë µ° ±µ¤†·`° ¨kº §š´«µ´k¶·`¸k¬g² ´«À §° Ƨ¨«°µÉ·»´k± ´«5Ç ¨«¾¸kƖ¤E5À¬o´kº†µÀ5§ ç ìí´«Âìí´k¶´k¨«´•¾¾ó5À¬ ´uº†µÀ5§ ë µ° ±µ¤†·`° ¨kº §¿Àɤ†5± °µÀ¾ê ë µ°± µñ¤†·»° ¨uº†§¿¾ oä ¾à ¼ ¼o§¨«àµ´kÂÇ ìí° µñ¶Ä ¾k®4¯o° ± ²/´k± ²æxå ½Ã’° ¬o¬o¤†¬Ã ·FÄo§¿µ¦ ÑƹÀ5¶ ´«µá ¾u®¯o° ± ²]´k± ²]µ¸«±»´k ìí´k´¿Ãx¾´•¾¾,¬ ´«·`5À5¬ ´kº†µÀ§ ë µ°± µñ¤†·»° ¨uº†§À4´k ÿº µº§3§ ë ´uã•Àº†5À5§3´k§´•¾´ È uè ß ± ²¤ Æ ° µ¤¾¼o¤ ¬F¾´«µ·»¸u§¿¨k´•¾´k§´«¬f¬

´«·Í¤ó·fº†¾µÀ ð`À¬¹Á ´«µ¨«¸•¾Ë§¿¨uº†·`ªÂ¾Ýÿº†Æ»Æ¹À ®¯‹¤†¬ ´k·7¤ó5À¬o´kº†µÀ§ ´k± ² ´k¬ ´«¾¿ê µ´«¬‹¶ §¿¨k´«µ¾3°5¶ˆÃ¿º Æ]·`´«±‹È õE ônM õnM@#Z*)B !"]1+(1, ð ê·é*ê·ééê"!Rì$# ¦%g'X
’ ê A&o¡,)'KW@$&7 `](XW I T 4Ò φ : R R ƒ4v P/{ „ˆ{ ¡Ë v¿Ð4z5Ž Î “•óԆ՝4Ö4” ¡ Òo“ Î ž Î ƒÑÐ4zÓҕԆÕ44Ö4” ¡ ҋ¡Ë{ž NO#1#(1'{8)ô}m1:F`'p $&('Y)*+ 2 Ÿ Î „ „†v{ „ 6 ¡Ë{ Ö4{utgÕ#¡Éž-,؎ Î lim + x0+ φ(x) = +∞ gi (x) ≥ 0 ∀i minn f (x), x∈R . Aµô MKCM GXôXI µA ô MKCM GG9I 'Y,9Z*B,†#1('}+ . 1+ 3 Mú:' 'Y+*+ : . WQ"1!<1X+`@+ É$&('K+ ]1+(' ('W1++¡"m(]1'$&7>

`](9W@Z]( }+ 6++7,ÒZ#Q('3S51'!"(,†+ ]15+¹Úx.* x ,†#1('²`;V+ ( ('KW(,9.1# ,9,< m(]('K$&7`]19W *W '(5#MxNP#(#1('(W (]1#„Z']1!ym( 'Y)+c,‚ +^,ò µ  !<]6+ ( $&7`]19W]1Z(a p. ' µ = 0 16+ ]( `n5#,‚nS5^ ,#c1() 1+ €$&('Y)*++Mjñ€(5!<]1#11+ 1( (!®`.+ : W„f* !<]6+ (5($&(')+, µ>g.†#t+Lr+ #1:U!"(')* f!"1(W1#(#1( (#t(1)1+a!"!c^!".1''KW1' @#t#(9Z*„!<pS51'` 1 .W<* µ @^''Y. #ƒ+L5+ ,9,x™* !"]1+(1{$&1'Y)+ ,t!<1') X+ 5+ ,%ƒ`':@) @ +

!f^!".#M‚o/,†F`1+5> ,(#(3<#,5#(Z*(#1+¡#%'K+L'ó)'+ +ó$&gSr+·S57,~,9€"' 5ó,!"'1!<19+ K+  (6+ ]1M minn f (x) + µ x∈R X φ(gi (x)) i mUuø!U é!þfÿGêŠÿqú Šÿ1XEÿG Z`Y[^`XE^fe o/Z1'39+{1' S,t5#(X+ w,@K`]1+('@]1',97'nƒm1(X+  ' ^ + +{.*5#('TS,u#v +!<'K !<1')  i('](]1]1MNP1,/#!c+L'/!<:@)Si'1.1+]((1 1#(1,@53't)^ ,/!"r+y(W †‚`@);†5#(6$&''Yr+FM ƒ(! m1]('^,‰!<)1W@,['(.6+ 35]( + ('TS5(yZ(!f^ + +L  w,‚ZZ m( ,~J$&3f9W,+ S5('†'

S57,8,9É](óc,†‚`1+,(#(3U,]1+$&DS51#(6+ 5d.*WS5^,"(WX>g1Wy, ,@5]1+ 'K+!f^ ](5#('1+ 1ƒ('1!<#1](5]1+M 5‚IÑ‚ ƒ„‚ †Üë߄‰섏ŽÝ0íj‘•ØïîæŠÕ‘•ðRîRŠrß óñ †ZZ!":@)Qƒ1)#1(5(#(.1+·S57,B#%'K+!f^ r>?+}^5,+MjeB5+¡",†F`1+,(#(3y$&1'5#6> +L5+ó+ 1,@9+&S57,Úòñ™0n+ ,†‚`6+ 3 Nó"Q,‚'Y#(0É *''1+5`d+(m1Y'm(5†,9,(X+ 3<1' S,M o¡# Sš“xž$oÖ4{ “•‡ 'K+!f^ ,c1`@7,9.†#¡}K`(~%m1(X+  'O^ + +v,†F`1+ ,jM@o¡#ƒ('3ZZ(,> Z1iZ('Y+ +5+d+]1+('P5#((5X+J„m(19+ 't0 +%'!"1#19+&S~ + !y'Kt!"(')*1MpeB)1

ô¨–¤Fìí´«5°§¨•¾ñº §ØîÂæ ï5ê|¼4´k¬*° 5Á¤§¿¯o¤†¾¦oÈ ôh ônM õnM@#Z*)B !"]1+(1, + 1 m1:Z 'WXkX+ ,9,‚'P)'#(*,j a!"('KW(,B,†#1('x`, „m(19+ 't0 +d`'Y> !<1'Wy9+&S.#M*oä)'#+ó+‚`ZZp5](#1](Z1 ,]6+™0 +,†‚`6+ 3<1' S5+ó5!<15+ 6+ 7 ,M o¡@# Î õ ¡ sž4Ð4zÝÐtÐosOS 1' S,J!<P!":@)8³! ˜,†@)1,MXo¹!<1( () 1++w!"(')*, ']()1]1+u(''#1))*'|,†#1('}+ ,]1c#U(''5#()V!<!<'K#' S,Rm(]1'$&7`](XW@g+ Ú (#y( )#1(9+f,†XW(ZZò$&('Y)*+ !"9+cB']() (1

,(5( n+ !f^ !"+Mpo®,++ 9+w'(#ƒ#²K+ (m(:d,†‚`6+ ,1#(3f9+·SnM9NPZZ™um(5r+ Z*d+Lr+ #1,f1 l¡,9; D@!<:@) #(1(M o y ›o“€{ ¡Ë v¿Ð4z|s-žžÑ{ ¡ “€‡ Î z #›o„†vx“¿w­;s›ž #„K+ 1 m1:+](89+·S.#~+(m1Y'](r+ ]1,1+ ( )('(,jM NP#tu$&7`]19W<'WXÉ 9.W −∞>?(#ƒ+L5+ 9c#™+(m(:u+Lr+1W" +!<> '€9+ . # ''6+` +∞>?(# *¡0W%+Lr+ .Wc™!"(11)1+5+x+ 5+!y9Wu*+L .# .Wi (!  + !y'K%9+ +c,†#1('}+J!"(M o™#~'5K+ !c^5,Vm1]('TSB1#(1ò$&7`](XW˜](D> + ](,]1(,Òm1†,9,19+ ](MwNPZZQVm1r+ Z* +Lr+ #1,˜V'11'3

Z('53X+ 8!":@)#1(U b%!<,• Dn' + !c^? p CFoE r|1MaeB1S5(W1#(#17, p. Wy+ 1m('m1†,@,19+3„(' S 5, #c,‚)*'KW9$&7`]19W1,(V,1(5#1+ 7 '|#1¡,‚m('Y+Z'',~#%0 + ,†F`1+ 3y!<:@) #(l> (,9,1' M 5‚IÑ‚jÑ‚ ö÷ˆ ì7ˆ æ섈 ÛRˆ”‘h‘ŒŠŒ“ ì o4Z1'53X+ !":@)#1(1,<()) j+LWX' *'`](f7 W('K+ 7,<5  @. Wy#¡K+ 1 > m(:,@ZQ,++ƒX+ ,~Z1'3"9+ ,~'1W( (,j *`W@ƒ5#(50Q!"((1)6++™!"(> ')*,ò,(5(5#1+ 7 '²,†#(1'}+&S57,i!"(V#~

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#1:<X+·S+:'É`':U+ F`' +²,9$&]1'1,](1 !<]1.6+·S57,jMjo™'YX`1+3"('` ( *.W„1# k+L`'˜,9,"'(W(/!<!<'K( x.*˜# (x, s) X+< ‚Sr+LV`;im1(9+ 'U0 +a `W@ x · s = µe MxNP#6+"V,†‚`1+('!<]19W9+U!<XZ*  ô*W1'1!8Zy`]6`„+ (5!<]1#11+ 18`> 'Y5#1+L™'1c kxs − µek + `' <.*#('Y+L #1@Z 11,B50'W¡.9W X .Wf1! +Lr+P`]1+ 1'1(# 9* x `W s `'Y!"('KW@,%,@)+ S¡,†#(1',(7 'h>?.#Mòÿ eB`1' h ™ ó+n>?( )#1(Z(

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“(8xÖ Î ¡ `'K+ #6+L+&S5^,~ µ  !"]1+(ó]15+ ]1,]1+(M —j^ m(²(''19+ !")( (. Wd²W,'+ Z%ó'Y5l>g^)*+ P!":@)5#(1(,t³ ! ˜,†@)1,wS5Zn> Z*a !",†#(Z( y5!y'' >?^)+J1' S ,R('!"]('6+  ,!<'6=n+  tS5Z 3Z Ú O ¡√n log ¢ # O ¡√n log n log ¢ . ('W6++(ÉM ƒW@+++v,]() ](( W<1'](51+3>g¡óS5ZZ~,!<'6=nK+Lv 'Y56>?^)*+d!":@)#1(5(,9,('µM n n 1 √ 2 n n ε  n ε ¬ ¬ ´kÆf´kÂ5´«¬

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¾Ä ¤Ã¿¶° ¬‹§º ±¤†À‹·`À¤¾¿¾c`À¶Ç¬ Æ ¸«¬¾3¯o¤§¿¨k¬oº† ÿº†Æ»´k¨•¾3·`¸«µ¾¸kÆ ´•¾kÈ ôõ ônM õnM@#Z*)B !"]1+(1, 5‚IÑ‚ ׂ † ÒӈÔՑŒ„×Ö-Ý0ˆ RÛ抌ÜcˆÝ “ Š ßù‰““ Š æ7ŠrŠrÜ o ƒó+n>g5() #(15+< .*#1' S5^ , |.Wò#„ZZ:'):@): (4x, 4s)  XW@Z* 1'K> '](X`"m1†,9,19+ (O+ ^ )S5^,VRm((X+ 'd0 ++:'x`':„+L`' +Mao ƒó+n>g5() #(1f'K> ,'!<#( %11)1+'(fJ,†F`1+,(#(38(W@5#(1u ˜R†+ '6+ Z3'j5#5!y#(
, Ú9oó)++ (x, s) X+ Z:' #(16+ ]1,ò1' S5^ +w'WX (x , s ) = (x + 4x, s + 4s) X+ Z*

{!<1'Wk!<J,!<'6> !<19+L1 `W@ x · s = 0 MjNP. (#JU,†F`1+,(#(3y5() #(15+™,(''( u!"(')*9^, Ú 0 0 0 0 M x0 = s 0 x0 , s 0 ≥ 0 x0 · s 0 = 0 ON '3#1†"('= ' S5^ ,Ò˜1!<1X+`@+ $&1'+ ]6+ 1'1,1+ O(1,‰(5#( #B'1.1+y˜,†F`1+5> ,(#(!"](XW p.W ~,++ (4x, 4s) !<1')*¡ (! S5: `W@d# (x + 4x, s + 4s) 9+%1! '(5#"!"((1)6++J!"(')(MaNP#1+J0WV,@7 #(†Z†' S57 ,V,9 a.WV1! „+(' S51 ']1](5+¡+ 1#17,!<1 .*1! ,Qm(, α>g#1 +( É.' α ≤ 1 ËM —*('.*#l> (4x, 4s) *'`U# (x, s) 9+ 8`+ ,#(:"(W1'6+

1,1+t8,†‚`1+ ,1#(3+™,@S5^,Ú M 4x = 4s  S5 ¹(#ä(W x · s + αs · 4x + αx · 4s + α2 4s4x = 0 X, `)  +,9^ 1W( '1+1)#1(( R}W­(#,X+-!"(') Z›'+L'Y‚> Z* (!æ1!<]19W@,(). 1+ 7,jMpo ,†F`1+ ,(#(3R,†#1('}K+ ]15+d','!<#1#(^ ,
Ú1'*XWX' S5^,R# + +( 9(#'+L'J,++!<1')*wm1 ,",9 α ]15+ ]1,(,<11+](„'1#ƒ`n#19W@'Y α 4s4x 9+(MOo ƒ ó+n>g5() #(1~}WÒm(,[.#1#`1+ 3'1(< 9W9+<)â!"( O^ + *i,97'† !<1Q,(''P.*+L#1@^ , . WV!<1XWnK+%'](7 ,kZ Z*B#" XWnZ p0 Wk. WkR,++

9+ƒ!"]1"!"((()6++ƒ!<1')*'1W(ÉM o ™ ó+ @>g1)#1(O'@S+|#) +  . WJ'WX (x , s ) 9++x,1(r+ 7 , X!"('WX> õ  x ·s = 0 MNO#6+P™m1]('+{(W%'](](5Z(8ƒ$&('K+ ]1+('(,d('= m1:S²!<++ (!ä+ ^)‚S5^,u(']1µ `n#19+ƒZ}#1^,„ZZ*a j. WRU,++ƒ0S²X+ƒ`'Y!<' W(R](r+ 1'(!fZ18S5Z Za *!<9+²# ('3#(3 ](')*^',95(ZZ< %u)^'K+L5](1 `W8,†#(1'1ZZ<`yum(19+ 'P0 + .#M9o™#ƒ'KW( '](]15+d6`(#1Í , Î õ ¡ sž4Ð4zÝÐtÐosOS ']1](5(,jM %ñ€(5!<]1#11+((˜ 0 .('KW1++d!<¡]15+ ]1,1+u !<1m(]1'#1.*+5+

^,Ò`' w]1')^' µe>?+( w( (,â˜1),X+ y,51,9+y!<:@) #(11,@ ](' 2 0 0 0 0  RÆ É ZÊ ä ´«±·Ä¾¤†¾¯o¤†¾¦‹® ¯ ° ±²F´«¨¤†¨,Àµñº†¬¹²F·`´«±†ìí´k´k4¤†¨,¤`¬=§Æº†º†¨uº §¿½`·`¦¶ §¨«´«µ´«Æ oÈ ÈÂæȧ¿¨u¤†Æ¤§¿¨ É´k§´•¾¸k¬ d ¤ ¨Ø´«5ÂÀ ë §¨«°À5¶°¬=ìí´«5Á ´•¾¾[° ë ¾À5·Ä ·`¬o¤†ÆÈ  ôô ônM õnM@#Z*)B !"]1+(1, '(5#u*W%S5('(X+ 35](M&‰t]17' #6+LB#uó1',]1#(1'.6+ 3 *.WR8!<1')*) :™+ n> ( )#1(J!yd1! `(#11+.1+3`@5#R`'Y!"' W(k!<1m(]1'#1++u]15+ ]1,9 p`W@%1!  $&(9+p )'Y+ !"( 1+ Z3'a,‚

+^,jMNPd$&^,y]1')+²'Y+aUGM $&DS51#(6+ Z1aM ñ€(!"]1#(6+ 1(¹ ™ ó+ n>g1)5#(("(W['+L'YZZa |k (!"' 5f1W( '1+(, !<1') „`*+ ,#(:('!<]1'6+ pKC‹ô rw]1#(MxeB`1'v#1@Z (6+ 7 ,@Z(Ò!<)1i+ ^n> 'Y‚S5)+,†XW(R+ ^) ^,yZ#(XW@}+L @1!­,1''a#d'+ ' 1'!"]('1+¡.K`+,#6> @^,jM ôG  LON àUßyàâå[àÒÙ ÙRè õQ à[Ùx à;Ú6ßyÛØ R  ØBà/á   Ú ÛØ R ÙõQ ÜÞÚR 2á áÞ×ÛÚ o¡#Q'ZZ,9Z*Ò]1')+8!8^n+L+^, 1WÒ'KWX[,9'Y#1,9^U0 + ,†F`1+3˜('TS  n x!<1'W + 1+#(3'1( P

(κ)>?!y+  =<!"(''1+5+!"(')Scc' 5v,!"'1!<19+LK+L5Z'](!<+M o¡#ƒ('TS  O(√n log n log ) ']1](5Z1y'W"!"(')*r+P+L''µ !"('KWn ]('ju)^'+ 5]( '($&(' S51ZZ ε M9o™#™1() !<]19W"1DS5¡ZZ*<gS5', .W8+L`'!"](5+](,1,8#1]('(v#6+L'KWX A·†5(^ 'Y5€$&7`](XW(,nIx(51+ ]1"X9+ '.+ :¡#('TS |³! ˜,†@)](5Mo;,†F`1+ ,1#(3%#(> ,#1,y'@S^,' p ôXõ rpõ D9GM9$&DS51#(6+ t#1''+( @#1#'É#¡(Wn5#(1u ˜}+](5(' . W<!"m1 ,< '

5Z']1!y+`n#(5X'TS5^,jM ñ€(,9X+ 57,~+ (.+™ ∗ n ε Mx + q = s x, s ≥ 0 x·s=0 $&('Y)*+ + v.' M P (κ) !y+ =ÉMwo +‚`ZZY,@Z;+ 1W@7,/$&('µ vWâ $&(')*+, ']1+(#1,y5#(0*Q!"((()6++ƒ!<1') n n`Wn ª A&GMöhnMC‚I F = (x, s) ∈ R : M x + q = s 6= ∅. o¹Z1'39+P!":@)5#((5(,f'K+L'€+L5WX'Y 'n!<|Z('++^,j .Wc1,9,O!" )( 11+](<1W](r+ ('³! ˜(U']6+ 1#(,J'WX (x, s) `1,9+( X!<1'W@ M x + q = s ]1 x · s = µ>0 O`W@<']6+ 1#(,[˜m1(X+  '"0n+Mwo¡#B'W(/('TS 5,9Z*â5#(,‚~,†‚`6+ ,1#(3['](]1 µe ô ∗ + 2n

⊕ GMCM(d (^'Y5$&7`](XW(,„](ó+ ^'S5)  X, < +L`'$&7`](XWB!"()* !"('KW@(,R ,9+,9^ k<,†‚`6+ ,(#13„+ ^'S5)  ,+ ,(''É+ 1' S51}+(4Ú p(W1~!<!<'KE DU',‚'!" </h D @!<,OJ9+‚Sr+Lu`~Jm1(X+  'O0 + aM • o÷m1(9+ '0 ++:'a+ F`'@) `f+Lr+ B8`]1+ 1'1.1#M • o÷!"(11)1+5+ƒ+ 5+!<9Wy.+L#d,†#1('() `u+L5+ 5Bf`](+('(1#M • NO#1(,6+Qi$&7`]19W1,1+Q'K+L'Z*â v = q `'+#(:i$&7`]19W1,](9+Q)‚S,[!<1M zO,X+ ,@^ J`'Y#1+ ( .*Q'KW(k+ ^'S5)  0 Ψ

$&7`](9W9+ Ψ(v) = P ψ(v ) 'Y,X> Z*„,1(57, .5#((„!")1~,@5)*+L+(W@$&!<y`(5#(7 ‚, ôW('(!fZM b™†XW( !<1) '.+ : WR1,@,tU$&(X+  *! + ^'‚S5)".*':+5`@.6+ 3 ψ>gu(M NO#1(,9Z3'É%!<1 )'Y,9Z:'µ `'!"9+wJZ1m(5']1(,9.(#t#(7 ,@5]((+ F`ZZ j$&('K+ ]6+ ('(,> Z3' ) :@),yU,†F`1+ ,1#(3y5#,#J'Y@S+ó):U) ô* }m1:M x·s µ n i=1 iUT@U  ýqýGC?Z] ?^fe Z]i Y&EüÿRo3ú û?i[bqý?ÿqage † (^'Y> €ê·é*ê·éjê"!„ì$# &%('X
’ ê A(t1^'d$&7`]19W I­) ψ(t) : R R ԆÕ44Ö4” ¡ ҋ“  ¡ Î ž 2 s{ z Ԇ~€„ˆ{ñ#z Î „

6 ¡Ë{ Ö4{uttgÕў-[, Ž Î žÑ” “’suto{ „,Ô¹›ozÓÒo“•›o¡Ë›s Î ¡Ìƒ4v /P { „ˆ{ ¡Ë v¿Ð4z5Ž Î “€h”s=žov{ z|”x8 “v Î t Î zíÐ;ŸŸgv,žÑ” “Ô¹{ zӓ•” “€{ zӓjV é ) “•{ z ‘¹{s” „†“•{ zÓwm{uto”s vó“ Î „†“•›ow!Ð4¡ Ò4Ð4¡ sutgvݝ#›o„OS Î ¡ žÑ›o¡ Ö4{k}N, Î t = 1 y ›o¡ “Ÿ Î ¡Ì4z|›4ŸÐ4zÓv|s whvx¡ vxw­# w Î Ö Î ¡•, Î wm{ zÓҋ„ˆ{ ψ(1) = ψ (1) = 0 . M ” “€{utg¡Ë{ž“€›oÖoÐџŸ ν ≥ ν > 0 ”s Î , Ž ›ˆ4Òwhvx¡
ƒ#{ ¡ t > 0 ~€„ Î p, q ≥ 1 žÑ›o¡És “ ¡És›ž-E  E i X^fý üb1XEa¹i   +   0 ν1 (tp−1 + t−1−q ) ≤ ψ 00 (t) ≤ ν2 (tp−1 + t−1−q ).   û ° vx¡
ƒ#{ ¡ t , t > 0 ~€„ Î ”s r ∈ [0, 1] ~€„ˆ{ 1 2 ψ(tr1 t1−r 2 ) ≤ rψ(t1 ) + (1 − r)ψ(t2 ). 2 1 A·G MCMC‚I A·G MCM õI o¡#('3¡$&1'+]1+

1'n9W@K+x,†F`1+(' !"(t φ>g53' F.Wc™`]1+ 1'18]1Pth™,†#(1']1Z(8!<1$&('6> '3@(8†‚`1,()S51a nd!y@),J()u(Wf6=n1m('P,X`6=n+ aS51''(˜f+^'Y‚S5)  M ôø GMCM(d (^'Y5$&7`](XW(,„](ó+ ^'S5)  X, o 5#(,‚€!":@)u'(W1 v = q ]1€) ô*' S5^ ,tóm(19+ 'p^ + +vAµô MKCMùøXIa'@SaM [ 1W(˜+ F`Z Z* ψ : R R †1^'d$&7 `]19WMÓn%NP,9,cQm(19+ 'u0 +5+ :'P!<]15+ +LF`' +² X A·G MCMùôXI Ψ(x, s, µ) := Ψ(v) = ψ(v ) ,]('1++('p) ô*Y'.*+&S5^,jM NP'35#(†`@#1' S5^,<!"( .W~](r+

('!"(r>?d(#%8) ô* }m1:MNOW (x, s) 9+ ón> +  U,9,|`8 ‚Sr+Lt¡m(19+ '|0 +a . x·s = µe `W@ v = e MNOZ Z("#(51+Z( 3`+ D~ ψ $&7 `]19W„#10R,X`6=n+  U!<+–+ D„f$&5)}+++ó''}K+  Ψ(x, s, µ) = 0 X#MXeB5](#6+ X.*u# (x, s) `(,X+ x`'!<1'KWn,u,@) + S5 |?,†#1(' `Œ }óh>?.# ,9, |g*WR'(5O# }L j!" #1+PS51'19+ µ .WQU$&(X+  Ψ $&7`]19WQ1W@#1(5fZ7 9+1+ € Ψ(x, s, µ) m((X+ '²0 +5+ :'a`':"1'+ F`'@)*r+²](¡U!"(11)1+5+™+ 5+!y9W~.*+L.#d`':",†#(6> '() ](5+(

a!"($&('1'`fy$&19+d,@:+5+t,†‚`6+ 1'!"](9W(,9(,MpoÞ,]1+$&]1'fZ7X+ 1+](%9WX+u ψ ) ô* }m(:SZ„#(11'3 p ]1 q !<]6+ 1(,y#(Z'KW#(#(,jM —*('!<17'.6+²!"]("U,](5)](1 W„']1+(#1,>gd(WX'+L'Y„† (^'Y$&7`](XWMNP ,†9W1B`'Y5#('.*+^,j .5#(1 ψ(1) = ψ (1) = 0 !"Y+5+ x·s µ + n i i=1 0 ψ(t) = Z t Zs 1 A·GMKCM G9I ψ 00 (z)dzds. 1 Ù¡™!"r+| 16+ 1+|+ 1,99+&S57,j ,9,|#6+|,@S5^,j . W ψ00(t) = tp−1 +t−1−q ν1 = ν 2 = 1 !<d#É)%Ck+ ^ 'Y‚S5)*'d!"(,†‚`6+ ('K+B11!"$&1'+]1+ 1'1,@,1'u(W@7 +5+1W](r+ ('!³˜( >

`('9S51'†'TS57,jM Û !<1.*+ #™(W $&7`](9W9+(MNP#1+ƒJ$&7`]19W9+ƒc+‚`ZZ,9Z eB`1'd$&7`](9WUψ,,9]1+'Y,S t](w)(5`'+·S tO5#(7,95](7,"Υ'(5#(t) 91#(]15+v!<5+v#1+ p,q ¢ !3º†µË¾¸«¾´«5´«À5¬ Æ=¼ À5¨«° ¬¹²¹ª ¾ñº §º†¬oº ‹ìí´«Ñìí° ±uÃ’Ä Æ=¯o¤ §¨«¬oº ¬ ÀÑ·»À5¬o¶Æ ¸«¾3¾Ä ¤Ã¿¶° ¬o§º†±°†¾u®ìí° ¬g¾°g§Ë·`´k± ´«·`5ª ¾´«¬ À|® " ¯o° ± ²*´k¨«´k¬dìí´’Ã’´k¨«´•¾Ø´kµ´k¶ ·»¸k¬g² ´kÀɤ†¼o¼o¤†¬q¤ ¨´k§´•¾¼4´k¬1À§À±g¤†¨u¤†ÆÅ·f¤†µñ¤ ¶ ¬o¤†Æ®o¯o¤Å¤ ìíá ± ±Á ¸«¬g²¹µÇ šãk§¤ Æ/¤†¨ ψ ê|µÇ |® ¯ °± ² #%$ æó¾Ä 5¤Ã¿¶°¬o§º†± ° ¾¾´u§§¨«á ÆÅìíà

|®‹Á¤†±²¹À5§3¾Ä 5¤Ã¿¶°¬ Æ ¸ ëoë ´k¬1ãk§¤ Æ*¤†¬ ¬¹²gÀ¾Æ àˆÁ ´«¾´kÂá ¬ Æ·»´k±=¤ ψ ¬¹Ä 55º ¯ ° ¨®ÀÂ55´•¾Á ´óÁ ¸«± ¾´k´k¬ ¯ ´«¨E¾ñ¤†µ¾Á¤»·»´k±†ìí´«5´«5ǹ´k¬Å¬ àÁ ´«Æ´k¶ˆÃ’´«¬#È Þ«£¤†5¦†Ã¿º†¼o¤ ¬F㫧¤†Æ–¤†¬ ¬¹²¹À ¾ 5º†¾¿¾Äo¬ Æ®¯ ° ±² ÆgÀ5´«5¸«±ª ¾À¤†¨ #&$ æ%ìí´«Â¾¸«¾´«Â¾kÈ ß ¾°ˆÁ†º†¼ ¼ À¹¼ À5¨«°¬g²¹ªÂ¾º§¿°Æg¯o° ¨ Υp,q ´k¨·fº†µ´«5¸«±o®4¤ ¨«°¬g¼o¤†¬ *À¾¿¾E¬ ´k·õµ¸u§¿¨k´«¾´«¨k´•¾¾,·`¦¶°¬ ¼ À5¨«°¬g²¹ªÂ¾¯o¤†¾¦o®‹¯ °± ² ¤†¨ #&$ `ìí´«Â¾¸«¾´k ¾ØÀ5§ Υp,q ¾´« Ò´k§ª ¾À|® Á¤ ± ²¹À§
¾¸«¬¹²g5´«±`ଠµ´«± Äo5º µÀ5§kÈ R R ôH É GMCM(d

(^'Y5$&7`](XW(,„](ó+ ^'S5)  X, ó!"(X)‚S5^,Ú Υp,1 (t) = Υp,q (t) = Υ0p,q (t) = &A GMKCMI A·GMKCMùøXI A&GMCM HI tp+1 −1 − log t + p−1 (t − 1) p(p+1) p 1−q −1 tp+1 −1 + tq(q−1) + p−q (t − 1), q > 1 p(p+1) pq −q tp − t q + p−q , ∀q ≥ 1 p pq ‰O(W@7,„]1#1 j.W .* q 1 M Υp,q (t) Υp,1 (t) Ù¡ƒ#O())ƒ!<15!<15+a+L`'$&7`](XW(,6+{`@#(5X'TS5^,j ‚,9,{#1+ +L''TS5^,j .W  # °°°q − e°°° +LF`' Q (! † (^'Y5UA·1! Z7X+ 1+O,†#1`6+ '(97'vR!"(11)1+5+ +L5+!<9W%.*+  .#v`':t,†#(1'() ](5+( X!"}t kv − v k +L`' t† (^'Y5A p = 1 I6M9NP#ƒ+

'Y"5#('X'.*+P]1!<!<WX#++L'#²^ + :ZZ+ F`' X' ('](. 1+ 3S5ZZ q=3 (1)!"](9W1,9M ñ€(,9X+ 57,„!<5+ƒU,†F`1+,(#(m3 ™ ó+# D@5() #((r¹+ Ú x·s µ −1 2 A·GMKCMùsXI M 4x = 4s s4x + x4s = −µv∇Ψ(v). o ƒó+ #D@1)5#((53'vAµô MKCMöõIg>?Z(R!< )++ ,<](r+ ('!<]1Z(Q1#J81)#1(¡# A&GMCMöEXI x · s = −µv(∇Ψ(v) − v) (W1'6+™!"(') +,†#(1'}K+  .' ™S5ZZR')*'Ɇ#1}+1++(M eBK`(' M P (κ) !<+ 5K= p}WiòA&GMKCMösId( )#1( (, 1W](r+ 1'³! ˜1ò']1+ 1#(,V!<1' > )* nòM K<oæ5()5#(1y!"(') ,]19+",‚++ (4x, 4s) 

XWÒ!"(X+ ]1/0WÒ#1(6+ ]1, ('']( ²# (x(α), s(α)) = (x + α4x, s + α4s) 9+Z* P.W[k,+5+<X+<!"]( Z1'7 ''1W1‰ !"((1)6++8+ 5+!<9WaM{o­+‚`ZZY,@ZÒ'Y+ O$&S5^,j |. W 1#6> #(('c1'J+ ^)‚S5^, ](5f#6+ ¡. W ;m1(X+  '0 +5+ :'J!"](5+B+LF`' ;m1†,9,(‚S5(a t`W@ M á o¡!",u1#~R+L`'Q'(m1†,9,1 τ ' ,9, µ ](D> Ψ(x(α), s(α), µ) < Ψ(x, s, µ) + ](,]6+Qm(†,@,19+·S57 , ó`Wn„('3]1ZZ ']17,;Òm1(9+ '~0 +a ó#6+L¹`@5#(+ ]17,¹ ™ ó+n>?( )#1(M ∗ [ #Äo5¤Ã¿¶°¬ Æ ¸ ë ë ´k¬@¤ ¨•¾ØÆ ´«5&¼4´«º¾¬ À ®Ñ¯ ° ±²d¤

5´«·`·f¤`º Â5ªÂ¾º§¤oÈ è  ·`ª¾´«¾¿¾á Æ ·fº µk®%¯ °± ²‚¤ ìíá ± ±Á ¸«¬g²g¾*¤ ('  aÆÈÉÈ oÈ5æ Ê ¡ M −I ¢ ·fº¾µÀÂð1¬ ´«·f§¨«À5¬ ± Ä º†µÀ§«È ¨»¸ ë ë ´«¬?¤ X S Á†º†Â¾°¨«¦hìíá ± ±Á ¸«¬g² ´kÆ ¸«¬g¾=¸kµ¿¾´«5·`´«¨«¨ká ÆÈ ß x·s § ¨«´k·`¸k´«¾´k§§¸«±»¸u§[¤»Æ à ¬o¬g² ´«¼o¼*¸«µ¾¯ ´«¾Çg§¿¸«±`¸«µñ¶´«Æ ¸k¼4´«¬/¤†¨k° µ¬¹¼o¤†¬¯o¤ §¨«¬oº ¬oÀìí° ±uÃ’Ä ÆŤ Ò´kÂà¸u§’¾ Ψ(x, s, µ) À§k®¯o¤`¯o¤†¬ ±g§¿½o² °¨«¬ Àƹª5Áº†¬†Ã’Ä Æ®¯ ° ±²]·»´k²¹À5Æ=Áº†Â¾°¨«¦ ¬‹¤†Æ`·`À#¤ ¨Ø¸«µ¾¸«Æ ´È Ψ v = ôs q GMCM(d (^'Y5$&7`](XW(,„](ó+ ^'S5)  X, 8  "  þ
Y )([ $ $ ï  êµé ™ ó+

@>g']1](5(™0n+ ,†F`1+ 3<'K+!f^  $d {z 9+&Ú ε > 0 ñ²F`' "Um1(X+  'ó0 ++:'Ú τ > ν —j55}+](5a*!"]1+ 1g Ú 0<θ<1 ( ]  j   < ! 1'W1,@,1' Ψ(x , s , µ ) ≤ τ (x , s ) µ = 1 $  $ $d ]1 *!"('W1,9 x s x s<ε x := x s := s µ := µ
 ì   nµ ≥ ε −1 1 0 * 0 0 0 0 0 T 0 0 0 +-,
ì € ]( ,95#!"}+  RA&GMKCMösI'YnS o™# '](]1.#%,9#(!"}+L  ì , ¿
 ì ì , ¿
 ì µ := (1 − θ)µ  +-,  Ψ(x, s, µ) ≥ τ 4x 4s α x := x + α4x s := s + α4s +-, +., ™]1#(#17,„„A·GMKC‚IvZ  +](ó#d'5K+ !c^v'(} +MšÙ™8#tK+ 1 m1:c5 „'WX 9++<,n+ ^,j v!<1'W@ Ψ(x, s, µ) < τ

v,@,<m(5†,9,(X+·S57,â µ * !"]1+(5+((x,Mٙs) {,9,J!"(')S5^ ,k A·GMKCMùsXId1)5#((r+8](J!<1.*+ #(#(^ ,i α '](](D> Ψ(x, s, µ) ≥ τ .5#1+ |0W | W ,++f9+f!"]( !"1(()6++8!"(')*J'1W(ÉM ñ o ™ ó+n> ( )#1(5+BXW@#1„')S5^,‰!"( . W/`](7'uò,++„9+„!y~Z1k'(W1- m((X+ 'd0 + µ >g.†#8+L5+ #(:R9+&S*, τ >g,†59W1#(6+ ](Z(aM€o™#"1](5#<K+ 1 m1:F`'x,9, ''^,t!"( ‚.*™'KWXJX+ ZPS5^n+ +5+ ^,j .'@™)^'K+L5]1€,91ZZa !<9+ ε Mo τ > ν $&('+]1+('5²uZ#(XW@}+LO!"Y++|`<5#(7,95](

@!"`('#(9Z*"†5(^ 'Y5|$&7`]19WU,r> +L5#("ó† (^'Y1 }W~1#%1!ÛS51'19+ó`':@)a!"(#(}+ 5+(M o÷,!"'6=n+ ('(!"#(](5J#%'YZZÉ']1](1,9Z3'€''Ú ÿ Ù¡¡'WX X+ Z*c`W@^,j X!<1'W@ Ψ(x, s, µ) < τ ](P µ !"]1+ 15+x# • (x, s) 'K+!c^*,"!<1$&('('3@( (1 − θ)µ>?tm(5(]1' S57,j j,@,ó!<19W@`('a3.6+²!<1 0 −1 1 ô ¨«´k¬h¤ ë °¬¾°¬?´k·`¸«5´•¾¸k§ó± ²¤†Æ °µ¤ˆ¾Ø¸k´u§¿´k¬?ƹá 5à ¬gÁ†º†5ÀÂÆÈ ß ± ²¤ Æ ° µ¤¾¼o¤ ¬qÆ à¨«Á ´•¾5´«¬gá Æ ´«µ´k§¬ ´«Æ ´k± ²ÅåÜ´k¸k±`Ò¦lçf5¸ ë

¸u§¿¯o°§§¿¨«¾k®;·`À%ÁgÀ§¨«° ¬g¾¤dÆ ° · ë 5´•ðÀ¾º§ó´«5´«·`¨«¸u§¿¸k¬ ¸«
·`´«±g¤ ¶Ä ¬ Æ@´k± ²@Æ ° ¬ ƹµ¸•¾–¸«µ¾¸kÆ ´•¾u® ¤ ·`´«5²¹¬ ´«Æ ë ° ¬g¾°g§3ÆgÀ§¨kº†·`ªÂ¾º§¤»§¿°Əº†À5±¾ñ¤†µ¾k® ¶ ´Ø±g¤†µñ¤†¬¾ñº†5¯o¤¾¦FÁ ´k´–¤»·»´k±†ìí´«5´«5Ç`Æ ° · ë 5´•ðÀ¾º§«È  ß ¨óÀ ¾¾[´«5·`° ¬o¶° ¾¿¾ñ¤†Æ=º†Â¾¤ 5º ¬ °§Ë¸«µÁ ¸«¬¹²¹­ ´«Æ®¹¬ ´k· 㫧¤†Æ]´«µµ´Ø¤»Æ °¬ ƹµ¸«¾¤†5± °µÀ¾·ċ§¿µñ¤–Á ° ¬‹¤¾Æ °¨«¬o¤ Æ4È ôE GM õ@M*NO'31']1](ƒ8m(19+ 'ó0n+  GMKCM*Z   M  Fÿn ì  ì   ýþ‚ý ï 0/ ÿ€î@ï ì t+ F`'  `W@P'1$&('TS5(ZZf!<1XWnK`(' +](. 1+O(' Ψ(x, s,

µ) ]( Ψ(x, s, (1 − θ)µ) M Ù¡J'WX X+ Z*y`W@^,j !"('KWn5 Ψ(x, s, µ) > τ n,@,1W ™ ó+ n> • (x, s) ']1](R  !<19W@`('dm1†,9,(4ò+L`'  v`W@ D¹','!< α `'Y5#1+L  (51+]( Dy!<19W@K`('p'1.1+²,95(ZZ Ψ(x(α), s(α), µ) *!"9+ Ψ(x, s, µ) M o­,†‚`1+,(#(3V#(,5#(,9Z*Ò('3#(†f#y1'3,]1)]15+f`n#(5X'TS5^,j €!y‚S5)˜!<1X)^ , (WB,,9]6+ α '](]1.#6+ a]1d!"(Z(m17'TS57,B!"( 9W@`1'|m1†,9,(i<+LF`' ~1W,> , ò']1]( ](' MNP#(1 11)!"](9W1,;Zr+ ,‚Z*¹!<y!"(ò+^)S5^,â)t# (' S y']l> ]15#!<+(M iUuø!U ~

üOn1XE^füZ Z=i ÿ »^]hYXEbqü a¹i ‚Y[û? 1 32 1 W( (, ν ν p ]1 q #²†5(^'Y5|$&7`](XWBA&GMCMC‚I|) ô*}m(:SZU5#(1( '3%,r> +L5,j σ := k∇Ψ(v)k MXNPZZ(<d#(,5#(Z*~#1+P$&S5^,%`@#1X' .Wf!"( 9W@`1'*]1 .WX`'K+ #1,R!"(„<+L`'  *~ym((X+ '¡0 + µ ]15+ ]1,]1+um1†,9,19+·S57,j a`W@ >?+™]( Ψ(x, s, (1 − θ)µ)>?+™.*5'}+·S5^ ,„†5#(MÉo v `'+#(:F`' !<1$&X'!"#6` Ψ(x, s, µ) (#d(#6+xS51'19+ µ n.W Ψ(v)> +ó]1 Ψ( )> +ó*5'}+·S5^ ,"†5#(Mjoó):@) ,"+ 1.*+( .W~1']1 >?`1'a$&'',#(µM NO'3#1†+

(,99+7,„#u1W9`'+#1:²(51+ 6+¹Ú Ψ(ϑv) Î Î Î Î €êûaê·éjêxüOì  ê ŒdÎ ψ o¡ „ˆ{šzÝÐ4„†v|s ν , ν , p, q žÑ›o¡És “ ¡És›žž zt, žžÑ›o„*whvx¡
ƒ#{ ¡ ϑ > 1 1 2 √v 1−θ 1 2 Gh GM õ@M*NO'31']1](ƒ8m(19+ 'ó0n+  O, {s{ “•” ¡ vݝ Î t ؎ ›u4Ò A·G M õnMKC‚I ë
í c<@% ý ïdƒNO'3#1†J+(,9X+ 7 , #1+8#y(51+1+ x!", ϑt ≤ 1 {(,9, ϑ > 1 !"Y++ a`W@ t < ϑt ≤ 1 Mpo ψ $&7`](XWk#10i,9`l= K+LQ!<++ ψ(ϑt) < ψ(t) t<1 `W@²#u1W( '3+'1]1"+K`n'K ~+('TS5(7 ' M o®!y,V16+ Z(Ò'1W( ϑt > 1 ](J+ 1W@7,k$&('w('3#1†1 €.W t ≤ 1 M{NP,@, 1 < p}W ψ(ϑt) ≤ ψ(ϑ) ≤ ν Υ

(ϑ) a`W@u,](5#((˜`W@^ ,M€NP'](„+ (.+J#1+J# ϑt ≤ ϑ (6+ 6+™`@#('µ !<, t > 1 M o¡#ó†1^'|$&7`](XW(,8) ô*}m1:SZ:' ,†‚`6+ ,1#(,j 9.W ψ(ϑt) ≤ ν Υ (ϑt) X + ‚`ZZ,9Z*c(#^ + :Z Zn!"(XWn]1) ^,u$&('3¡Z(m1']15+(Mo¡# Υ $&7`](XWc, ,@5]1+ 'Y,(S+$&1'.#1*'`8,@S5^,j j.W
Ú ψ(ϑt) ≤ ´ p ν2 ³ p+1 ϑ ψ(t) + ϑΥ0p,q (ϑ) 2ν1 ψ(t) + ν1 Υp,q (ϑ) . ν1 2 p,q 2 p,q p,q ϑp+1 tp+1 − 1 ϑ1−q t1−q − 1 p − q Υp,q (ϑt) = + + (ϑt − 1) = p(p + 1) q(q − 1) pq ¶ µ p+1 (ϑp+1 − ϑ1−q ) p+1 t − 1 t1−q − 1 p − q + + (t − 1) + (t − 1)+ = ϑ1−q p(p + 1) q(q − 1) pq p(p + 1) (p − q)(ϑ − ϑ1−q ) ϑp+1 − 1 ϑ1−q − 1 p − q + (t − 1) + + + (ϑ − 1) = pq p(p + 1)

q(q − 1) pq tp+1 − 1 (p − q)(ϑ − ϑ1−q ) = ϑ1−q Υp,q (t) + (ϑp+1 − ϑ1−q ) + (t − 1) + Υp,q (ϑ) p(p + 1) pq Ù¡5#(*'TS5^,y$&1' .W ª ª ψ 00 (t) ≥ ν1 (tp−1 + t−q−1 ) ≥ ν1 max tp−1 , t−q−1 = ν1 max 1, tp−1 , t−q−1 !<++ ]( 1 ψ(t) (t − 1)2 ≤ 2 ν1 tp+1 − 1 t − 1 ψ(t) − ≤ . p(p + 1) p ν1 GC A·GMöõnMöõI A·GMöõnMùôXI A·G M õnM G9I GM õ@M*NO'31']1](ƒ8m(19+ 'ó0n+  NO#1(,„1}K+ ]1]1`1'p+ ‚`ZZQ}5.*+&S5^,~8Z(m1'](5+gÚ ¶ t−1 + Υp,q (ϑt) ≤ ϑ Υp,q (t) + (ϑ − ϑ ) Υp,q (t) + p (p − q) (ϑ − ϑ1−q ) + (t − 1) + Υp,q (ϑ) = pq (p − q)(ϑ − ϑ1−q ) ϑp+1 − ϑ1−q (t − 1) + (t − 1) + Υp,q (ϑ) = = ϑp+1 Υp,q (t) + p pq ¶ µ p ϑ − 1 1 − ϑ−q p+1 (t − 1) + Υp,q (ϑ) = = ϑ Υp,q (t) + ϑ + p q 1−q p+1 1−q µ = ϑp+1 Υp,q (t) + ϑΥ0p,q

(ϑ)(t − 1) + Υp,q (ϑ) ¡o # É)%CQ$&('+]1+('+",]1+#1(~X+ 1 'K`k,nS5^,j v. W‰Z!"('KW ψ(t) †1^'8$&7> `](9W@54Ú ψ(t) = Z t Zs 00 ψ (z)dzds ≥ Z t Zs ν1 (z p−1 +z −1−q )dzds = ν1 Z t Zs Υ00p,q (z) = ν1 Υp,q (t), A·GMöõnM5I .' p q ]( ν #ÌÉ)%CR$&1'+ ]6+ 1'Z(‰5#(1( '3 ''Y ):,j x''6+`„*#(' S5^,Ò/A·GMöõnMùôXI Z1m(5']15+M£/g%W Ú 1 1 1 1 1 1 1 Υp,q (ϑt) ≤ ϑp+1 ψ(t) + ϑΥ0p,q (ϑ) ν1 s A&GM õ@MöøXI 2ψ(t) + Υp,q (ϑ) ν1 ON Z Z3' ψ(ϑt) ≤ ν Υ (ϑt) ô*W('(!fZd`]6+ 1']6`(' ,@S5^,„U'(!"!yf''} +L +(M Î €êûaêûpê ‚@ ì ÿ ì@í ˜în=< ê ) t!{ z|‡gto‡z|{ whw Î Ô¹{ zӓ€” “€{ z|{ vówm{ zxz|{

“¿“Fz ” “•{utg¡Ë{ž , ν ν > 0 s ž 2 p,q -, 3 Î p, q ~€“€ozԆÕ4#‡hžÑ›o¡És “ ¡És›ž EŽ ›u4Ò!whvx¡
ƒ#{ ¡ ϑ ∈ [1, 1 + ν3 ] {s{ “•” ¡ 4 ·A G M õnMöHI ë
í c<@% ý ïdRo ' ÂٙK+L' >?#(Z*'W 5(}K+ ]1]6`('",† 9W1Þ,98+^)S5^,4#!<}K+L 8# 'YZ Za.+L5](r+ ](,1,1g+ Ú Υ (ϑ) Υ (ϑ) A·G M õnMùsXI Υ (ϑ) ϑ +ϑ = lim = lim =1 lim = lim (ϑ − 1) 2(ϑ − 1) 2 2 ]( ϑΥ (ϑ) ϑΥ (ϑ) A·G M õ@MöEXI ϑ(ϑ +ϑ ) lim = lim = lim = 2, ψ(ϑt) ≤
4 ´ p ν2 ν4 ³ ψ(t) + (ϑ − 1) 2ν1 ψ(t) + ν1 (ϑ − 1)2 . ν1 0 p,q p,q 2 ϑ1 ϑ1 ϑ1 0 p,q ϑ1 00 p,q ϑ−1 ϑ1 00 p,q ϑ1 1 p−1 p−1 ϑ1 GXõ −1−q 1 −1−q GM õ@M*NO'31']1](ƒ8m(19+ 'ó0n+   .'w$&1'*5#('+ ^ ,˜# Υ (ϑ)

$&7`](XWò)1K`'K+·SQ`+ ,#(:‰A·GMKCM5I%†#11$&7]6> (,6+MwNP!"Y+5+8']1+(#1(, ν ]1 ν #(K+ }K` , 5+L,j O. Wò!")1 ϑ ∈ [1, 1 + ν ] (6+ ]1 A&GMöõnMKCFhXI ϑ ≤ ν , ϑΥ (ϑ) ≤ ν (ϑ − 1), Υ (ϑ) ≤ ν (ϑ − 1) . NO#1(,6+tUZ1m(5'](5(,1+t8'(!"!<Z*c.1'W1+5+ 1}+5`%,nS5^ ,R#u''}K+ r+M o ,++~Z(m1'](1,/+ 1!"](#11+(„!":@);+`@.1+3,/˜+†ZZX`'K+ #1:y16+  y# 'YZ ZÉ$&!<Z*ËÚ P €êûaê ð êxüOì  ê M {4ÒÑ{ ¡ Ψ(v) = ψ(v ) , T žžÑ›o„ A&GMöõnMCC‚I Ψ(v) ≤ ³ ´ A&GMöõnMCFõI v ≥ 1+ ³ ´ A&GMöõnMCôXI v ≥ 1+ p,q 3 p+1 4 4 0 p,q 3 4 p,q n i=1 2 4 i σ2 2ν1 Q ›oÖoÐ;ŸŸÐ+,Ž Î v max > 1 °

vx¡
ƒ#{ ¡ ϑ > 1 {s{ “€” ¡ ”s v min qσ ν1 max pσ ν1 min < 1 σ ≥ ν1 µ , Î žžÑ›‹„ − 1q 1 p −q p (vmax − 1)2 (vmin − 1)2 + p2 q2 ¶ 12 . ´ p ν2 ³ p+1 0 Ψ(ϑv) ≤ ϑ Ψ(v) + ϑΥp,q (ϑ) 2nν1 Ψ(v) + nν1 Υp,q (ϑ) ν1 ”sqwhvx¡
ƒ#{ ¡ ϑ ∈ (1, 1 + ν ] wm{ zxz|{ “¿“ 3 Ψ(ϑv) ≤ , ´ p ν2 ν4 ³ Ψ(v) + (ϑ − 1) 2nν1 Ψ(v) + nν1 (ϑ − 1)2 , ν1 A&GMöõnMKCG9I A&GMöõnMCI A&GMöõnMKCFøXI Î Ž ›oz Î Î Î Î Î ν3 ν4 žÑ›o¡És “ ¡És¹› žŠwm{#{4ÒÑ{utg¡Ë{ž tÌ{4ÒoÖoÐ4zӓ•›gtos!z|{ whw!Ðџ ¡suto{ „ˆ{€y
z|‡ žÑ›‹¡šs “ ¡ ~ s›žž Î z . ë
í c<@% ý ïd{o™#u('3U(W1'3+ '1]1.1#%*#1'TS5^,y$&1' . W ψ 00 (t) > ν1 (tp−1 + t−1−q ) ≥ 1, Gô A·GMöõnMKC‚HI G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .#

}W ψ(t) = Z t Zs 1 1 1 ψ (z)dzds ≤ ν1 00 Z t Zs 1 ψ 00 (s)ψ 00 (z)dzds = 1 1 = ν1 Zt ψ 00 (s)(ψ 0 (s) − ψ 0 (1))ds = 1 0 2 ψ (t) . 2ν1 Zt ν1 (1−t−q ). q A&GMöõnMKCFsXI NOZ Z3'p†5#((#(]1 ('€,‚@S5^,R8,9}`9+²Z(m(5']15+Mjoä!y@),<Z1m(5'](5.(# t < 1 11+](3Ú ψ 0 (t) = Zt ψ 00 (s)ds ≤ ν1 Zt 1 (sp−1 +s−1−q )ds ≤ ν1 s−1−q ds = A·GMöõnMKCFEXI ñ{1,@9+7,< v `(,X+ ,@) *+LK+MÑÙ¡ v < 1 ,9, |ψ (v )| ≥ (v − 1) @.J() j,9,ƒ^WX(#J#%1W( '3+'1]1y#(]15+ƒ+('TS5(7 ' !<15+™¡S5Z ZB')*'f(+ }K`M v ≥1 pP ‰O(W@7,„]1#1 j.W *}
W Ú ψ (v ) ≥ |ψ (v ) | σ = k∇Ψ(v)k = ¯ ¯ ν A·GMöõnMöõhXI σ ≥ ¯ψ (v ) ¯ ≥ (v − 1) ∀ i, q .(W@#1(k ˜i+

1)1#(](5('|,@S5^,B"!y5@),~Z(m1'](5+(MoÛ*!<),~(W('3+5> '( ](".*':X„Z#19W@}+.*+:M o (W(), (W1'3+ '(5](;‰!<5@),;]1‰.*5!y) ,â†5#(6` ¡#i^ + ': ,1++3U1)8RA&GMöõnMCFIw'1!<!<u+ †ZZX`'K+ #1:'Y,SnMnNO#1#(1'p!" )(Wn,U(W1'3+ '1]16+ Z1'Y++ ^,jM NP#(1BZ(m1'](5(,„1}+]1]1`1'p+^)S5^,y!"(Z1m(7 'µ .Wy!<19W@K`('a3"f+ F`'  !<,†#(Z1  µ * !"]1+(ƒ. ('W6++ (1 − θ)µ>?+²`(5#(7, `Wn™ WXQ`@#19W@^'€(WX> !y5.# Ψ(v) ]1 Ψ( ) M*o®A·G M õ@MC I11)!"](9W@Z ϑ =

>?+².1'W6++ 1}K+`%,@S5^,j .W 1 1 1 0 i ν1 q i −q i i 0 i 2 n i=1 0 1 −q i i 2 √v 1−θ Ψ Ã v p (1 − θ) ! ≤ i 2 √1 1−θ ν2 τ ν1 (1 − θ) 0 p+1 2 + ν2 Υ0p,q (1 − θ) − 21 s 1 2nτ + nν2 Υp,q (1 − θ)− 2 . ν1 (1 − θ) A&GMöõnM õ@C‚I iUmU B W@þ»^fü^]ú Z]i ÿ »^]hYXEbqü a¹i ‚Y&€ûqþ 1 32 oâ+ ‚`ZZ,9Z*U#1+x`@#(' S5^,j .Wc1W³™ó+n>?']1](P 8!"(XW@+|+ ^) ^,c,†#(6> '() *dm(19+ 'x0 + .# `W@P†#(}K+ 6++ µ !<1''1+5+ x ]1 s `'+ #1+L+LF`'m1†,9,19+·S57, GG G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# Ψ(v) >?+(M*NP.1#J1W™ ó+ @>?+ }^50B5() #(15+™')^ ,y!<1 6`(#11+ ((3Ú A·GMùô MKC‚I

M 4x = 4s s4x + x4s = −µv∇Ψ(v). 1 W( x(α) = x + α4x s(α) = s + α4s ]( v(α) = q #(,™5,'#5p*!"]1+ 11,1+gÚ dx := v 4x, x x(α)·s(α) µ ME‰O(#1(57,[ZR A·G Möô MöõI v ds := 4s. s NO#1(,9,('{™S5('†'](5(,9,1' 5()5#(17,Q8,†‚`6+ ,(#13~'Y,@ZR}5.*+ :ÉÚ A·GMùô MùôXI .' Λ = µV S M V S M@o¡#¡'YZ ZjZ(m1']11,y,†9W1R,†F`1+,(#((,y P (κ) !<+5> K=n,~'Yn+ ^'‚S5)XZ:'](ƒBAµõnMùG MöôI'1!<!<Z:¿' Ú Î>2 « X 6 „ˆ{ ¡
ƒosuto{ „wm{#›oz݃ÑЋs Î , , Î €ê ð ê·éjêxüOì  ê M {4ÒÑ{ ¡ 4x ”s 4s d d y {uƒ4vݝ t . ΛM dx = d s , −1 M dx + ds = −∇Ψ(v), −1 ∗ 2 . « «6 „ˆ{ ¡
ƒosuto{ „ˆ” T žžÑ›o„dvݝ Î t Î ž Î t Î zÝÐџŸgv Î

žV 65 x Гxsž4Ð4zÝÐto›o“¿“ 65 s Ë cÌ −κσ ≤ = d d ≤ σ Ë Í Ì kd · d k = k4x · 4sk ≤ (1 + κ)σ Ë 'ŒÌ kd k + kd k = kd + d k − 2d d ≤ (1 + 2κ)σ Ë ŽÌ kx 4xk = kv d k ≤ kd k ≤ σ Ë5ì Ì ks 4sk = kv d k ≤ kd k ≤ σ ë
í c<@% ý ïPd oA&)jIƒ]1d#„A&FIƒ9+¡+K`n'K B,†F`1+,(#(,òA·mFI?>gZ3'µMjoA·mIg>?.(#J`(Wn7 , 2 x x 4xT 4s µ 1 µ s ∞ 2 1 4 T x s s 2 1 4 ∞ 2 x s 2 T x s 1 −1 −1 x vmin −1 −1 s 1 vmin 2 x s 2 √ 1+2κ vmin √ 1+2κ vmin ](#1 .W kd + d k = k−∇Ψ(v)k = σ }W%Ú x s ·A GMùô M G9I .'v#y',‚'!<#(++%(W1'3+ '(5](V# AµXIuX+fZ*'v')' Mpo A&ZjIJ''}K+ %1W@#1(u˜

,†‚`1+,(#(!"]19WyAµXI?>g*,<f5!y,",†#1†++a1W( '3+'( ]((,<$&('.*#1' `'µMno™#"AµXI 9+y('3 1W('3+ '( ]((,X`@`'1~#(#(' O.W M P (κ) !y+   = D[(# ;AµõnMùG MöôXI kdx k2 + kds k2 = kdx + ds k2 − 2dTx ds ≤ σ 2 + 2κσ 2 = (1 + 2κ)σ 2 ∗ GÑ G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# '(!"!<ò''}K+ ÏDj ó[!<@) ,9.# ()[*#1'TS5^,/$&(' .W;Z*!"' W( a, b ∈ R (6+ ]1 ab ≤ *}
W Ú (a+b)2 4 dTx ds = n X i=1 [dx ]i [ds ]i ≤ n X ([dx ]i + [ds ]i )2 4 i=1 NO#1#(1'{8'1!<!<c!<)1Q''}K+ +óZ1'Y+5+ ^,jM p(W1 = σ2 4 A·GMùô M5I ½ vmin ᾱ = min 1, √ σ 1 +

2κ ¾ A·G Möô MùøXI , ( ,9,P#('3#(3¡'(!"!yJA·)jI€]1²A·FI€X+·S™!<++€!<)( α ∈ [0, ᾱ) (51+]( x+α4x > 0 `W@|¡!<=n!y'K€!"(11)1+5+x'](]1.#1 tX# ‚W α ≥ ᾱ MNP##(@Z*f!<]1 (! (']("UZ #(9W@}+ . # (531ZZB(5()!"](XWn5u`Q#17,9](7,jM o¡#6+c,S5^,V!"(`@#(5X'  É.Wk!"(XW@`1'wm1†,@,1[„+L`'Q1Wk']1](%^ +La `W@² g(α) = Ψ(v(α)) − Ψ(v) $&7 `](XW9+ƒ`@#1X'TS5^,jMo÷,†F`1+,(#(3<+]1+ 1'pX#4Ú Î €ê ð êûpê >JîX ì  ê ŒdÎ Ψ(v) ≥ ν , žžÑ›o„ A·G Möô MöHI C C g(α ) ≤ − σ ≤ − (2ν ) Ψ(v) , max −1 1 q−1 q ∗ Î Ž ›oz Î t α 4 ∗ z|”€y ”sŽ ›ssut Î t

4 1 q−1 2q q−1 2q ¡ ¢ 0 = −1 + ν2 ν4 α (vmax + αν3 σ)p + (vmin − αν3 σ)−q . -, {ñ4ÒÑ{ ¡ z|{ “wm{#›oz݃ÑÐos Î žÑ” ¡&“FÐ4zxz{ z|‡ Î C žÑ›o¡És “ Î ¡És`y {uƒ4vݝ $ ν1 √ 2ν2 (1+2κ)(ν1 +p)+ν1 1+2κ(p−1) ž ¹t՚z Î žov|s{gŸŸgv|ž . ”s ν12 √ (1+ν1 )(2ν2 (1+2κ)(ν1 +q)+ν1 1+2κ(q+1)) Gø A·G Möô MùsXI A·GMùô MùEXI G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# ë
í c<@% ý ïdcAµz{1 `p ôXõ rI%o¡#É)™õB$&1'+]1+ 1'w$&('.*#1' `' f g(α) $&7 `]19W@R# 'YZ ZaZ1m(5'](ƒ}.*+:&Ú 7 n ³ n ´ X X g(α) = ψ(vi (α) − ψ(vi ) = −Ψ(v) + ψ(vi (α) ≤ i=1 i=1 ≤ −Ψ(v) + l¡ô* Y'TS5^,~ n X i=1 ≤ −Ψ(v) + 1 1 (vi + α[dx ]i ) 2 (vi + α[ds ]i ) 2 ≤ n ´ 1 X³ ψ(vi + α[dx ]i ) + ψ(vi + α[ds ]i ) . 2

i=1 A·GMùô MKCC‚I A·GMùô MKC‚õI ϕi (α) := ψ(vi + α[ds ]i ) ri (α) := ψ(vi + α[dx ]i ) $&'W9+  R)Èà(5(m1Y'.*+:f$&7`](XW(,1+(MNP,9,gÚ ϕi (α) = ϕi (0) + αϕ0i (0) + A&GMùô MKCFhXI Zα ³ ´ 0 0 ϕi (s) − ϕi (0) ds A&GMùô MCôXI ON '(!"„5#!"''R÷,†F`1+ ,1#(3Þ1W( '35](1,1+â,@S5^, #-1W( '1+Z(­#1(5('3 !<19W@5](1,94Ú A·GMùô MKCG9I ϕ (0) = ψ(v ) A&GMùô MC I ϕ (α) = ψ (v + α[d ] )[d ] A&GMùô MKCFøXI ϕ (0) = ψ (v )[d ] . NO#1(,6+¡Z(}5`U8$&19+ a(W1'6+ Z%,nS5^ , *.W 0 i i 0 i 0 i 0 i s i 0 i s i ψ(vi + α[ds ]i ) = ψ(vi ) + αψ (vi )[ds ]i + Zα ³ 0 Ù¡5':3Ú s i ϕ0i (s) − ϕ0i (0) 0 ´ ds. Zα ³ ´ 0 0 ψ(vi + α[dx ]i ) = ψ(vi ) + αψ (vi

)[dx ]i + ri (s) − ri (0) ds. 0 0 É A·GMùô MKC‚HI A·GMùô MKCFsXI kß ¨Ø´«± ¸u§¿¨¼ À5¨«°¬g²¹ªÂ¾º§Ë§¿°µº ¬]´«¨Ø¤ ¨´k± ² ´•¾Â´k¬=¯ ´k²®¤ ¯ ° Ѥ†¨ #&$ –ìí´«Â¾¸«¾´«Â¾¯‹¤ §¨«¬oº Ã’Ä Æȹ£¤†5¦†Ã¿º†¼o¤ ¬=À ¾¾ §´«· 5´«¬ ¬ ´Åµñº1§¨«á Ƨ¸«±o®š·»À5Á ´«Ë¼4° ¬¹² ° 5Ä Â¾¤ ¼ ¼§¨kº ·»°5º§–º µº ¬h¤ ¨ #%$ æ]ìí´«Â¾¸«¾´k¼4ÇËÀ5§F¯o¤ §° ¬ 5¦1¼4´kãk§¿5¸k§¿¾ ¾Äo¶¬oº ¬ Æ=5´«Á ´k¨«´•¾¬ À|È ß µ¸k§¨«5´•¾´«Æ1î  uï5ê|¼4´«¬*¾¤ 5º ¯o¤†¾¦Æf·`´«±oÈ  WÉ GXH G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# o¡#u^ + :ZZp,]6+™1W('3]16+¡# i )l= 1,9%†5#(1(#6`XJ,@S5^,j j. W n ³ ´ X ψ(vi + α[dx ]i ) + ψ(vi + α[ds ]i ) = i=1 = n X i=1

ψ(vi ) + α n X i=1 ! Zα ÃX n ³ ´ ψ 0 (vi )[dx + ds ]i + ri0 (s) − ri0 (0) + ϕ0i (s) − ϕ0i (0) ds = i=1 0 à ! α Z X n ³ ´ 2 0 0 0 0 = Ψ(v) − α k∇Ψ(v)k + ri (s) − ri (0) + ϕi (s) − ϕi (0) ds = i=1 0 = Ψ(v) + g1 (α), .' g1 (α) = −ασ 2 + Zα ÃX n ³ i=1 ! ³ ´ ri0 (s) − ri0 (0) + ϕ0i (s) − ϕ0i (0) ds ´ A·GMùô MKCFEXI A&GMùô M õhXI o ,†F`1+ ,1#(3,9Z<Z1m(5']15+) ^,8# r (α) ](w ϕ (α) )1K`'K+L,9 n (. (#ƒ#17,9](7 , '(5#%#%'ZZ a'(!"!y š Ú €ê ð ê ð êxüOì  ê {4ÒoÕў Ô¹{ zt, Ž ›u4Ò z|{4Ò;{ ¡ Î Ψ(v) o¡ „ˆ{šzÝÐ4„†v|sl, ᾱ y {uƒ4vݝ Q 0 0 i ; o n √ žÑ”€y
z|{ “¿“•{ z[ƒ#{ –¡ v¿Ð4zÓÖ Î 1+2κ min 1, σ vmin 0 i . T žžÑ›o„V &A GMùô MöõnC‚I ë
í c<@% ý ïdaoÞA·G Möô

MKC‚IP'1!<!<<A&ZjIO9+&S+]1ó~A&GMöõnMöôIO'(!"!y+O$&('.*5#('`%,@S5^,j .W µ ¶ 1q ° −1 ° ° ° √ °(x 4x, s−1 4s)° = °(v −1 dx , v −1 ds )° ≤ 1 ≤ 1 + 2κ 1 + qσ . ᾱ ν1 ° −1 ° ° ° °(x 4x, s−1 4s)° = °(v −1 dx , v −1 ds )° ≤ √ µ ¶1 q 1 σ 1 + 2κ √ qσ q 2 2 ≤ kdx k + kds k ≤ ≤ 1 + 2κ 1 + . vmin vmin ν1 NO#1#(1'{8'1!<!<+Z1'Y+5+ ^,jM eBr+ƒ!y!"($&'!<#(.*+·S5^,y8Z(m1'](5+gÚ €ê ð íê €êxüOì  ê M {4ÒÑ{ ¡ Ψ(v) o¡ „ˆ{šzÝÐ4„†v|sl, α ∈ (0, ᾱ) .&T žžÑ›o„WV max {ψ 00 (vi + α[dx ]i ), ψ 00 (vi + α[ds ]i )} ≤ ν2 ω(α), i Gs A&GMùô MöõõI ¤ A·GMùô MöõôXI G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# Î Ž ›oz ³ ´p−1 ³ ´−q−1 √ √ ω(α) =

vmax + ασ 1 + 2κ + vmin − ασ 1 + 2κ , ”s n ³ n ³ ´ X ´ X 0 0 0 0 ri (α) − ri (0) + ϕi (α) − ϕi (0) ≤ αν2 (1 + 2κ)σ 2 ω(α). A·GMùô MöõG9I ë
í c<@% ý ïd8o™#i1'3Z Z-Z#(XWn}K+ ++R'(!"!<ò!"Y+5+R (0, ᾱ) 9+ 15`''^!fZ¹']6`3 α #!<,9 A·GMùô Möõ I (v + α[d ], v + α[d ]) > 0, }Wy}5.*+·S5^ , .W ¡ ¢ A&GMùô M õøXI ψ (v + α[d ] ) ≤ ν (v + α[d ] ) + (v + α[d ] ) ¡ ¢ A&GMùô MöõHI ψ (v + α[d ] ) ≤ ν (v + α[d ] ) + (v + α[d ] ) . —j1'.#1*'` . WRQA·G Möô MKC‚I'(!"!<RA&mFIX+·S8!<+5+ √ A·GMùô MöõsXI α kd k ≤ ασ 1 + 2κ √ A·GMùô MöõEXI α kd k ≤ ασ 1 + 2κ ,@S5^,„U'(!"!yc('3"''}K+ +M oq!y@),V''} +LdZ#(XW@}+L

.#"*#(' S5^ ,V$&1'w ϕ $&7 `]19Wi)1`'+&S ~`> *+ ,#(:A·G Möô MKCG9Iv,]('1+(,6+¡]1™U) È à(1m1Y'#!<}K+ ,†#(]1](r+ ](,X+ ]1+(']6¹+ Ú i=1 i=1 x s 00 i x i 2 i x i 00 i s i 2 i s i p−1 p−1 i x i −q−1 i s i −q−1 x s n ³ n ³ ´ X ´ X 0 0 ϕi (α) − ϕi (0) = ψ 0 (vi + α[ds ]i ) [ds ]i − ψ 0 (vi )[ds ]i = i=1 i=1 = n X ψ 0 (vi + α[ds ]i ) − ψ 0 (vi ) α[ds ]i α[ds ]2i = n X ψ 00 (vi + χi [ds ]i ) α[ds ]2i , A&GMùô MùôhXI  .' 0 ≤ χ ≤ αe v`W@ χ ∈ (0, ᾱ) v}Wâ','!<#1*+:ò !yyZ#(XW@}+ ++y('3 (W1'3+'1]1&Ú i=1 i=1 i 98 n ³ n ´ X X 0 0 ϕi (α) − ϕi (0) ≤ ψ 00 (vi + χi [ds ]i ) α[ds ]2i ≤ i=1 i=1 ≤ αν2 ω(α) n X i=1 [ds ]2i = αν2 ω(α) kds k2

A&GMùô MöônC‚I käd´«±ˆÃx´k± ² ´k¨«¨«áoÆ4®¯o° ± ²=´«¬ ¬ ¸k;´kµÇg§¿´k¼ ¼Å¼4´u㫧¸u§3À§[¤ ¶¯‹¤¾¦‹®´k¨ó¤ ¨«°¬g¼o¤ ¬Å¬ ´k·c¼4´«ìí° 5²º §° ÿ¤»¤`Æ ° · ë 5´•ê ðÀ¾º§¿ÀÑ´kµ´u¶·`¸«¬g²=¬o¤†±²§º ± µ´«¬o¶ˆÃ’¸«¾kÈ : GE G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# o÷!y,"+  f(. (#%*':U!":@)5#((51' ,‚.*+&S5^,~!<1" A&GMùô MöôõI n ³ ´ X 0 0 ri (α) − ri (0) ≤ αν2 ω(α) kdx k2 Z1m(5']15+Mjg5!<]6+ƒ.*5#('`"QA·GMùô MKC‚I'(!"!y„A·mIX+·S+²,nS5^ , .W i=1 n ³ n ³ ´ X ´ X ¡ ¢ 0 0 0 0 ri (α) − ri (0) + ϕi (α) − ϕi (0) ≤ αν2 ω(α) kdx k2 + kds k2 ≤ i=1 i=1 ≤ αν2 ω(α)(1 + 2κ)σ 2 .

NO#1#(1'{8'1!<!<c!y@),~''}K+  +óóZ1'Y++ ^ ,M ‰t]1(5()!"](XW@Z( #1+², +^,j j.W g1 (α) = −ασ 2 + Zα ÃX n ³ 0 i=1 ¤ ! n ³ ´ X ri0 (s) − ri0 (0) + ϕ0i (s) − ϕ0i (0) ds ≤ ´ A&GMùô MùôôXI i=1 2 ≤ −ασ + (1 + 2κ)ν2 σ 2 Zα A·GMùô MùôG9I sω(s)ds =: g2 (α). o g (α) $&7`](XW-,]6+ 5#((Q$&'KW@+-) àÉ11m('.+ :[ [0, ᾱ) 9+(5`''^!"aMo )(5`'K+L,Ú A·GMùô MùÑô I g (α) = −σ + (1 + 2κ)ν σ αω(α) ¡ ¢ A·GMùô MùôøXI g (α) = (1 + 2κ)ν σ ω(α) + αω (α) > 0 o¡#u1'3U)1K`'K+`@#(5X'+L`'É!"(W3#13@).1+7,~:'Y nW g (0) < 0 ]( g (α) #1K> + }`R# ᾱ ,9™,†XW(#11+ ]1Z(aMÉoÛ!<@)

,~)(5`'+ƒ#(K+ `@K+LU!"Y+5+ g (α) 5#(0 ,9`l=  [0, ᾱ) 9+(5`''^!"a }WR<!"!f^ !<+ó`WQ#%X+ (r`''^!­5#(]1']1 ']1`3 ᾱ 9+Z*a €`W˜`'Y!"' W( ,†#(Z 7'53 α .('KW(ò`1#1$&†' M€NOW 'W1 α X+ Z*Ò# ('53") (K`'K+²5#(7,95](,](( 0 }W α # 0 2 0 2 00 2 2 2 2 2 0 2 0 0 2 ∗ ∗ ∗ h³ √ ´p−1 ³ ´−q−1 i √ + vmin − ασ 1 + 2κ A&GMùô MöôHI (W1'6+™1W]15+ 1'!³˜Q!"(')  M bu]6+ƒ(51+1+™,@7 '†9Z†#1+1+7,„!"(ÉÚ.* α ≥ ᾱ j,@, yA·GMùô MùôXIx'(!"!yd':%Z(m1'](5+5#('X'K+L+ α >?  Mo-(.

(#(1ZZ„16+# 9 α < ᾱ 9(#1#(1' $&'Y',#1^,R<+‚`ZZ,9Z*aMo™#c(W1'1+tW†,]1(,BZ1m(5'](5](.(#85#(7,95](7,'(#c ,†‚`1+,(#(3<'1!<!<É Ú h 0 = −1 + (1 + 2κ)ν2 α vmax + ασ 1 + 2κ ∗ ∗ ∗ G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# ê M {4ÒÑ{ ¡Ë{ž p > 1 ”s γ > 0 „ZutW8x“€{ ““óžÑ›o¡És “ Î ¡És›ž . €ê ð ê ' êxüOì  Œ1Î Î t∗ ­¬ Î t(1 + t)p−1 = γ {4Ò;{ ¡ z|{ “f{4Òє „†“•{ zÓw Šwm{#›oz݃ÑÐos t∗ ≥ ° Ðos „ˆ”sutg“;,؎ Î t ∈ (0, 1) Î t(1 − t) ∗ ρ 1 + γ(p − 1) −1−p t∗ ≥ , Î žžÑ›o„ ­¬ $ l, Î žžÑ›o„ = γ {4Ò;{ ¡ z|{ “f{4Òє „†“•{ zÓw m4Ò žÑ{ γ 1 + γ(p + 1) ë
í c<@% ý ïd{oä* !<]6+

1(,95u+1++²!<1,†+](1,„!"Y+5+ t > 0 + 1' S517'µ }W ∗ µ 1 (1 + t∗ ) = γ 1 + t∗ p + 1)1#1`c#u(W('1+ 6+¹Ú ¶ > 1. A&GMùô MöôsXI A&GMùô MöôEXI A·GMùô M GXhXI µ µ µ ¶1 ¶− p1 ¶−1 γ p γ + γt∗ − t∗ γ + γt∗ − t∗ t∗ = γ + −1 = 1 − −1 ≥ 1 − −1 = t∗ γ + γt∗ pγ + pγt∗ γ + γt∗ − t∗ = > 0, (p − 1)(γ + γt∗ ) + t∗ A·GMùô M GC‚I .'É$&('*5#('K+ ^,y γ + γt − t #1+`@+ + n''1+5`u# ∗ ∗ (1 − t)α ≤ 1 − αt ∀ α, t ∈ [0, 1] A·GMùô M G9õI (W1'3+'1]11+(M*ñ{1,9X+ 7,„!"r+ƒ ·A G MöônMùGXôI (W1'6+ 1+(M=]v(.1'W6++ 1}K+ ](5('€!"(W3#13@)1+7,~:'

WR#d(W1'6+™,]1+™W†,4Ú A·GMùô M GG9I γ t = −1; t = > 0. h(t) = (1 + γ(p − 1))t2 + (1 + γ(p − 1) − γ)t − γ = 0 1 2 NO''13#(. 1+ 3 W }W t #1+`@+ f!<+5+ ∗ 1 + γ(p − 1) h(t∗ ) > 0, t∗ ≥ t 2 = γ , 1 + γ(p − 1) @C A·GMùô M G;I A&GMùô M GXøXI G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# `W@P!"(, + ^,f#²1'53J''} +L5+(Mo !<5@),9.#`(#(17,UZƒ t̄ = !<1''KW1' t∗ t̄(1 + t̄) = 1 − t∗ p µ 1+ t∗ 1 − t∗ ¶p t∗ = 1 − t∗ µ o¡',‚'!<#(#(^ ,~#d('53<''} +L5+óU,++²(W1'6+ ‹Ú t̄ ≥ `W@ 1 1 − t∗ ¶p t∗ 1−t∗ 0Sx`'+ #(:+( = t∗ (1 − t∗ )−p−1 = γ. A&GMùô MùGXHI A&GMùô M

GXsXI γ , 1 + γp A&GMùô M GXEXI γ t̄ γ 1+γp t∗ = = ≥ . γ 1 + t̄ 1 + 1+γp 1 + γ(p + 1) NO#1#(1'{8'1!<!<c!<),]6+ƒ''}K+ +óZ1'Y+5+ ^,jM eBr+<!<f!<1k+^)S5^,òZ1m(7 ' α >g+"# α < ᾱ (51+ Z1â(Mxo+‚`ZZ,9Z*¤ + (Wn7 ,~$&('µ .W Ψ(v) ≥ ν ](1 jW v > 1 Mñ€(,9X+ 7 ,R# ³ ´ √ A·GMùô 5M hXI 1 + ασ 1 + 2κ ω (α) = − + (1 + 2κ)ν α v =0 2 ](™# ³ ´ √ A·GMùô 5M nC‚I 1 ω (α) = − + (1 + 2κ)ν α v − ασ 1 + 2κ =0 2 (W1'6+ (,1+MoâZ*'@')'c'': ω (α) ]( ω (α) $&7`](9W1,c †‚`1,9`3@1, }WJ#(W1n> '1+(,91,˜1W](r+ 1'³! ˜ !<1')*^ ,V`a S51'†' S5U(#1(,1+81)5 α ]( α ,M ‰O1#(17,˜Z  √

√ A&GMùô M õI α 1 + 2κ 1 + 2κ t = , γ = 2(1 + 2κ)ν v S5('†'](5(,6+MjNP#1(,9,('{#u1'53"1vW( '1+ A&GMùô M ôXI t (1 + t ) =γ 'Y,9Z*R}.+ :Mno¡',‚'!<#1`8#d('3#(3U'(!"!yc('3UZ(m1']1]6+¡,nS5^,j *.W A&GMùô 5M G9I γ t ≥ . ∗ −1 1 ∗ max p−1 1 2 max 2 2 min 1 −q−1 2 ∗ 1 1 1 p 2 max max 1 1 p−1 1 1 1 1 + γ1 (p − 1) õ ∗ 2 G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# 6ó+ 'Y,9}K+L^ +LQ}.*+·S5^,j . W α1∗ ≥ γ1 vmax vmax √ √ = ≥ p σ(1 + γ1 (p − 1)) 1 + 2κ 2(1 + 2κ)ν2 vmax + (p − 1)σ 1 + 2κ 1 √ ≥ ≥ p 2(1 + 2κ)ν2 vmax + (p − 1)σ 1 + 2κ ν1 √ ≥ ≥ 2(1 + 2κ)ν2 (ν1 + pσ) + ν1 1 + 2κ(p − 1)σ ν1 √ ≥ σ −1 2(1 + 2κ)ν2 (ν1 + p) + ν1 1 + 2κ(p − 1)

A&GMùô M4I  .'#ó1W('3+ '(5]((,f5()5 v ≥ 1 XUA·GMùô MKC‚I{'1!<!<8A&mFI|X+·St]( Ψ(v) ≥ ν !<++ σ ≥ 1 ,†F`1+,(#(!"](XWX1 M o÷!<,<(W1'6+ƒ(6+ ]1.*5':8(' S 5 'a,‚@S5^,j j.W A&GMùô 5M øXI ν ¡ ¢σ √ α ≥ . −1 1 max ∗ 2 2 1 (1 + ν1 ) 2(1 + 2κ)ν2 (ν1 + q) + ν1 1 + 2κ(q + 1) l¡ô* }m(:c5#((5X+ó# − q+1 q 1W1'6+(,X`@`'1# ω (α ) + ω (α ) = 0 (W1'6+5> + ('µ }W M (t#1(5}+5`f#u()) p11)!<]19W1,1+d#6+™,‚nS5^ , j.WQ *]1 v Ψ(v) ≥ ν $&('+]1+('1,y!"(''1+5+²QA·GMùô MùôXHIv(W1'6+™1W]15+ 1'³! ˜ α !<1')* É Ú ω(α∗ ) = 0 α∗ ≥ min {α1∗ ,

α2∗ } 1 ∗ 2 ∗ −1 1 ∗ q+1 α∗ ≥ Cσ − q , .' max > 1 A&GMùô M5HI A&GMùô M5sXI NO#1([Z1m(5'](fZr+ ,‚Z* !<J,†XW([Z1$&gS5(#1.1+&S57,òB+ ]1+('vZ#19W@}K+L+ME@#(7,9]l> 7,~'(#%8,†F`1+ ,1#(3~ :U'(!"!<É Ú Î €ê ð ê GpêxüOì  ê {4ÒoÕўIÔ¹{ zt,!Ž ›u4Ò h(t) žÑ” “’suto{ „7ƒ4v P/{ „{ ¡Ë v¿Ð4z5Ž “€žÑ›o¡ Ö4{k} Ԇ՝4Ö4” ¡ Ò-, Q C = min n ν12 ν1 √ √ , 2ν2 (1+2κ)(ν1 +p)+ν1 1+2κ(p−1) (1+ν1 )(2ν2 (1+2κ)(ν1 +q)+ν1 1+2κ(q+1)) Î wm{ zÓÒo„ˆ{ . A·GMùô M5EXI h0 (0) < 0. h(0) = 0, o ~€¡Ë{ž@4z|›4ŸÐ4zÓv|s1whvx¡ vxw­#w Î Ö Î ¡ Î t > 0 y ›o¡ “Ÿ Î ¡”sdŽ ›u4Ò Q {ñ4ÒoÕў]wmÔ¹›o{ ¡Ëz3›o“•“€›o›oÖo¡ Ðџ¡”ŸoÐ+Ö4,Ø{Ž žo›uÖ44‡`Ò Ô†Õ4h(t) 4Ö4” ¡

Ò .@T žžÑ›o„ ∀ t ∈ [0, t ] {s{ “€” ¡ h (t) ∗ 00 ∗ h(t) ≤ ô h0 (0)t . 2 A&GMùô MùøhXI G MöônMjb™†#(1'() ](™8m(19+ 'ó0n+ .# ë
í c<@% ý ïdxeQ`1' h(0) = 0 }W h(t) = Zt 0 0 h (s)ds = h (0)t + 0 Z t Zs 0 00 0 h (z)dzds ≤ h (0)t + 0 Zt sh00 (s)ds = 0 = h0 (0)t + [sh0 (s)]t0 − Zt h0 (s)ds ≤ h0 (0)t − h(t), .B+( )(#1](™^ + Q,nS5^,„8,@}K`X+™''} +L5+(M jg)]1#(#(7 ,„$&('a](.*9WR)  ô}m(:+²]1ƒ1() !<]19W9+¡U5#,‚#%('DS5]13'Ú A&GMùô MöønC‚I 0 g(α) = n ³ n ´ ´ X 1 X³ ψ(vi (α) − ψ(vi ) ≤ −Ψ(v) + ψ(vi + α[dx ]i ) + ψ(vi + α[ds ]i ) = 2 i=1 i=1 1 1 = − Ψ(v) + g1 (α), 2 2 .' 2 g1 (α) = −ασ + Zα ÃX n ³ 0 i=1 ri0 (s) − ri0 (0) ´ + ³ ϕ0i (s) − ϕ0i (0) ≤ −ασ 2 + (1 + 2κ)ν2 σ 2

]( ¤ Zα ! ´ A·GMùô MùøXõI ds ≤ sω(s)ds =: g2 (α), A·GMùô MùøôXI 0 ³ ´p−1 ³ ´−q−1 √ √ ω(α) = vmax + ασ 1 + 2κ + vmin − ασ 1 + 2κ . o g (α) $&7 `]19W~+ ('TS5(5}+ aU'(!"!<c$&1'+]1+ 1'1+( }W 2 A&GMùô MùøG9I &A GMùô MùøÑI /DS5^,„Z%(Z Zc#%1W( '3+'( ](Zc#u!"](9+²,++cA·GMùô M5HI(1)!"](XW@+( }W„,nS5^ , UZ#(XW@}+L):8+]1+('Z1B5#(1( '3"Z(m(5']15+ *!<#(19+ A&GMùô MùøøXI −σ −σ C g(α ) ≤ α ≤C σ =− σ . 4 4 4 ñ{1,@9+7,RQA·GMöõnMùôXI'(!"!yc('3"''}K+ ¹+ Ú p A·GMùô MùøXHI σ ≥ 2ν Ψ(v). G 2g(α∗ ) ≤ g1 (α∗ ) ≤ g2 (α∗ ) ≤ ∗ 2 2 ∗ 1 g20 (0) ∗ −σ 2 ∗ α = α . 2 2 − q+1 q q−1 q

GM GM*b™!"'6=n+  NO#6+ƒ$&1'.*#1'`8,@S5^,„8+]1+ 1'p!y5@),<1W( '3+'1]1]1+gÚ A&GMùô MùøsXI q−1 q−1 C q−1 C − σ q ≤ − (2ν1 ) 2q Ψ(v) 2q . 4 4 ON #1#(1'{f+ ]6+ ('K+²Z('++^,jM €ê ð ê J êØ9äì;"  ì;" < í înï ê ›o¡ “•›s]Ž Î ¡#osOS#zÓÒ;›gtg¡ v ,[Ž ›u4Ò Î wm{ Î ƒ#›o““ α z|”€y ”sŽ ›ssut*¡Ë{ w Ô¹{ zÓ~ l, <; +8 ∗ .) “•” “¿z|{ ¡ ՚z–›y
“vxw!Ð4zÓv|s fƒ#{ Πў ”s‡4ŸŸgv»ŸgvÝto›o¡ Ò x“Ðos›žŽ ›gthw!Ð4„!{ ¡ ¡ ÒovFv|s { z|” 4Ò Î žÑ›o„ z Î “Ÿ Î ¡ Ðz“ Î zÝÐџ Î ¡I¡Ë{ w { z|‡o„ˆ{1„ ut x“•{ “¿“ α ~€“óŽ Î sutg¡
Ð4zÓ¡ Î ž ,Ž Î ¡Ë{ w Ö Î z Î whvxzÓÒ;{ ¡ {4Òƒ4vxwm{ ¡#tgvs@›y
“¿vx~ w Î zÓvÝt¹ÐzíЋs†v zÓvx¡Ë{ ~s{ Î „Ž

wmgƒosuto{ „†„{ z
ž gtgÖ4{ “¿z|{ ¡ ՚z
žÑ{ „{s ¡Ë{ž Î z5ž Î zÓw Î sdz|”€y ”sŽ ›ssutg“ 2 Z W8 6 -, $ . 2P)1!<1ó!<1]1#(µ .Wy!"+)Rc$&19+a11)!"](9W~#t(W@#1(k˜ZZB11+(,9Z(aMÙ¡ j`W@™# !<+ 5K=R#(K+ }`„5#((!") *ô K+™](™+ F`' ,~`'Y!<1'W@, ]1')^' ,  ,9^κ =Υ0 $&7 `]19W9+yM`'#6+·S5^,j v,9, C = min n , o Mvb™,9]6+L; !y,+ƒ(!"'}K+1++ kv − v k +L`'U16+ ](òA p = 1 q = 3I(# C = >g1)B)aM 1 1 3p+1 6q+6 p,q −1 2 1 24 iUjiU B ûa ü ^Ðk‚a†Y,bqi o¡#J('3#13<#(,5#(,Q1() !<]19W1|'YnSR,†9W1V!"(U+ ^)S5^ ,Q) {#%('TS ƒ'](](D>

#!y+(Moä`]1'1(²(5()!"](XWQ8,†‚`6+ ,(#13~'Y,+ó†'+ Ú Î €íê  ê·éjê >JîX ì  ê ŒdÎ τ ≥ ν , žžÑ›o„d¡Ë{ w “v4ŸŸO,–whvx¡ “ −1 1 & q−1 q+1 8q(2ν1 )− 2q (ψ0 (θ, τ, n)) 2q C(q + 1) = -,  '» ¼ 1 n log , θ ε A·G MùGMKC‚I O, z|”€y”sgŸ { ¡ ›ozÓÒ Î ¡mwm{#›oz݃ÑÐos “%ž Î y #¡Éž Î wm{ zÓҋ„ˆ{ Î ƒ Î zÓvx“ñÐos „ˆ”s ε ~€¡
Ð4z;žov|s{gŸŸ Î Ž ›oz ψ (θ, τ, n) 0 Î 65 {4Ò;{ ¡ z|‡o“¿z|{ ¡És”]‘¹›4ŸŸ›oz݃ Î z Î –Ö Î 4Òov|s :2 . Y YrX 6 ψ0 (θ, τ, n) = ν2 τ ν1 (1 − θ) p+1 2 , +ν2 Υ0p,q (1−θ) − 12 s 1 2nτ +nν2 Υp,q (1−θ)− 2 . ν1 (1 − θ) A·G MùGMöõI ë
í c<@% ý ïd|bd]6+ƒ)'+ó,1''a!<1`@#1 '@^ ,
Ú o¡# (x , s , µ) X+

Z:'²)^'K`i.9W@#1y,1''²1'35(']1²˜m1(X+  'U0 +a OW‰ ,9}`9+²9++²(']1&S57,jM 0 0 4 GM GM*b™!"'6=n+  Ù¡"#%'5K+ !c^ó5  j,9,²'1$&('TS5(ZZR.*9WŠƒó+#D@'](]1™#17,> ]1(™. # *. Wy`](7' '1W(aM o¡#%1'53<9+ . #%)'TS5^,y!<1 W„*"#J'K+ !c^ó µ = 1 ](5+](,93'€)^ ' ](ƒ!" )(R'](]1Z( µ .('KW1++ (1 − θ)µ>?+ó`1#17, ,@, k ']1](²^ +LB A·G MùGMùôXI (1 − θ) nµ ≤ ε $&('+]1+('+ó,1''É`n#(5X'@^,jM* XK+ !c^5+`]6`4 Ú A&GM GMùGXI k log(1 − θ) + log(nµ ) ≤ log ε,

!<Z3'É+ 5()1#(]1™^ +La $&1'.*#('` .W − log(1 − θ) ≥ θ ,nS5^,j *.W A·G MùG5M I nµ kθ ≥ log(nµ ) − log ε = log , ε `W@ó'1$&('TS5(ZZ § log ¨ ',‚'!"!<',(''  µ * !"]1+(ó]15+ ]1,]1+™m(5†,9,(X+ ( M o¡#u!<@) ,<X+ .#d+(,9X+ 7 ,R#u'YZZ+ (mL,p'(!"!<+¹Ú Î Î Î €íê  êûpêxüOì  ê M {4ÒÑ{ ¡ t > 0 ›ozÓÒ ¡Is¹›‹„ˆ›gt “–, wm{ zÓҋ„ˆ{ A·G MùG MöøXI t ≤ t − βt , β > 0, γ ∈ [0, 1), k = 0, 1, . , k̄ Ψ(x, s, µ) > τ Ψ(x0 , s0 , µ) < τ 0 k 0 0 0 0 1 θ nµ0 ε k k+1 k γ k T žžÑ›o„ » ¼ t k̄ ≤ , β(1 − γ) s‡o“;ó , Ž Î Ö Î z Î wm{ zÓÒ„ZutW8x“•{ “¿“ ρ ≥ 0 {s{ “€” ¡ t > ρ, Î žžÑ›o„ A·G MùGMöHI 1−γ

0 k A·G MùGMùsXI » 1−γ ¼ t0 − ρ1−γ k≤ . β(1 − γ) ë
í c<@% ý ïd,‰P(W@7,„]1#1 j.W„ β ≥ t j,@, 1−γ 0 ·A G MùG MöEXI `W@'($&(' S51ZZym1 ,y k = 0 ]15+](,dS5†.1+5#(:Z*n 5 %`@#19++`@'K "+ 1' S517'(, #„1W1'3+'1]1(,jMxo­+ ‚`ZZ,9Z*ò+(.*+f$&1'+ 15#(7,j P.W 0 < β < t óM ٙV >D) ,<']1](5Z1B!<]18+ ('TS5(57'1,„#d(W1'3+ '(5](1, j,9, k t1 ≤ t0 − βtγk ≤ t0 − t0 = 0, 1−γ 0 ¡ ¢1−γ γ 1−γ t1−γ = t1−γ 1 − βtγ−1 ≤ k+1 ≤ (tk − βtk ) k k ¡ ¢ 1−γ ≤ tk 1 − β(1 − γ)tγ−1 = t1−γ − β(1 − γ), k k ø A·G MùG MCFhI GM GM*b™!"'6=n+  .'É$&('*5#('K+

^,yU#(+ 8`+ ,#(:U$&('K+ ]1+('K+ó`'Y!"X+²# &A GM GMKCC‚I (W1'3+'1]11+(M~NO#ä#6+˜S5('(X+ µ ".W t ](r+ ]1,ä!<)(­'](]1Z1æ'($&1' S51ZZ > `'am1†,9,(aMNO#6+¡+ 1)1#1`J,‚@S5^,R#%''} +L5,+M β(1 − γ) o A&GMöõIf#(,#(Z;Z('++^,j O. W[im1(X+  '80 + µ>g3' (1 − θ)µ>?R']19`B¤ +L`' ,9(Z Z['(# |!"9+ ψ (θ, τ, n) M|o™9W@f ƒ ó+# D@'](]15+U,1'' Ψ(x, s, (1 − θ)µ) + ( @7,j *.W~c+L`'f,@1ZZQ'(W1a !<9+ τ Mo A·G MöôXIw#(,5#(Z*#6+L„!<1!f^n> +L++^,j j.W„+ 'Y'*+ :8'WB'](]1.5#

É!<1''KW('a(ÌW ƒ ó+ # D@']1](™U+LF`'K>  m1†,9,1](J'1X'ZZ (2ν ) Ψ(v) Mañ€(,9X+ 7 ,V!"5+u t = Ψ(v ) 5#+ + .'* v X+ ,‚+w#ó(W@!<P^ +L",†F`(,1#(³3 ƒ ó+# D@'](]1(,9Z3'j,nS5^,jMXo™#²('!<n> )++ ,y](5+('!<]1Z(3Ú A&GM GMKC‚õI C t = Ψ(v ) ≤ Ψ(v ) − (2ν ) Ψ(v) = t − βt , 4 .' A&GM GMCôXI C q−1 β = (2ν ) , γ = ](%+ ^)‚S5^,j €.W t = Ψ(v ) ≤ ψ4 (θ, τ, n) M€p(W1 ρ2q= τ 1,@,f#"!<]19+%Z1'Y+++ '(!"!<f!<5@),y''} +L +ó','!"#6`f,@S5^,j j. W (1 − t)α ≤ 1 − αt, ∀ t ∈ [0, 1] 1−γ k 0 C 4 1 q−1 2q q−1 2q k k k k+1 k+1 1 k 1 0 0 q−1 2q q−1 2q

k γ k q−1 2q 0  &  ' » 1−γ ¼ − q−1 1− q−1 1− q−1 1−γ 2q 2q 2q q+1 8q(2ν ) t0 − ρ ψ (θ, τ, n) − τ 1 0 = k≤ ψ0 (θ, τ, n) 2q . ≤ q−1   C q−1 β(1 − γ) C(q + 1) 2q  (2ν1 ) (1 − )  &A GM GMC(G9I oä,‚++ó,]6+²']1](5#! A&GMùG MC(G8]1óGM GMIv5#(5#+ U)Sfc+ ]6+ 1'Z(„#(11'3UZ1m(5']15+M NO#1#(1'{f+ ]6+ ('K+²Z('++^,jM p(W1~!<r+ θ ∈ (0, 1) ,5+ 1 ]1 τ = O(n) q = log n p ≥ 1 ,r+L (M NO,9,  ψ (θ, τ, n) !<19W@](f%) ô}m1:SJ'Y@S O(n) *W@1&) ˜a @}WU#¡'5K+ !c^ '](]1#(!< O(√n log n log ) Mú('(;''*+Z‰(# ˜'1S5ZZ/']1](5#!fZ1m('](„

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$&('53UZ1m(5']15+™,@S5^,Ú iU !U >= » q+1 8q max {3p + 1, 6q + 6} (ψ0 (θ, τ, n)) 2q q+1  ¼» A&GM GMKC I ¼ 1 n . log θ ε ü†Y,ÿ1ügb@ û?i Y,b@i ? o¡!"9+x#6+P!y|tZ1`1#(1+3Z(<1!<'}+1+5+ 7,j ¡$&gS5(#11+O(5()!"](XW(*+`@. 1+ 3,c#(](#1( '+ ',!"'1!<19+ K+LZ '](!<,9 fgÚ A&G5M nMCFI f (x) ≥ 0, x ≥ 0, x · f (x) = 0, .' x ∈ R f : R R Mpo¡#U'W1i Z'](!<,ÒA&!<9+d](')*^'PR`m(:d1W(@> '3+'1]1(,nI<òWX,5'Y+ƒ]('6+~#(!c+L'/+ 17'1+](¹1'3$&)^'*,jMwo™#V'+ ' }K+  'Y@S^ 'aU,†F`1+ ,(#(3y)

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(4x, 4s) M 5 wgexi s.ex{nlnzŒf›œug ' s.œ}j>c>h ʺb]lnwghjmÈ€ºu>wxegs«h7fde ›|vhj  •djmœ| yw z7egdexjm‹Ìz7ex{n|›y}{´nl £v£ c>y<€}œz7yu>wj¿’(•viŽÎ α bvycmys«‚xj>egcrjUy ª α = max α0 ∈ [0, 1] : ∀α0 ∈ [0, α0 ], Ψ(x(α0 ), s(α0 ), µ(α0 )) < τ 0 , È7{´ £ hŒwxj>lnu>ÈjËy|dh jm•df¢’(•viŽ°vln{}´ y.bvc>exfvh7ikjmlc™z7‚xb]‚gu £ ex|—i¢yb]l}j>j ¤geg|kjmc>œz7hŒuÝÅdj qpvnwÒjËy}c(jmlwgtÌb]ln|kj¿’>œ|›y}i (x0 , s0 ) b]l|j ½ Á±žnžnž%à £ eg|d|ve<z7exu>wŸy inc>|k´negwxexjm‚ £ eg| (1 − α)µ τ0 , eg{´nex|Ôjmex›œ}j  ‚xu ‚xu¯€nexwgexu>u>¦vi x(α) = x + α4x s(α) = s + α4s µ(α) = (1 − α)µ £ e«y α2 ’(egzŒnz7‚xu(jg aTbvc>exfvh7ikjmlc¬z7‚xb]‚gu £ ex| i¢y}b]lj>j®b]ln|kj¬j˜&€nlz7u>œ{kœc>y”y”i&€nexj>iegwg‹ ϕ = 1−α h7{kyw}Î

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jmexzĒ(egu(¦vz
 α0 ∈ [0, α] €}y{´ h7u 5 iilnc x(α0 ) · s(α0 ) = 0 ½
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Á±žn „ kÃ

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