Stochastic Modelling and Optimization using ...

3 downloads 0 Views 434KB Size Report
eds. MPS-SIAM Series in Optimization, 2002. Stociastic Modelling and Optimization using ЖИКЛНПСИУЛСЧ. M.A.H. Dempster, J.E. Scott & G.W.P. Thompson.
   "!#$ %'& ()%+*-,#.%/10 !'(2#3 5476%"86!99:&;8=@?BAC?AEDFFD G5HCAC?JI5?@KLG )M>DN< GPOQ ?7RTSU=WV+=NX.Y5RZ=NG GPQ[\5]N ^ D_ `D ¨D_F`©¡G Q"G D" 1 Ÿ ªO G«X.Qž"ª G«¦¬RTDD G G  ­®°¯ GP"Q"°.±–¦ ¥ D )MOG H ¥ D_ )M"O Gk=;±;QžG Q[YtQ"QN P¨D_"G Q[²³ "G O ´ ‰DOa;ށÊ/Ò;ÐTË°ÒØ+Ö;ËÍÕ-ʕÉ+Ø+ËÍÌ1ÅÇ)úyǤÕ+ÖÏ2úCÖWÏ2úüØ+ÖWËÍÕ-Ì1Ê/ÒûÈÇ)ÐÏÔÒÇÜÏ/ɫʎÕ+ËÍÎ5Ú;ÅÍÇkÇàÊ/Î5Ú;ÅÇ ÚWÉ+Ï/È;ÅÍÇÑÎøLýUÖÇ1É+ÇʕÉ+ÇhÖÏ2úyÇ1×/Ç1ÉÑÞWÕ.Ç1×/Ç1ɞÊ/ÅÕ+ËÍØ+ÓWʕØ+ËÏÔÒWÕmúUÖÇÑҚØ+ÖÇÒÏ ÐWÊ/Å;ÜÏ/ɞÎtÓWÅÍʕØ+ËÏÔҜÈÇÑÌÏÔÎ5ÇÑÕ Ê«ÅÍËÍÎkËØ+ʕØ+ËÏÔÒ>ø‚þUÇÑÊ/ÅÍËÍÕ.Ø+ËÍÌÕ.Ø.Ï ÌžÖ;Ê/Õ.Ø+ËÍÌ¡Ú;É+Ï/Æ/ɞÊ/ΫΫ˰ÒÆ-ÚWÉ+Ï/È;ÅÇÑÎkՃΫÊÑÙ¤ÖWÊ2×/Ç[×/Ç1É+ÙÅÍʕɞÆ/Ç[ÐÇ1Ø.Ç1É.Ý Î«ËÍÒWË°Õ.Ø+ËÍÌ®ÇÑÿ_ÓWËÍ× Ê/ÅÇÑÒ_Ø+Õ1ÞÊ/ÒWДË܁úyÇkÌ1Ê/ҐÊ2×/ÏÔËÍДËØúyÇkÐϤÒÏ/Øhú7ËÍÕh֖Ø.ÏËÍÒ;Õ.Ø+Ê/Ò'Ø+˰ʕØ.Ç5Ø+ÖÇÑÕ.Ç«ËÍÒ ÜÓ;ÅÍÅmʕثÊ/Ò_ÙTÚ ÏÔË°Ò'Ø-ÐWÓɞ˰ÒÆØ+ÖÇÎ5Ï ÐÇÑÅÍÅ°ËÍÒÆÚ;É+ÏÌÇÑ՞Õ1øÄeÅÍÕ.ϖô¡õ öÕ-Ø.ÇÑÒ;Ð Ø.ϐÖWÊÑ×/ǤʖÖWËÍÆÔÖWÅÙ É+Ç1ÚÇ1Ø+ËØ+Ë×/Ç-Õ.Ø.ɞÓWÌØ+ÓWÉ+Ç/Þ Ê/Ò;Ð ËÍØeËÍÕ7úyÏ/É+Ø+Ö Æ/ÏÔËÍÒƚØ.Ï)Õ+ÏÔÎ5Ç-Ç  Ï/É+ØØ.Ï)Çà Ú;ÅÏÔËØ7Ø+ÖWËÍÕeÜÏ/ÉÇ kÌ1ËÍÇÑÒ'Ø ÚWÉ+Ï/È;ÅÍÇÑ΅Æ/ÇÑÒÇ1ɞʕØ+ËÏÔÒ> ø LË°ÒWÊ/ÅÍÅ٘Ø+Ö Ç ™Ê/ÅÆ/Ï/ɞËØ+Ö; Î BÕÜÏ/ɞ Î ƒÏ/Ü5ʒÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍ̖ÚWÉ+Ï/Æ/ÉÊ/ΫΫËÍÒÆ ÚWÉ+Ï/È;ÅÍÇÑÎ ËÍÕtÐWË  Ç1É+ÇÑÒ_Ø[Ø.ώØ+ÖWʕØtÏ/ÜUËØ+Õ-ÐÇ1Ø.Ç1ɞΫ˰ÒWËÍÕ.Ø+ËÍÌ-ÇÑÿÓWËוÊ/ÅÇÑÒ_Ø1ނÊ/ÒWÐØ+ÖǚΫÏÐÇÑÅ°ÅÍËÍÒƤÕ.Ù Õ Ý Ø.ÇÑÎÕ+ÖÏÔÓWÅ°ÐØ+ Ê /Ç5Ø+ÖWË°ÕeË°Ò'Ø.ÏÊ/Ì1ÌÏÔÓWÒ_Ø1ø®ÄeÐWÐÉ+ÇÑ՞Õ+ËÍÒƜØ+ÖÇÑÕ.Ç-ÚÏÔËÍÒ'Ø+ÕÑÞ úyÇ5ÐÇÑÕ+ÌɞËÍÈ

Ç >Þ Ø+ÖÇkÕ.Ø.Ï ÌžÖ;Ê/Õ.Ø+ËÍÌkΫÏÐÇÑÅ°ÅÍËÍÒÆ)ÌÏÔÎ5ÚÏÔÒÇÑÒ_Ø¡Ï/ܬØ+Ö Ç !#"%$'&()*+&-, Õ+ÙÕ.Ø.ÇÑΤÞ@úUÖ;ËÍ̞֎úyÏ/.É

Õ[ËÍÒ ÌÏÔ0Ò /+ÓWÒWÌØ+ËÏÔҎúUËÍØ+֎ÇÑËØ+ÖÇ1É[2 Ä 14 ö 3’Ï/6 É 5ö8 þ 7yõWõ9Ý 1ö Ê/ÒWЖÊ/ÅÍÅÏPúUÕhØ+ÖÇ5Ç kÌ1ËÍÇÑÒ'Ø¡Æ/ÇÑÒWÇ1ɞʕØ+ËÏÔÒ Ï/ܳÅÍʕÉ+Æ/ÇÝ$Õ+Ì1Ê/ÅÇtÕ.Ø.Ï ÌžÖWÊ/Õ.Ø+Ë°ÌhÚWÉ+Ï/Æ/ɞÊ/ÎkΫËÍÒÆ5ÚWɞÏ/È;ÅÇÑΫÕ1; ø :–ÇtÆÔË×/Ç[Õ+ÏÔÎ5Ç[ÐÇ1Ø+Ê/ËÍŰՃÏ/ܳÕ+ÓW̞ÖÚWÉ+Ï/ÈÝ ÅÇÑΫÕ@Ê/ÒWЮÐWÇÑÕ+ÌɞËÈ< Ç =>? >ÞPÊ/ÒtËÍÎ5Ú;ÅÍÇÑÎ5ÇÑÒ'Ø+ʕØ+ËÍÏÔÒ¡Ï/ÜWÒÇÑÕ.Ø.ÇÑA Ð @yÇÑÒWÐÇ1ÉÕÐÇÑÌÏÔÎ5ÚÏÔÕ+ËÍØ+ËÏÔÒ¡úUÖ;ËÍÌžÖ úmÏ/BÉ ÕmÇÑËØ+ÖÇ1ɬËÍÒWÐÇ1ÚÇÑÒWÐWÇÑÒ'Ø+ÅÙ®Ï/Ü>Ï/ÉyË°ÒkÌÏÔ0Ò /+ÓWÒWÌØ+ËÏÔÒkúUËØ+C Ö 5Ê/Ò;МÖWÊ/ÕmÈÇ1ÇÑÒkÓWÕ+ÇÑЫØ.Ï Õ.ÏÔÅ×/Ç-Êš× Ê•ÉžËÍÇ1Ø$Ù¤Ï/܂ɞÇÑÊ/Å@úyÏ/ɞÅÍЉÚWÉ+Ï/ÈNÅÇÑΫÕ18 ø :–ÇtØ+ÖWÇÑÒ Ð;ËÍÕ+Ì1ÓWÕ+Õ7ÖÏ2ú}× ËÍÕ+ÓWÊ/Å°EË DÑʕØ+ËÏÔÒ¤Ø.ÏÏÔÅÍÕÌ1Ê/Ò ÈÇ-Ó;Õ.ÇÑЎØ.ÏÊ/ËÍЎØ+ÖÇ«ô¡õ ö Î5ÏÐWÇÑÅÍÅÍËÍÒƚÚWÉ+Ï ÌÇÑÕ+Õ1ÞÊ/Ò;АÕ.Ç1ØhÏÔÓØhØ+ÖÇ5Ú;É+Ï/Æ/É+ÇÑÕ+ÕhúyÇ«ÖWÊ2×/ÇkΫÊ/ÐÇ Ø.Ï2ú¬Ê•ÉžÐWÕ5Ê/Ò ËÍÒ_Ø.Ç1Æ/ɞʕØ.ÇÑÐûÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍ̚ÚWÉ+Ï/Æ/ɞÊ/ΫÎkËÍÒÆ ÇÑÒ'× ËÉ+ÏÔÒWΫÇÑÒ'Ø-ËÍÒØ+ÖǚÐWÇ1×/ÇÑÅÏ/Ú;Î5ÇÑÒ_Ø-Ï/Ü '"$!&()*+&F,Uø Á ML¿N GIH>JKAL 0

ôeÙÒ;Ê/ΫËÍÌhÕ.Ø.Ï ÌžÖWÊ/Õ+Ø+ËÍÌ¡ÚWɞÏ/Æ/ɞÊ/ΫΫËÍÒWÆÞ Î«ÏÐÇÑÅ°ÅÍËÍÒÆ-ÅÍÊ/ÒÆÔÓWʕÆ/ÇÑÕÑÞ;ÐÇÑÌÏÔÎ5ÚÏÔÕ+ËØ+ËÍÏÔÒ>ø

O

PRQTSUWV

X Y

Z4[]\^Y`_UaZ4[bX Ydcfeb\hgCebUWV

V

X Sg

IhG« •D_  •DF¬#jie%>k (2# !/8ml no_&!9 ®(2#+%/10 !'(2#$ 6%"86!99plAq3«= Ssr7·¡"F G•D7  Qž"DQ¡ D MG«·¡G   t u

QP?™ ?†…

v?w

qyx

{}|Z ‰ qyx q | r„ | r |€ vƒ‚ ‚ t~B

r w

z

x w‡w | w … q ‡r x ‚-w x b„ w w2‹ z

x

… q| r ‚

‹

‚‡‚

q | r‡x

b

q| r ‚

ˆ ˆ

‰0w | q r ‰‚

qž¸r

!Š°(0Š !Š°0( Š

·¡GP"GŒx DQ" h]N PG QQ¬·¡ Q" G n!# !  !#:0¢ ( ‘ ·¡FF³M@GD_]]"  ›N ‰D_"G OTM±TC D £> GœWª )MG t_¦ß]>D_ Q®¦§" qDOQ GPGœ GkQ; 1>DQž Q" )ªFD_ `G ">DŽ"G->D_"ªD_Fy#$:p 9 l l26Ñ 0% n q$G?™?ßDš·7G GP»  k ‰W"%  r-¦e"GŽD_]]F •D_" L?‰IhGP"G‰ D•±ûMG QGP¯ GPDFUQ"ª 1E ‰G¤]G "NO QkMGP.·7G GP  nml12(2 %'& (2# !/m8 l1(¦‚"G« ‰;OG FLDO–Wª¤ Q Ÿ "G ] "G QG Q[Q" )ªFD_GPO–Q1D'G Q¡¦‚"G«ªOGP"F±;O>D_D¤] "; G QQ QD_ ‰]FG O D_¤ G”QG P O Qž1DG”OD_G_¥ ? q$>  ‰"G  Q¤OQž P  MGP.·7G G   G Q •DFG Q O ±;>D  5Qž; 1>DQž «]  "D ‰  (2#3  l-¨0N’ ©'!* l2&>#@]"MFGP ` ¦ q+ƒ¸ rU·¡ 1` •DF ªFD_œ"G5OQ" GP"G5G ›N]@GP PD_  G ›N]F PF°± ‚

q’x

q’‘`r‡x ‚

‚

q’‘`r(„F‘`r

q’‘`r

ˆ

‰0w ˆ

‰‚

q˜‘`r

a

‘

‘





Ÿ ŸA„

q  a r

·¡GP"G q’‘`r«OG  GPQ¤"G–] M>DMF°.±ŒDQ"QN P¨D_"G O ·¡" GŽG2¯ GPWh‘pªO G ¤Q" GŽ] M>DM F°.± G•DQª"± GT²¤? Ih G¦§" )ªFD_  ¥q3a?ršQ¤»W/·¡}D_Q¤"G (2# !&³n!6Fn´q$ T1%'9t !;2#*rœ¦§  OGP"G +V Q ‰G œWª¯_DFG •?’›[G "GŽ·7G–>D•¯ G  G ¯ GP P)¯D_"¨D_MFGhx ·¡ 1 QµQ"D"GPO+¶’M@G2.·eGPG  DFF "G DF^•D_"  Q ?¡YtF°G >D_"¯ GPF±·7G) •D ¦§" )ªFD_G5"G)OG2G  ‰w Qž œGœ;ª ¯_DFGP-ªQ–D¯D_"¨D_MFG x q’‘`r¡¦§ -G•D_ 1f  ‘ ŸpDOû"G ŒDOO &L%'&!&>#$ 2¼ !#3’ ©/:j# oš P QžD_A Q x q˜‘`r x q˜‘©¡N 1»'D_¦§GPFF¨D_DOA˜GPQP©[ª Q"^ P^P¤  ± Á Q"»W3kž!&;8 N°!6ߦ§ "  "G Q]@O ""G«]" M;V FGP¡ ęQeON±ND ‰ tQž"ª Pª "G?ßY®QhD–DF°GP">D_"¯G®šQ" F°¯Nš"GtOG2GP" Q ®G œWª¯_DFG •N6(l7¦§ª P" r ÉWqyx

 € ¦ ° q’‘`r qyx ’q ‘`r„B‘`r vƒ‚ ‚ t ~ ˬ ¬®0¯ ¬m®0¯4± Q ?B•?̅ q’‘`r‡x ’q ‘`r q˜‘`r4͗… q˜‘`r.x ‚‡‚ ‚ ˆ¥‰‚ -‚ w w r

w

ˆ

qjÏr ‘ 

ŸAÎ

X  Q G•DQž± G ª   Q" Ô·q>D'ÐÉ{QD P ¯ G(› ¦§ª P"  ¦Ñx ?ÒÅeG O G "Q”OGP  ]@Q""  ]N PG G O Q[M± MªFO )ª]”D)]GP G2·¡Q"G5FG DeF/·eGP¡D]]›N D_" ‰w ¦6É M±Ž"±;¤D)QG0œWªGP G € ¦hO Ô Ó@G "GPœ6%1>%/( !¼(Õx ¦Ö ¦§ œ·¡ 1 Oª>D_F7Q Fª "  Q× ¦Ö € q’‘`r«DG  M D G O`?IhGPQ"GDG ªQG O ¦§ " %1#$:9!:j# o” ;#§(2D•¯G‰ G‰ DQ"GÆÉ q’x w

€ šÖ r

q˜‘`r.x

‚-w

ˆ

zÑÞ

w

Ø

z

Ø q§

ˆ ˆ

‰0w °

q’‘`r‡× šÖ q˜‘`r

¬® ¯ ±

?™G? ¦§ kQ" ‰G¤] ]@QDFQx ¶

qÝÜr €

‰‚

w

¦Ö

q˜‘`rÚÙ €

ˆ

¸?„ ΠΠ΄ÛÎ

 G "GP  ª "Q"G¦§ª 2˜Q

 N¦§G•DQMFGr2·7G •DT G GPD_"Gt* l"!'(2jº ::#joN#:(®M±TªQ OG P"  Q®¦ßG PG Q"Q ûM 1DG O¦§"  Gßq§ªWM@ ª OG OrOªDF«Q" Fª  Q ? >  ‰D± ]D_ P •DFt]" M FG QTÅeG  OG QTOGP  ]@Q""  >DQ®M@GPG ûQ/·¡ ŽMGœ¦$DQžGP®D Q"F¯;GœOGP"G  ‰Qž «G œ;ª ¯_DFGP-] MFGP ‡DQtªQª>DFF±  F°±D‰¦§GP·pGPQ-¦ß Pª Q«DGš0G œWªG O "D PªD_"G F°±TD_]]"  ›N ‰D_"T G ɜD"DFFG F3D_FF±[ GeQ" D_ Ge0G œWªG G Qm%¦ ÅeGPOGP"Q OGP  ]@Q"" – PNO G Q[D"G5  " ‰DFF°± )ª 1”Q" ‰DFFG e">D G  G D_F°V.] ª"] Q"G« GP"NO Q¡D]]FG OŽ"  GkO GPGP" Q" -G œ;ª ¯_DFGP¡]" M FG Ú q ?a rÑ? IhGOG DQ¦¡ GŽ.·eŒQD GŽ¦§  )ªF¨D'EG ›;"G O "G DOF°±’ûGû 9 ;#$(2# !/m8 lš •DQGDFF±D_Qß" Gh]" "GPQ"Q"GPQ ?ßIk¦§  D_FF±k(G › ] "G QQ7"Qƒ¦§ UÕ D â¡V QD Gh] MFGP < ·7G[ªQ"G®D ã #$6"!#$ %'I & äæå ´ ä§ç]åè ¸ „ Î Î0Î0„-⫹WF°D'Q¡G ›N]F °F±< ·7Gš GPG "DFF°±–»–GP" Q®§ ¦ l©l2&>#e#$6 ll1(2< ·¡ 1pD/¯G D  NOG˜¦§ ûG•D 1CQGP’Ê  éWç.„ìè ¸ „ ΠΠ΄-â DOpDpD" EM@G2.·eGPG £G D 1pNO G  ˆ "GP]"GPQ"GPW"“  … · éWç˜ê D_ O … éWç.? =ND•¯ G "GŽGP¯G ¤G GŽQ/·¡  y ªGT¸ ? Ih GŽªQ"ªDF7""G G‰G  ‰ F ±ìq$D GPQ"  DMF.± " G•D 1 FG•D_¦Iq§ ŽG0œWª°¯_DFGPW"F±  G D 1 Q" PG >D_" r2D )M ª ª Q"F± OG £G5G«] M>DMF°.± ¦yDFF`;OG Q¡–"G«GP¯ GP®""G G_? X UQ¬_ßQ 2F±šG PG QQDž±>D_ƒ"G¡]>Dž"  QyMG  ‰2 G £> G UD_ßG•D_ 1‰]G ;O`Dw Q" OGP Q" ]"; GPQ"QÕx4q | rÕå q’x „-x q | r„ Î Î0΄-x ò q |Aò r-r[¦§ ®·¡ 1û·7GªQ"G G D_ Æ  x ç q | ç r å qyx „ Î Î0Î0„-xçBq | çjr-r-”"GP]ˆ "GPQ"G w )"‚ G ‚ ] "; G QQšª]ŒTDOE FªOQ1D_ GCèÑ? A˜Gk D”/· G ›N]ˆ G QQ® w GkFG D"F°±  Qž"DGPO˜ 9 ;#$(2# !/m8 lano'&!9 -(2#.%/10 !'(2#3 6%"86!9p 9 lDQ  

t u

w

x



zÐ{ | w

‰| 

~

€

t~F

~ vƒ‚

qyx ‚

q| ‚

r„ | ‚

r¢z—{ | | ~ z

Q ?B•?ʅ

{ |  |

| 

t 

  u 

x

w‡w | w q r.x ‚ ‚-w | í w … í q .r x w w …

… ò w

q|

ò

r.x

q|

z

… z

… í

??

w

… ò x

ò

‚

q|

r‡x

b

‚

w ‹ ç ¢ xç-q | r ‹

q| ‚

t



€

|  

íƒq’x%í>q | í(r(„ | ír;zÎ0Î Î v



€ v

ò q’x ò q |Õò r„ | ò rÎ Î0Î  ˆ

r

‚

‚‡‚ | í ‚ q r.x q | r ‚ ‚ ‚

? z

q|

r.x

‚



ˆ

í í …§í‡í>q | r‡x%í>q | r

z

?? ?

rRz

… òò q |

Î0Î Îz

b è

‰0w | q r ‚ ‰‚ | í í>q r

ò

???

?? ˆ

r.x ò q |

ò r

‰

ˆ

? ‰

í>q |

ò r

a#„ Î0Π΄Fâ6„ ˆ

qݿr

·¡GP"GŽD_DŒDFFƒ   Q"D"QœFOŒDF  Qž«Q"ªG F±?–›®GP"Gh{ | | u qB™Er-OGP"G Qš P O° >DF¬G(›WV ]G PD_ T·¡° G Q"]G 25Ž"G) ª ""GP«Q1D'G)¦ƒ"GO>D_D ]N PG Q"Q | P O°G O  "Q-]>DQž Q± ?’Y®DLDQM Qž1DOD"O D O Q"]Fš¯_D¨DM FGOGP"G "  Q" ‰G0œWª¯_D'V FGP[¦§" )ªFD_ A Q q§·¡GP"G«¯D_"¨D_MFGPQG•D_ 1TQD G¤?¸ „ Î0Î Î0„-â Í ¸k D•±ŽMG«Q"]F r7·¡ 1·eGk  GP"G ¦§ ŽM "GP¯;°.±  ? Åe Q GP>D" OG P ]@ Q"  D O ÅeG  OG Q–OG P ‰] Q }M>D_Q"G O Q FªNV  E GP";OQš D MG G ›;"G OGPO TG   GPF±˜>D OFG¤G )ªF"Qž1D G¤ DQ"G_?TIhGŽ šªF"QD G ÅeGPOG QhOG P ]@ Q"  DF "" Q7»W/·¡D_Q-! & l1(2.# l-nÕÅeGPOGP"QhOGP  ]@Q""   q9Åe" G_D_®Q® ;OG F³G D 1T ‰G D’ MQG ž¯D'ŒQ5 D_OG«Q-¦§ FFÔ·7G Oû ‰ G O ¨D_"G F±”M±’D–OGP Q L GœD'¡G•D 1;OG5Ž"G5GP¯G ["GPG-G 1% l! 2Ý l2&># 6%/ l1(1# ( "0çFq | ç r¢å qy…§ç q | ç r„ Î Î0Î0„ž…¤çEçFq | ç r„ çBq.™*„ | ç r(„ çFq | ç rFreD"GP]"GPQ"G D_ QªDMFG-¦§ Q"'V FªNšG °GPy¦>ˆ "GOG2w GP" Q ß0G œWª¯_DFv G ƒ] MF‰ G CM±)Dt P ¯ GPW" DFOG2G  ‰Qž UQ F¯G P<  ® Q ‰Gœ"GP]"GPQ"G D_ ûQ"ª°1DM FG«¦§ tQ Fª " M± DŽ«=NS¬V Q"]G PÔ £> )Q F¯G W  q§¦§ ®G › D ]FG_FG®¦§ " ‰D_  qjÅe G-GP[D_F ?°