Interior Point Methods With Decomposition For Multicommodity Flow Problems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Interior Point Methods With Decomposition For Multicommodity Flow Problems J. Fr´ed´eric Bonnans, Mounir Haddou, Abdel Lisser and Raja R´eba¨ı. N ˚ 3852 Janvier 2000. ISSN 0249-6399. ISRN INRIA/RR--3866--FR+ENG. ` THEME 4. apport de recherche.
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(178) lw}y r¶ wV| y h kuk^ *yz{ ÄtKr£|a ( ,+0 ) ( ,N6 ) }w | ( , ) S * * * * * * * * ; * S) * * ( 3 ) 1 * #( P * ) 1 ( * P ( P ) 1 * ) * T P ( 3 ) 1 ) ( ( P -) 1 ( P ( P ) 1 ) T P - ) ª ; S) G G w} - y r¶ - wV| y h knk^ yz{ Ktazrs|K ( ,N- ) ; {l.~KtKyz- ky h kpl ; - S) - - ( 3 ) 1 ª ; S ) C J ª G ; q[{vsky h k k tI k z[zyk+l ( ) ª q[{vskny h knvsrs|Kk^w * z[zy* k+l ) K¨ h k k. . ?. . -; . S) )*_+.- 3. ;. S). '. S) - )*_+.- - 3. '. wV| >. . . * *. ª. S) ( G. - . ) CDJ ª. ; ¤¸{ T 9 ) ,5&.--.- & : * * * ; {l~Ktayzk ( > & ? & ) z{v£tayzrs{|.{}¤y h knz[zyk+l ( , ) ª ¹. . V !. % /0 +"$ u%. %( . *I3. ¡ 7. $47¿
(179) . ' 90 n%. g$h ktKvsyzrx {ll.{ rây]|Kk+y]¨°{ ± ~ { ¢ vsk+l.w}~K~4k^w ¨ h k+| r ; k k+|[yu {l.l{ rsyzrsk+ h w kw¯ {l&µ l {||Kk+y]¨°{ z± ª g$h k+k#~ { ¢ vsk+l.Uw kk+|Kk w}vsvsk r * tKvsyy{{vsk|K{y ¢ k^ +w}tazkn{}¤Äy h k r v¦w k . +w}vsk ¢ tay
(180) w}vsz{ ¢ k^ +wVtKzk.r£ynrs|K{ynk^w}»yz{ h w}| vskuy h k {|[§Irx y ¢ k y]¨®k+k |¹zk k w}v# {ll{ r¶yzrsk+ {|»y h ku w}lk|Kk+y]¨°{ ± . -}m+2zª g$h k kw ku|[tKl.k {tKnl{ k+vs#rs|¹wV~K~Kvsrsk {~Kyzrsl.r£´w}yzrs{|y h w}y rs|[{v£k#lutKvsyr¦ *{lulu{ rây]pz[zyk+lF. 8# 6+2z.EDFDqHGK-C6=2á. È qHG?-w72z. È qHGK- ¢ 2zª g$h k È rs|Kkw lutKv¶µ yr¦ *{ll.{ rsy]|Kk+y]¨°{ ± µ¸§I{}¨ ~ { ¢ vsk+lrsy h k.l{zy#{V¤¸yzk |¹zyt rsk ~ { ¢ vsk+l!rs|)lutKvsyzrx {l.lu{ rsy] {~Kyzrslrs´^wVyzrs{|%.E@k+|3672z . 8n N -72w}| . $È -C672zª g$h rs~ { ¢ vsk+l {|Krszyz{V¤ y h k k+yzk lrs|IwVyzrs{|{V¤Iy h k®l.{zy¬k^ {|K{l.r¦ w}v¨$wn{}¤Itazrs|Kny h k$ww}rsvxw ¢ v£k y w}|azlrszrs{| +w}~Iw rsyzrsk+$rs|{ k yz{ {tKyk
(181) wpy wA* lw}y r¶uy hK {tK h y h k|Kk y]¨®{ z± ª g$h k ¢ k+zyn {l.~Kvsk Krsy] ¢ {tK| ± |K{}¨$|u¤R{ lutKvsyzrx {l.l
(182) { rsy]n~ { ¢ vsk+l.®rs°~ {}[r k ¢ w}|.r£|[yk rs{ . óxç¬ÕYóxô.
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(184) "!#$ %. ,^. ~4{rs|[ywVv£{ r£y h l .E@ G?-072á4y h {tK h IwV#k+y^4|K{.k<* rsk+|[y#rsl~av£k lk+|[ywVyzrs{}| h w ¢ k k+| { ¢ ywVrs|Kk ª8 w rxw}|[y${}¤@w lw z± w °~ {^*k^ yzrsk²wVv£{ r£y h l¿rswV~K~Kvsrsk rs|P.E@?@ -72z4tKzrs|Kw~ k^ *{| rsyr£{|Kk {|^ tK[w}yk w rsk+|[y ( D ) {vsk ª {|[y w rsv£¯yz{¡y h k~ k^ {| rsyzrs{|ak ~ k+k+|[yzk rs|.E@?@ -72 y h w}y r |K{yyw ± kw wV|[yw}kn{}¤®y h klutKvsyzrx {ll{ rsy]zy tI yt kYy h k
(185) {|Kk~ {~4{k rs|:. DF-=2 k K~Kvs{r£yny h rsy tI yzt kª k^ *k+|[yzvs w}y {~ k+k+|[yzk rs|:. w} w 2®w.z~4k^ rxwVv£rs´^wVyzrs{|){}¤$w}|)r£|[yk rs{ ~4{rs|[yw}vs{ rsy h l©y{nlutKvsyzrx {l.l
(186) { rsy]°§I{}¨ ~ { ¢ vsk+lª>$k$tazk+ ¢ {y h w~ k^ *{| rsyr£{|Kk {|^ tK[w}yk w r£k |y°z{vsk wV| wnz~4w k °h {vsk+ ± p¤Ew yz{ rs´^w}yrs{|Yyz{{vsk#wpvsrs|Kk^w z[zyk+l¿{}¤Yk¾t4w}yzrs{|K®w}y k^w h rsyzk w}yzrs{|{}¤y h kw}vs{ rsy h l)ª. . 4 ( . È k+y ( & )°¢ k
(187) w r k^ yk wV~ h a¨ h k k rs#wzk y{V¤* E|K{ k+w}| rs#w&zk+* y{}¤ w ^Äw}| vsk+y ¢ k
(188) wk+y${}¤:! {ll{ rsyrâk+y{ ¢ k {tKyk y ha {ta h y h kn|Kk y]¨®{ z± k+~ k+zk+|[yk ¢ uª w h {l.lT { rsy]@9ĤR{ 9 :» h wV#wta|Krx¾takz{t k
(189) wV| wtK|Krx¾tKknzrs| ± E ªº»k k+|K{yzk ¢ ' $D P B y h k$lw}y r¶{ ¢ y w}rs|Kk ¢ k+vsk+yzrs|K
(190) wV|²w z¢ rsy w {}¨»¤ {l8y h k$|K{ k*µáw $rs|I *r k+|I k ( ' lw}y r¶:{V¤&ª yrs¨°k+vsv ± |K{}¨$|
(191) y h w}yy h rslw}y r¶r£¬¤¸tav£v {}¨ wV| ± ª ) w h {vstKl|{V¤ P6rs k+vxwVyzk y{w}|¡w wV| h w}${|Kvsuy]¨®{|K{|a´+k { ( ,+& F, ) {[k<* rsk+|[y#rs|y h {zk {}¨$#w}z* {K rxw}yzk ¨$rsy h ( k+~Ik^ *yzrsk+vs ) y h k{ rsrs|¡w}| y h - k k zyzrs|IwVyzrs{||K{ k {V ¤ ª$º»kn h w}vsv *{|Kr k y h wVy#y h k
(192) w #{V¤ hy k
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(194) w. +w}~Iw rsy] 5 ( P' ) ¤¸{ w}vsv {ll.{ r£yrsk+^ª º»k k+|K{yzk ¢ 5 P' D T a¤¸{ 9 :»Äwk^ y{ {}¤ztK~a~Kvsr£k k+lwV| °¤¸{ {l.* l{ rsy] 9* wVy$y h k|K{ k+{}¤y h kp|Kk+y]¨°{ z± k I|Kk ¢ S * # * r¶2¤ ' )
(195) * & ( ) 5 $ ) r¶2¤ ' ) E & h * k+vszk+¨ k k & ¨ h k k rs$y h kwVvstKk#{V¤y h k#§4{}¨{V¤y h L k 9 A0Cu {l.l{ rsy]ª È k y#tK#wVzztalk#y h w}y#wV|tK|Krsyw {zy / ( 0 / ) h k^ yz{ ) rsr£k+|¹w}| y w}yy h rs
(196) {y {[k+|K{y k+~Ik | {|¹ {ll{ rsy]ª&º»ku h wVv£v rsyzrs|Ktar£ h y]¨®{lJ { k+vs°{}¤ ( È ¯³ ) SKy h k|K{ k µzw #l.{ k+vYw}| y h k
(197) w µE~4w}y h {|Kkª. !
(198) g$h kp|K{ k µzw ¤¸{ lutKvxwVyzrs{|{}¤ ( È ¯³ ) +w}| ¢ kk K~ * rs| / 1 )*_+ H F ! $#$#$#$ F - % ' T* ( ' ) # * )*,+ ) ' T * ( ' ' ) P ) ( ' 'O' ) . ¨ h kk *. 0. k zzk wV°¤¸{vsvs{}¨$NS. 5*& 5 & 9Q) ,5&.--.- & ::& ,&9()P&.-.-- & ::&. *. ; rs#w ' µ]k^ *yz{ sr °y h k:§I{}¨ +w z sr k ¢ {l.l{ rsy] 9.{|¡w [ ( ) . Õ¬Õ Ü9T+ì
(199) ^ê . ( ).
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(202) =O $ 1 $! . , m. -. -. ; rs$y h k +k r tIw}vY w}~Iw rsy]k yz{ ( B' 16 ) ª ' | y h k¡|K{ k*µáw .¤¸{ lutKvxwVyzrs{}|Yy h k k^ rsr£{|6w rxw ¢ vsk+w ky h k§I{}¨ {}¤
(203) {ll.{ r£yrâk+p{|6k^w h w }ª g$h rsn¤¸{ lutKvxw}yr£{|¡w}z{K rxw}yzk uyz{¡k^w h {llF { r¶y] 9¯§I{}¨¿ {|Kk wVyzrs{| ( aª 'O' ) w}| |a{F |[µ |Kk w}yrsrsy] ( Kª'O' ' ) {|Kzy wVr£|[y^ª g$h k
(204) k^¾tIwVyzrs{|a ( Kª' )$ k*¤¸k y{y h k² +wV~Iw *r£y]¡ {|Kzy wVr£|[y^ª g$h k |K{ k*µáw ¤¸{ lutKvxw}yrs{|#{V¤ ( È ¡³ ) ( )Yh w} EQO )6: ( E , ) #Q {|azy w}rs|[yz®wV| O ) ( : # , ) w rxw ¢ vsk+ª«K{ $ k^wVvY|Kk y]¨®{ z± #tI h w}$y h k w r£ r£y r¦ *y$y wV|Kzl.rszzrs{||Kk+y]¨°{ ± Ky h k|[tKl ¢ k {V¤ |K{ k ¼ +wV| ¢ k$vxw k y h w}|%,wV| y h k#|[tKl ¢ k {}¤w °vxw k y h wV|aª ' ¤Y{|Kk#w}ztKl.k$y h wVyy h r£ |Kk y]¨®{ z±h wV,^# {ll.{ r£yrsk+^ y h k°zrs´+k${V¤Iy h k°~ { ¢ vsk+l +w}| ¢ k°vxw k y h w}|B,^$ {|Ky w}rs|[yz wV| nw r¦w ¢ vsk+^ªqB{I[y h kvsrs|Kk^w ~ { ¢ vsk+l ( ) rs$k .v¦w kk k+|.¤¸{ wlw}vsv4|Kk+y]¨°{ ± ª º»k~ {~I{zky{uz~4k^ rxw}vsrs´+ky h knlk+y h { k+k v£{~Ik rs|y h k~ k rs{tK$zk^ *yzrs{|¡y{uy h rs$~ { ¢ vsk+l)ª È k+yntK#|K{yzky h w}yy h kpz[zyk+l ( ) *{ z k+z~4{| yz{.y h kpz[zyk+l ( ) ª ' | k+k Ärs|y h k
(205) +w}k
(206) {V¤y h k |K{ k*µáw ¤¸{ lutKvxwVyzrs{|{}¤y h kvsrs|Kk^w lutav£yrx {l.l
(207) { rsy]n|Kk+y]¨°{ ± µR§I{}¨P~ { ¢ vsk+l ¨°k h wkCS $ $ ; ¤¸{ wVvsv ' 7}&.--.- & : 9HS S) w}| 3 S) \ /( ) ª $ $ $ ; ¸¤ { - Vw vsv ' H7 ,5&.- --.- & : 9HS P S )#P&E S) E ,nw}| / ) / ª ;E S) w}| / S) aª. º»k k ¤¸k y{¡y h £r n~Ik^ *r¦wVvsr£´w}yzrs{| V{ ¤¼~ k rx y{ µz { z k yz{ w}vs{ rsy h l y{¡|K{ k µzw p¤¸{ lutKvxw}yr£{|{V¤ ( È ¡³ )$¢ y h k ³8n«¹w}vs{ rsy h l ª.
(208) g$h k)vsrs|Kk^w lutav£yrx {l.l
(209) { rsy]¡|Kk y]¨®{ z± ~ { ¢ vsk+l1 +wV| ¢ k ¤¸{ lutKvxw}yzk rs| yzk * l2{}¤~IwVy h ª g$h k k* rszrs{|uw rxw ¢ vsk+ ¢ k^ * {lk°* y h k°§I{}* ¨6 {l.~I{|Kk+|[yz* {|
(210) ~IwVy h ªYº»k k+* |a{yzk * ¢ @\ y h k$zk+y{V¤ r£yzrs|I y k+vsk+l.k+|[yw ~4w}y h ¢* k+y]¨®k k+| y h k{t k
(211) |a{ k
(212) w}| y h kzrs| ± |K{ k E r£| uª°º»k k+rs|IwVyzk ¢ y h k w r£|4w}v°{}¤h\ ( ) \ ) ª%8|¹k+vsk+l.k+|[y('${V¤#y h kk+ya\ rs² h w w yzk rs´+k ¢ ¹w ' µ ¢ {[{vsk^w}|k^ yz{ 7 $ 9 w}yzrsᤸ[r£|a3S * [ ,r¶¤ ¢ k+vs{|K$yz{ur4& ( $ ) ) ( }m ) p{y h k ¨$rsDk g$h kpw rxw ¢ vsk+* .T &-.-- & Iw k|a{}¨ wVzz{K rxwVyzk y{uk+vsk+lk |y w ~IwVy h w}y h k y h k+|¯yz{w ª°º»* k k |K{yk ¢ $ y h ku {l.~I{|Kk+|[y${}¤®y h k
(213) * {ll.{ r£y] 9¡ +w z r£k ¢ y h k~Iw}y h h w w yk * rs´+k * ¢ $ ª g$h kutKl {}k '°{V¤®y h k² ' {l~4{|Kk |y $ r£y h kuwVvstKk{}¤°y h ku {l.l{ rsy] 9 S ) $ ) - g$h k * * $ J > ) h h ' ! ' ! µ $ $ y{ywVv§4{}¨· } { # ¤ y k { l l { s r ] y ) 9 £ r n y k E µ ^ k z y { ª » º k k + } w s v ° v y h wVyy h k ' $ J > r£$y h k k+r tIw}v +w}~4w rsy]k^ y{ ª k^ yz{ 3. óxç¬ÕYóxô.
(214)
(215) "!#$ %. ,^. g$h kw µE~IwVy h ¤¸{ lutKvxw}yrs{|{}¤y h k ( È ¡ ³ ) {|azrszy${V¤Uy h k¤¸{vsvs{}¨$rs|K~ { w}l/S. . * *. rs| / 1 )*,+ H F ! ' T* * $ #$#$#$ -F % ( ' ) # )*,+ ) 5 & * ' T * 9 7 ,5- &.-.-- & : 9 [ ( ' ' ) ( ' 'O' )2 $ ( 'O? ) 0 ,& * ¨ h k kn¤¸{ wVvsv9 :& r£$y h k ' ?>p {}¨°µ]k yz{ . . -. *. *. * ) & 0 &@' \ &. *. *. (87 * ) (J$ ). (7 ). ( ). (J ). *. k I|Kk ¢ $ ) ,5&¤¸{ ' ) ,5&.-.-- & ªº»k +wV| ^k {|Krs´+k#y h k# w}l.k®y][~4k${}¤Y {|Ky w}rs|[yz$w}rs|y h k|K{ k µzw ¤R{ lutav¦wVyzrs{|¬ª ' | k+k y h k#k^¾tIwVyzrs{|K ( Kª'O' ) w}| ( Kª' ) k+~ k+k+|[y k+z~4k^ yrsk+vsy h k§I{}¨ {|azk w}yzrs{| {|Kzy w}rs|[yz#wVz{B *r¦wVyzk y{k^w h {ll.{ r£y]w}| y h k: w}~Iw rsy] {|Kzy w}rs|[yzª g$h kk+|Kk w}v°¤¸{ lutKvxw}yrs{| ( ) rsz~4k^ rxw}vsrs´+k ¤¸{ y h k ( È ¡³ ) w µE~IwVy h ¤¸{ lutKvxw}yzrs{| ( )n¢ k+yzyr£|a3S * * * * * * * * * * 3 >) ; ¤¸{ wVv£v 9 7H,5&.--.- &: 9HS / S) ( ) 1 / 0( / ' ) \5 ) P ) w}| E ) ,ª ; 3 ) \ /( ) w}| E ) ª g$h k | ¨°k
(216) +wV|YÄw}#rs|¡y h k
(217) +wVzk{}¤®y h k|K{ k*µáw :¤¸{ lutKvxw}yzrs{|Yw w}~ay#y h k
(218) w}vs{ rsy h l k r ¢ k rs| y h knk+|Kk w}v4¤ w}l.k+¨°{ z± {}¤vsrs|Kk^w {~ayzrslrs´^wVyzrs{|¨$rsy h {tK~Kvsrs|K
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