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Multimodularity, Convexity and Optimization Properties

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(1)Multimodularity, Convexity and Optimization Properties Eitan Altman, Bruno Gaujal, Arie Hordijk. To cite this version: Eitan Altman, Bruno Gaujal, Arie Hordijk. Multimodularity, Convexity and Optimization Properties. RR-3181, INRIA. 1997. �inria-00073508�. HAL Id: inria-00073508 https://hal.inria.fr/inria-00073508 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Multimodularity, Convexity and Optimization Properties Eitan ALTMAN Bruno GAUJAL and Arie HORDIJK. N˚ 3181 Juin, 1997. THÈME 1. ISSN 0249-6399. apport de recherche.

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(223) + :q` B ±o^ ˆK†g^z r{Sm'{ s}| /T | —V~²~!8 r{œm'{ s}| —V~²~·& r{Sm!{ sƒ| /Tq~ | U~U8 r{Sm'{ s}| /T£~²~†¬ r{Sm!{ s ~ | U~!8 r{Sm'{ s ~²~ & r{Sm!{ s ~ | m'{ s}| /T | —V~U89m'{ s}| —•~ˆ~U8 r{Sm'{ s ~²~ ¬ r{Sm!{ sƒ| /Tq~ˆ~!8 {œm'{ s ~ˆ~J+ Z'[K^ Á–vhfqn°|ˆK^gºj–iz~}|¬nªpH§2xk~}~xŽ«-f

(224) §2vsxƒ] nt[K^ †gxkˆ£k^lm|¬nªp xk§ ;ikˆK„Hfh|ˆK†g^>m!{ s| /T£~®¬·m!{ s ~¨ Z'[K^ fh^ †gxƒˆŒ„Â|ˆK^ ºrjKik~}|¬nªp¾§2xƒ~~xž«-f§2vsxƒ] nt[K^ § ik†nnt[Kizn Æ|n|fˆKxƒˆŒ„K^g† vh^Dizfh|ˆK—KµikˆK„Ånt[Œ^°§ ik† nnt[–izn¯p :q` Bum!{ s| /T£~†€zm'{ s ~ | m'{ s}| /1 | —V~!8bm'{ s}| —V~¨ Z'[K^ftik]_^ikvh—kjK]_^gˆ“n³[Kxƒ~„Kf'§2xƒvE · vh^ ‚K~´iz†g|ˆK—n/T Œ«-[K|†‡[9^gfqn‡ik¯Œ~}|fh[K^ fonh[K^ ‚Kvhxrxk§ñ¨  #  g JI  g  #+ !g ¸ -, I4 !ã¿,   EV¥'E  ©ªˆnt[Œ|}f­fs^g†nt|xƒˆ¦“«^¤«-|}~~KjKfs^']jK~¬nt|]_xm„KjŒ~´ikvs|nªp!ntx0xk‚Œnt|]_|}¹ ^-ic†gxƒfqn­§2jKˆK†nt|xƒˆ_¯–ikfs^g„xƒˆ_icfs^gºjK^ ˆK†g^ xk§¥§2jKˆK†nt|xƒˆKf'«-[K|†‡[ «-|~~¦vh^ ‚Kvh^ fh^gˆ“nciºj–ikˆ“nt|¬nªp%xk§|ˆnh^gvs^gfsn³|ˆ i_—k|£k^gˆ]_xm„K^ ~OŒfhjK†‡[Àikf¸«¤xƒvh²~x“ik„ |ˆ9iºjK^gjŒ^§2xƒv'^VlŒik]_‚K~}^

(225) : fs^g^  §2xkv-i_]_xƒvs^ ‚Kvh^ †g|fh^|}ˆKfqn‡ikˆŒ†g^xk§nh[K^ ‚Kvhxk¯K~}^ ] B¨ xƒˆKfs|}„K^ vi fh^gºjK^ ˆK†g^°xk§¸ˆKxkˆKˆK^g—ƒiznt|¬£ƒ^

(226) §2jKˆK† nh|}xkˆKf}m +"  ‰nh[–iznftiznh|}fq§ p¼nt[K^_§2xƒ~~}xŽ«-|ˆK— ikfsfhjK]_‚Œnh|}xƒˆŒf ` wm |f³]!jK~¬nt|]

(227) xr„KjK~}ikv ¨ . . . . . . . . . . . . . . . . . #". !. . . %'&. )(. ݥݿúMLä:,5/,. . . . $". . .

(228)       

(229)    "  + #. `ž” . . . . ” wm {  6*,+.+-+.*  ~€zm 476ª{  L,*,+.+.+-*  ~J* ¯(!E –xƒv-i—ƒ|¬£ƒ^gˆÀfh^ ºrjŒ^gˆK† ^nG  O Œ«¤^„K^ Á–ˆŒ^ nh[K^ †gxƒfqnE—^{  ~¸ikf. —^{  ~& 

(230)  ( m •{  6J*,+.+-+.*  T~+  6 ;^ n 6ÀikˆK„ ¯L^0nª«¤x

(231) ‚Lxƒfs|nh|£k^ vh^Diz~}f ¨ ¡ ^(„K^Á–ˆK^(nt[K^  lk. 0*43 /lk.G  { ~ O «-|¬nt[ vtiznh^ 6 ikˆK„ |ˆK|nh|´iz~¦‚K[–ikfs^ ikf  8 d{ 8@(h~ 6 |  * :q`ž’!B  { ~& 6 ! | " ) «-[K^ vh^  s  |fonh[K^~}ikvs—ƒ^gfqn(|}ˆ“nh^g—ƒ^ vcfh]

(232) ik~~^gvcnh[–ikˆSxƒv³^gºj–iz~ntx s ¨¢³xknt^nt[–iznont[K^!fh^nG  { ~J*KB ¬ ¬3( O ikvh^ ^lrnhvh^g]_^‚Lxk|}ˆ“ntf-xk§Eikˆ iznhxƒ] †gxkˆntik|ˆK|}ˆŒ—

(233) nh[K^ ‚Lxƒ|ˆ“n’{ 62*'62*,+.+-+.*+6:~V¨ Z'[K^aik|}] xk§µnt[K|f fh^g†nt|xƒˆ«-|~~w¯L^ntx‚KvsxŽ£ƒ^_nt[–izn(nh[K|ffs^gºjK^ ˆK†g^a]

(234) |ˆK|]_|}¹ ^gf!nh[K^§2jKˆK†nt|xƒˆ —Y ‚Kvsxž£r|„K^g„Snh[–izncfhxƒ]_^† xƒˆK„K|¬nt|xƒˆKf :2|}ˆŒ†g~jK„K|ˆK— ` ikˆK„ ” B-ik¯LxŽ£k^[Kxƒ~„¦¨cZ'[Œ|}f0fs^gºjK^ ˆK†g^«ikf jKfs^g„ ¯“p¼±oi4yq^g²9|}ˆ 3ò’5 ¥ikˆŒ„»«¤^_jKfh^

(235) fh^£ƒ^gvhik~®‚Kvhxk‚L^ vsnt|^gf xk§µnt[Œ^

(236) ¯Kik~´izˆK†g^ „fs^gºjK^gˆŒ†g^°^ fsntik¯K~|}fs[K^g„ nt[Œ^gvh^ |}ˆX¨EZxo^ fsn‡iz¯K~}|fh[nt[K^®]

(237) ik|ˆ!xk‚Œnt|]_|}¹giznt|xƒˆvh^gfsjK~¬ntfg4«^µˆK^ ^g„!nh[K^w§2xƒ~~}xŽ«-|ˆK—³nh^g†‡[KˆŒ|}†gik~M¥^ ]_]aim¨ ¤®È

(238) 

(239)  2]`_7]0™jƒm 0 0 .00.0/ %"'+      qp/ qp¦jlk/+  m  0' 0 .0 0.0/ %"+     j  % *0/  ?-    /-0¨ ° œ ÌXÌ:±²]@;^ nnµ & {œµI66J* l*Nµ ~  ¨ÆZ'[K|f_‚Lxk|}ˆ“n

(240) ¯L^ ~}xkˆK—ƒfntx»ikˆHizntxk] .E{œµT~]

(241) ik„K^¯“pÅnt[Œ^ ^lrnhvh^g]_^ ‚Yxƒ|}ˆ“nhf s  * s * ^* s ¨wZ'[K^(ˆjK]¯Y^gvs|}ˆŒ—axk§¥nh[K^(^lrntvs^g]_^ ‚Lxk|}ˆ“6 ntf'xz§nh[K^ iznhxƒ] |f'†‡[Kxƒfs^gˆ fhjŒ†‡[Ânh[–izn

(242) ik† †gxƒvs„K|}ˆŒ—¼nhx+nt[K^%¯–izfh^ D & {d8 6 *K/ 6 6 * ^*K/ 476 *N ~ s & s  8z 6 ¨ Z'[Œ^ xznt[K^ v |ˆK„K|†g^gf izvh^izvh¯K47|¬nt6 vhikvspk¨0Z'[K|f0|}]_‚K~|^gf0nt[Kizn s³ & s ³ §2xƒvoik~~  ¨0© §w«¤^†Diz~}~ ‰nh[K^‚Kvsxzyq^g†nt|xƒˆ xk§  xƒˆ“ntx  ik~xƒˆK—nt[K^(Á–vsfsn³†gxrxƒvs„K|}ˆKiznt^z n{`/ ³ ~´&  {£/ ³ 476 ~|¬§†(  7 4 6 n{`/ 6 ~´& 8 6 n{œ 6 ~´& B n{œ ~´&  476 n{ s  ~´& { s L * ^* s  ~ Z47'6 [K^gfs^gfa^gºj–ik~|nh|}^ fa|]_‚K~¬p¾nt[–i4n  { s  ~n & n{ s 6 ~izˆK „ n{ s 6 ~J* ^*  { s ~!§2xƒvs]@ikˆÂiznhxƒ] |ˆ  KjKfs|}ˆŒ—ant[Œ^„Œ^ Á–ˆK|¬nt|xƒˆ xk§Eikˆ9i4ntxƒ]¨ 8 ~fhxŒu:2¹*B¸¯Y^g~xƒˆK—ƒf'ntxnh[K|}f³iznhxƒ]ikˆK„|¬§ %'&. (. ". .  . . . . )(. &. (. &. . . . . . . . . . . . . . . . . . . . ". . . . . . .  . . . . . . . . . ^ s  | !6 s 6 |  6. . {`µI6J*Nµ L* 2*Kµ ~

(243) & U(8.  .

(244). nt[Œ^gˆ. . . |. . . . . . s *. . . {œµ L,* ^*Kµ ~·& n{œµI6*Nµ Lª* ^*Nµ ~ & U(E8 ^ n{ s  6 & U(E8 ^ n{ s  L .  .

(245). .

(246). . .  . . . . ~ | !6n{ s 6 ~ | . 6~ |. . | . . . | . . . . {s ~. . n{ s. ~J+ Û2ÜÝ;Û2Þ.

(247) 

(248)     ! 

(249) " $#&%"' ()*+ ,- .%/'+.0 ¢oxŽ« . `Ž‹. . . . m  {œµI6J*Nµ L*2*Nµ ~´&. . !(E8. ^ m { s  ~ | '6²m  6 € !(E8 ^ m 476ª{ n{ s  ~ˆ~ |  6 & !(E8 ^ m 476ª{ n{ s 6 ~ˆ~ |  L  & m 476ª{œµ L* 2*Nµ ~+  .

(250). . . . . . {s 6~ |.  . . . .  . . . . .  . m {s ~. . . .

(251). |. !6ˆm 476J{ n{ s 6 ˆ~ ~ |. .

(252). . . |. . . . . . . | . . . . m 476J{ n{ s ~ˆ~. . m 476J{ n{ s ~²~. ^]`_u]> " /?0.0/ %"'+ G0 

(253)  "  D /      "  /   .D  j / (  0//    

(254)   ¸ B*,(¹  " /   £p(  %Nk/+'+ ®o¨ ¨ S¨   

(255) ¨ ^p/  rm {   6 {  ~J*,+.+.+-*    {  ~ˆ~&   m  { 62*+6^*,+-+.+.*+6:~J+ :q`Ž B

(256)

(257) ° œ ÌXÌ:± ¡ ^ [–iD£ƒ^ §2xƒv-ik~~ E 2m 7{   6 {  ~J*,+.+-+.*   {  ~²~& m  7{62*,+.+-+.*+6:~+ :q`Ž‘ B : Z'[Œ|}f §2xƒ~~}xŽ«-f :q`ƒ`*B;§2vhxƒ] ;^g]_]

(258) iS”m¨ò•mikˆK„¼§ ik† n(nh[–izn m  |fi

(259) ˆK^_xƒˆ»^Dik†‡[ izntxƒ]¥ikˆŒ„ ik—kvh^g^ f «-|¬nt[ m S§2xƒv!nt[K^^lrnhvh^g]_^‚Yxƒ|}ˆ“nhf

(260) xz§³nh[K^ izntxk] ¨ B €m|ˆK† ^ m  {62*+62*,+.+.+-*+6u~!|}f|ˆK† vh^Dizfh|ˆK—|ˆ  ¯p ;^g]_]

(261) ia‹m¨}`ƒmnt[K^ ~|}]_|¬no|ˆ  ^Vlm|}fqntf : |¬no|f-‚Lxkfhfh|¯K~¬p|}ˆmÁ–ˆK|¬nt^*BV¨ ¡ ^ †gik~}~¦nh[K^ fh^ ºrjŒ^gˆK† ^nG   {  ~ O nt[Œ^vhikˆK„Kxk]

(262) |¹g^ „À¯Kik~´izˆK†g^ „9‚Yxƒ~}|† pk¨  ÈÌ œ È

(263) ^]-[2]> " /J0.0/ %"+ G0 

(264)  "   7j  // ¸ BV*,(¹ .   ( m •  { { J ~ , * . + + . + * { ² ~ † ~ ¬   m  { 62*+6^*,+-+.+.*+6:~J+ : ”= B 6     

(265)

(266)  6 ° œ ÌXÌ:±²]'Ãc^ ÁKˆK^ m · { *+6:~ &¯ ¶ m · {  6 { ~J*,+.+-+.*  · { ~²~J+ m · |}f-‚Y^gvh|xr„K|}† :2|}ˆ B¤«-|¬nt[À‚L^ vh|xm„¼`ƒ¨®Ãc^Á–ˆK^. m · { *'6:~&w ¶ m · {   4 · %6 { ~J*,+.+.+-*   { ~²~+ Z'[K^ ˆÀ«¤^[KiŽ£k^. m · { ™*'6:~&wm · { *+6:~ «-[K^gvs^ •& 87C 6^* : ”m`B  ÈÌ œ È.

(267). . .  . . ݥݿúMLä:,5/,.

(268) `@.       

(269)    "  + #. ©ªˆK„K^ ^g„¦. m · { ™*'6:~·& m · {  4 · %6 { .~J*,+.+.+-*  { .~ˆ~ & m · {  4 · %6 { | C 6u~J*,+.+-+.*  { ?| C 6:~²~

(270) &¯m · { *'6:~J* «-[K^ vh^0nt[Œ^0~´izfsn¸^ ºj–ik~|nªpa§2xƒ~}~xŽ«-f¸§2vhxƒ] nh[K^c§ ik†nnt[Kizn   4 · { | C 6u~&   { ~ n&¡(*,+-+.+.*-C+¨

(271) m · |foiz—“ik|ˆ%‚L^ vh|xm„K|†(« ¨ûv ¨ ng¨ Œ«-|¬nt[ ‚L^ vh|xm„¼`ƒKikˆK„|}f-|ˆK†gvs^Dikfs|}ˆŒ—a|ˆC¿fhx_nt[Kizn'nt[K^0§2xk~}~xŽ«-|}ˆK—

(272) ~|}]_|¬n ^lm|fsnhf :2‚Lxƒfsfh|¯K~¬p|}ˆŒÁKˆK|nh^*B m

(273) { *+6:~E& ¶ 

(274) m · { *+6u~J+ · {Àxƒvs^gxŽ£ƒ^ vg®«^%[KiŽ£k^ nt[–i4n ^m · {  *+6:~}& m  · { 62*+-+.+-*'6:~­«-[K^ vh^  ¯L^iÀvtikˆK„Œxƒ] £zikvs|´ik¯Œ~}^zwjKˆK|  §2xƒvs]

(275) ~¬p

(276) „K|fsntvs|}¯ŒjŒnt^ „ |}ˆ ¸ BV*(¹D:2nt[Œ|}fµ§2xƒ~}~xŽ«-f :ñ`ƒ`B“§2vsxƒ]);^g]_]

(277) i”m¨ò•mrikˆK„°§ ik†n¤nh[–izn m  |f¸i

(278) ˆK^oxƒˆ ^Diz†‡[9iznhxƒ]ikˆK„ ik—ƒvs^g^ f³«-|¬nt[®m §2xkv¸nt[K^ ^lrnhvh^g]_^‚Lxk|}ˆ“ntf-xk§nt[K^ i4ntxƒ] B¨µ±³^gˆK† ^k r m

(279) {  *+6:~&  m  {62*+62*,+.+.+-*+6u~J+ : ”ƒ”!B

(280) xƒˆKfs|}„K^ voˆŒxž«ãnt[K^ ¯–iz~´ikˆŒ†g^g„Àfh^ ºrjŒ^gˆK† ^!§2xƒv¸ÁŒlm^ „ ¨wZ'[Œ^gˆ ( m { 6 { ~J*+-+.+-* { ~²~  · ·   6 ·. ¬ ( m {   4  4 ·  { ~*,+-+.+.*   { ~J*,+.+-+.*  · { ~ˆ~ 6 ·. ¬ ( m

(281) { 8 C 6 | *+6:~+ 6 · Z'[K^~´izfsn³|}ˆŒ^gºj–ik~|nªp§2xƒ~~xž«-f'§2vsxƒ] ikfsfhjK]_‚Œnh|}xƒˆ ” ã§2xƒv'nh[K^(§2jKˆK† nh|}xkˆKf m ikf'«¤^g~~ikf³ikˆ9izvh—ƒj  ]_^gˆ“nofh|]

(282) |~}ikv'ntxnt[K^ xƒˆŒ^jŒfh^g„ |ˆ :O”m`B¨ © § 6Æ|}f|vhvhiznt|xƒˆ–ik~ ®ik‚K‚Œ~pr|ˆK—+nt[K^ ^ vh—ƒxr„K|†°nt[Œ^gxƒvs^g] xz§ ¡ ^ pr~'izˆK „ ®xƒˆb¢o^ jK]

(283) ikˆKˆ : 3û•5 B­«¤^ [–iD£ƒ^.   ( m

(284) {d8C 6 ! | *+6:~& 2m

(285) {  *+6:~J+

(286) 6 · –vhxƒ] wºj–iznt|xƒˆ :O”ƒ”!B“«¤^([–iD£ƒ^ 2m

(287) {  *+6:~& 

(288) m  {62*+62*+-+.+-*'6:~J+rZ'[K|f¸|}]_‚K~|^gfnh[–izn¸|¬§ 6 |}f³|vhvti4nt|xƒˆ–ik~ . { 8 C 6 | *+6u~&  m  { 62*+6^*,+-+.+.*+6:~  ( m : ”k‹ B

(289)

(290).

(291) 6 · © § 6|}f¸vhiznt|xƒˆ–iz~Ornt[K^ ˆ 6  &  I/«¤^gvh ^ ikˆK„ /aikvh^cvh^g~}iznh|£k^g~¬p ‚Kvs|}]_^ ikˆK„ / €3(ƒ¨­Z'[K|f'|}]_‚K~|^gf nt[Kizn®nt[K^ofh^gºjK^ ˆK†g^}{   4  4 ·  { ~J*,+.+.+-*   { ~J*+-+.+-*  · { ~²~­|f¤† xƒˆKfsntikˆ“n¸|¬§ ]_xm„ (¦ ¸

(292) /:* { | (h ~ I/T¹   . . . . . . . . . . . . &. . (. . . . . Û2ÜÝ;Û2Þ.

(293) 

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(295) " $#&%"' ()*+ ,- .%/'+.0. `ž•. §2xƒvµik~~ –¨­Z'[K^gvs^ §2xƒvs^kVm · { *+6:~­|}f¤ik~fhx†gxƒˆŒfsn‡izˆn¤xƒˆ_nt[K^ fh^o|}ˆ“nt^ vs£zik~fizˆK„°¯“p_‚–ikfsftik—k^³nhxnh[K^-~}|]_|ng m { *+6u~!|}f†gxƒˆKfqn‡ikˆ“n_xƒˆ¾nh[K^gfs^|}ˆ“nt^ vs£zik~fg¨#¢oxŽ« ®ˆKxknh^°nt[–i4n –vtik†<{ 8 C 6:~ ¸

(296) /:* { | (h~ I/T¹ §2xƒ

(297) v³^lŒik†nt~¬p9xƒˆŒ^!£zik~jK^xk§ C xƒjŒnoxk§/†gxƒˆŒfh^g† jŒnt|¬£ƒ^£zik~jK^ fcxk§ C ¯Y^g†gikjKfs^ °izvh^i/Àizvh^vh^g~}iznh|£k^g~¬p ‚Kvs|}]_^k¨ ¢oxŽ« K«¤^![–iD£k^ ž 476. ( (  m { 8 C 6 | *+6:~& / m

(298) {C I/u*+6u~& rm

(299) {  *+6u~J+

(300)  · 6

(301) · 6 wºrjKiznt|xƒˆ7:O”ƒ”!B¤† xƒˆK† ~}jK„Œ^gfonh[K|f³†gikfh^ikf'«¤^g~~O¨ . . E ˆ‡. . . . ‘#Š2Œ. ‘ ?‹ ?–. ©ªˆnt[K|f®fhjK¯Œfh^g†nt|xƒˆ¦«^-^ fsn‡iz¯K~}|fh[

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