Multiplicadores de Lagrange

 

DKSQJC\MMDJK Lg `äjcj cl gjs `ugtpgdmbcjrls cl Gborbkol ls uk prjmlcd`dlkj pbrb lkmjkrbr gjs `âxd`js y `àkd`js cl nukmdjkls cl `ùgtpgls vbrdbagls suilbs b rlsrdmmdjkls, lsl `äjcj rlcuml lg prjagl`b rlsrdkodcj mjk k vbrdbagls b ukj sdk rlsrdmmdjkls cl k + F vbrdbagls cjkcl F ls doubg bg kù`lrj cl rlsrdmmdjkls, y muybs lmubmdjkls pulclk slr rlsulgbs `âs nâmdg`lkl. Lg `äjcj cdml qul gjs pukjs cjkcl gb nukmdýk tlkl uk lxrl`j, mjkcdmdjkbcj mjk f rlsrdmmdjkls, lsâk lkrl gjs pukjs lsbmdjkbrdjs cl ukb kulvb nukmdýk sdk rlsrdmmdjkls, mjksrudcb mj`j ukb mj`adkbmdýk gdklbg cl gb nukmdýk y gbs nukmdjkls d`pgdmbcbs lk gbs rlsrdmmdjkls muyjs mjlmdlkls sjk gjs g js `ugtpgdmbcjrls. Gb cl`jsrbmdýk usb clrdvbcbs pbrmdbgls y gb rlogb cl gb mbclkb pbrb nukmdjkls cl vbrdbs vbrdbagls, sl rbb cl lxrblr ukb nukmdýk d`pgàmdb cl gbs rlsrdmmdjkls y lkmjkrbr gbs @ugtpgdmbcjrls mjkcdmdjkls pbrbclGborbkol qul gbs clrdvbcbs pbrmdbgls mjk rlsplmj b gbs vbrdbagls dkclplkcdlkls

cl gb nukmdýk slbk doubgls b mlrj. Lk gj gjss prja prjagl gl`b `bss cl jpt`dzbmdýk, jpt`dzbmdýk, lg `ä `äjcj jcj cl gjs `ugtpg `ugtpgdmb dmbcjr cjrls ls cl Gborbk Gborbkol, ol, ggb`bcjs bsà lk ejkjr b Ijslpe Gjuds Gborbkol, Gborbkol , ls uk prjmlcd`dlkj pbrb lkmjkrbr gjs `âxd`js y `àkd`js cl nukmdjkls cl `ùgtpgls vbrdbagls suilbs b rlsrdmmdjkls. Lsl `äjcj `äj cj rlcuml rlcuml lg prjagl`b prjagl`b rlsrdkod rlsrdkodcj cj mjk k vbrdbag vbrdbagls ls b ukj sdk rlsrdmm rlsrdmmdjkls djkls cl k + f vbrdbagls, cjkcl f ls doubg bg kù`lrj cl rlsrdmmdjkls, y muybs lmubmdjkls pulclk slr rlsulgbs `âs nâmdg`lkl. Lsbs kulvbs vbrdbagls lsmbgbrls clsmjkjmdcbs, ukb pbrb mbcb rlsrdmmdýk, sjk ggb`bcbs `ugtpgdmbcjrls cl Gborbkol. Lg `äjcj cdml qul gjs pukjs cjkcl cjk cl gb nukmdý nukmdýk k tlkl tlkl uk lxrl` lxrl`j j mjkcdm mjkcdmdjk djkbcj bcj mjk f rlsrd rlsrdmm mmdjk djkls, ls, lsâk lsâk lkrl lkrl gjs pukjs pukjs lsbmdjkb lsbmdjkbrdjs rdjs  cl ukb ukb kulv kulvb b nukm nukmdý dýk k sdk sdk rl rls srd rdmm mmdj djkl klss mjks mjksr rud udcb cb mj`j mj`j ukb mj`adkbmdýk gdklbg  gdklbg  cl gb nukmdýk y gbs nukmdjkls d`pgdmbcbs lk gbs rlsrdmmdjkls, muyjs mjlmdlkls sjk gjs `ugtpgdmbcjrls.

2

 

Àkcdml DKSQJC\MMDJK..............................................................................................................................2 JAILSD^J]......................................................................................................................................= @brmj ljrdmj 9...............................................................................................................................3 BXGDMBMDJKL]................................................................................................................................> Mjkmgusdjkls...................................................................................................................................4

@ugtpgdmbcjrls clGborbkol

JAILSD^J]

1

 

Jailtvj Olklrbg9 

Jalklr lg mjkjmd`dlkj klmlsbrdj pbrb pjclr rlbgdzbr bpgdmbmdjkls utgdzbkcj lg `äjcj cl gjs `ugtpgdmbcjrls cl Gborbkol.

Jailtvjs Lsplmànmjs9 

^dsubgdzbr bgoukbs suplrmdls muâcrdmbs y murvbs cl kdvlg pbrb cdstkjs vbgjrls cl gb vbrdbagl z.



Dc Dclk lkt tmb mbrr b r rbv bväs äs cl gjs gjs sd`u sd`ugb gbcj cjrl rlss gj gjss puk pukjs js (x (x,y ,y)) sjar sjarl l gb murv murvb b mjrr mjrrls lspj pjkc kcdl dlk kl l b gb nukm nukmdý dýk k rls rlsrd rdmm mmdý dýk, k, cjkc cjkcl l gb nukm nukmdý dýk k prdk prdkmd mdpb pbgg tl tlkl kl lxrl`js.



Dklrp Dk lrprl rlbr br orâmb orâmb`lk `lkl l gjs rlsug rlsugbcj bcjss jalkd jalkdcjs cjs l`pglb l`pglbkcj kcj lg `äjcj `äjcj cl `ugtpgdmbcjrls cl Gborbkol.



Bprjxd`br gbs sjgumdjkls clg prjagl`b b pbrtr cl gb jaslrvbmdýk lk lg sd`ugbcjr

@ugtpgdmbcjrls clGborbkol cl gbs murvbs cl kdvlg cl gb nukmdýk prdkmdpbg y gb murvb mjrrlspjkcdlkl b gb

nukmdý nuk mdýk k mjkcdm mjkcdmdjk djkbk bkl, l, bcqudr bcqudrdr dr ebadgd ebadgdcbc cbc lk gb rlsjgu rlsjgumdý mdýk k cl prjagl prjagl`bs `bs cl jpt`dzbmdýk lk uk b`adlkl mj`pubmdjkbg.

=

 

@brmj tljrdmj 9 ]lb n   (x) ukb nukmdýk clndkdcb lk uk mjkiuktj badlrtj k-cd`lksdjkbg {x ∅ Qk}. ]l clndklk s rlstrdmmdjkls o f f   (x) 0 8, f02,...,s, y sl jaslrvb (sd gbs rlstrdmmdjkls sjk sbtdsnlmebs) qul9

]l prjmlcl b ausmbr uk lxtrl`j pbrb e

gj qul ls lqudvbglktl b

Gjs `ugtdpgdmbcjrls clsmjkjmdcjs  ζf   sl cltlr`dkbk b pbrtdr cl gbs lmubmdjkls mjk gbs rlstrdmmdjkls y mjkiuktb`lktl sl jatdlkl uk lxtrl`j pbrb e qul bg `ds`j tdl`pj sbtdsnbml gbs 0 rlstrdmmdjkls rlstrdmmdjk ls (d.l. o f f08  ), gj qul d`pgdmb qul n   eb sdcj jptd`dzbcb   8 ), @ugtpgdmbcjrls clGborbkol Lg `ätjcj cl `ugtdpgdmbcjrls cl Gborbkol ls olklrbgdzbcj pjr gbs mjkcdmdjkls cl Fbruse-FuekSumflr.

Lk gjs prjagl`bs cl jpt`dzbmdýk, gjs `ugtpgdmbcjrls cl Gborbkol, kj`arbcjs bsà lk ejkjr b Ijslpe Gjuds Gborbkol, sjk uk `äjcj pbrb rbabibr mjk nukmdjkls cl vbrdbs vbrdbagls qul kjs dklrlsb `bxd`dzbr j `dkd`dzbr, y lsâ suilb b mdlrbs rlsrdmmdjkls. Lsl `äjcj rlcuml lg prjagl`b rlsrdkodcj lk k vbrdbagls lk ukj sdk rlsrdmmdjkls cl k + 2 vbrdbagls muybs lmubmdjkls pulclk slr rlsulgbs. Lsl `äjcj dkrjcuml ukb kulvb vbrdbagl lsmbgbr clsmjkjmdcb, lg `ugtpgdmbcjr cl Gborbkol, pbrb mbcb rlsrdmmdýk y njr`b ukb mj`adkbmdýk gdklbg dkvjgumrbkcj gjs `ugtpgdmbcjrls mj`j mjlmdlkls. ]u cl`jsrbmdýk dkvjgumrb clrdvbcbs pbrmdbgls, j adlk usbkcj cdnlrlkmdbgls jbgls, j sus pbrdlkls mlrmbkjs, gb rlogb cl gb mbclkb. Lg k ls, usbkcj bgoukb nukmdýk d`pgàmdb, lkmjkrbr gbs mjkcdmdjkls pbrb qul gb clrdvbcb mjk rlsplmj b gbs vbrdbagls dkclplkcdlkls cl ukb nukmdýk slb doubg b mlrj.

3

 

Xjgdkj`djs cl Gborbkol ]upjkdlkcj qul sl mjkjml pjr gj `lkjs

sl prjpjkl

pjr gj bkj9

@ugtpgdmbcjrls clGborbkol

<

 

BXGDMBMDJKL] Lil`pgj 29  Lkmjkrbr lg (j gjs) pukj(s) lk gb lgdpsl xy +  y  cx ⇑cy0 , bsu`dlkcj qul lg clkj`dkbcjr ≨ 8 lk lg `àkd`j. > x

⇑Cn

∑28 y

0V1x+1y º)P

⇑1x+1y

(

cx08

∑= y  ) 0 8 ⇑ y²x = x ∑ <  y



 

Lil`pgj 1 ]upjkob`js qul qulrl`js lkmjktrbr gb cdstrdaumdýk prjabadgàstdmb cdsmrltb mjk `âxd`b lktrjpàb. Lktjkmls

Xjcl`js usbr gjs `ugtdpgdmbcjrls cl Gborbkol pbrb lkmjktrbr lg puktj cl `âxd`b lktrjpàb (clplkcdlkcj cl gbs prjabadgdcbcls). Xbrb tjcj f   clscl 2 ebstb k, klmlsdtb`js

gj qul kjs cb

@ugtpgdmbcjrls clGborbkol Clrdvbkcj lstbs k  lmubmdjkls, jatlkl`js

Lstj `ulstrb qul tjcj  pd   ls doubg (cladcj b qul clplkcl sjgb`lktl cl ζ). \sbkcj gb rlstrdmmdýk pf   0 2, lkmjktrb`js −f   p

Lstb (gb cdstrdaumdýk ukdnjr`l cdsmrltb) ls gb cdstrdaumdýk mjk gb `byjr lktrjpàb.

Lil`pgj = Cltlr`dkbr gjs puktjs lk gb lsnlrb

qul lstâk `âs mlrmbkjs bg

puktj gb cdstbkmdb bg puktj

9

pbrb ebmlr `âs slmdggb gb jplrbmdýk sl `bxd`dzb j `dkd`dzb lg mubcrbcj cl gb cdstbkmdb9

?

 

gb rlstrdmmdýk9 Cl bmulrcj mjk lg `ätjcj cl gjs `ugtdpgdmbcjrls cl Gborbkol, sl rlsulgvlk gbs lmubmdjkls "

"y"

 

" y lg rlsugtbcj ls9

(2) (1) (=) (3)

gb `bklrb `âs slkmdggb cl rlsjgvlr lstbs lmubmdjkls ls clibr x,y,z lk nukmdýk cl sustdtud`js lk gb lmubmdýk (3).

cl gb lmubmdýk (2) jatlkl`js pulcl rlbgdzbr gb jplrbmdýk.

sl jaslrvb qul

≨ 2 pjr qul sd

y guloj

kj sl

gj `ds`j sumlcl mjk gb lmubmdýk (1) y (=)

@ugtpgdmbcjrls clGborbkol   sustdtuylkcj lk gb lmubmdýk (3)

lg vbgjr cl lktjkmls gjs puktjs (x,y,z) sjk 9

 y

sl pulcl jaslrvbr qul lg puktj `âs mlrmbkj lktjkmls ls

Mjkmgusdjkls

2-Lk gb rlsjgumdýk cl prjagl`bs, lsl `äjcj kjs byucb ebggbr `âxd`js y `àkd`js cl nukmdjkls cl `ùgtpgls vbrdbagls suilbs b rlsrdmmdjkls.

5

 

1-Gb ämkdmb cl gjs "`ugtpgdmbcjrls cl Gborbkol" ls ukb njr`b cl rlsjgvlr prjagl`bs cl jpt`dzbmdýk mjk rlsrdmmdjkls. Qlsugb slr sùplr ùtg =-Lk lg mb`pj cl gb lmjkj`àb, gb nukmdýk gborbkodbkb ls usbcb pbrb rlsjgvlr prjagl`bs cl jpt`dzbmdýk. Lg `äjcj cl gjs `ugtpgdmbcjrls cl gborbkol ls usbcj pbrb clrdvbr uk `âxd`j j `àkd`j gjmbg lk ukb nukmdýk suilb b mdlrbs rlsrdmmdjkls. 3. Lk gjs klojmdjs, kjs byucb b cllr`dkbr b `âxd`dzbr alklmdjs.