7.0 Equality Contraints: Lagrange Multipliers
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1 Systes Optzato 7.0 Equalty Cotrats: Lagrage Multplers Cosder the zato of a o-lear fucto subject to equalty costrats: g f() R ( ) 0 ( ) (7.) where the g ( ) are possbly also olear fuctos, ad < otherwse we have the possblty of a over-detered syste of costrats. If the g ( ) fuctos are lear or sple, the oe varable ca be ellated for each equalty costrat,.e. varables ca be ellated, thus trasforg the proble to a (-) varable ucostraed zato proble. Cosder the followg eaple R 3 wth oe equalty costrat: ellatg 3 we have f() 3 g( ) whch whe substtuted back to the objectve fucto gves us a ew objectve fucto R fˆ ( ) ( ) whch s ow a ucostraed zato proble. Of course, ths ca t always be doe easly sce the equalty costrats ay be coplcated or eve plctly defed. A geeral procedure for corporatg the equalty costrats to the objectve fucto was developed by Lagrage 760. I ths ethod a ew ucostraed proble s fored by appedg the costrats to the objectve fucto wth socalled Lagrage ultplers. We wll ow descrbe ths ethod.
2 Systes Optzato 7. Lagrage Multplers If we have a objectve fucto R wth equalty costrats, such as (7.) the we ca troduce ew varables called Lagrage ultplers, λ, ( ) (7.) to create a ew objectve fucto s called the Lagraga, L(, λ), defed as L(, λ) f() + λ g ( ). (7.3) We ow ust ze the Lagraga over the R + space of the orgal varables plus the ew Lagrage ultplers λ. Therefore, we have ellated the equalty costrats at the epese of creasg the deso of our proble fro R to R +. We ca ow apply the optalty codtos as before. Recall the frst order ecessary codto that the u be at a statoary pot. Therefore, f we take the gradet of the Lagraga fucto we arrve at the followg ecessary codtos: L f g () λ j j 0 j ( ) j λ λ L g λ ( ) 0 ( ) (7.4) These sultaeous equatos are solved for (, λ ), that s, we have + equatos + ukows. Note that the secod set of these equatos are just the orgal costrats! ths eas that Also, sce at the statoary pot of the Lagraga, (, λ ), we have g ( ) 0 L(, λ ) f( ) (7.5) but t s ot ecessarly the case that f( ) 0. That s, a proble where we have equalty costrats, the u s ot ecessarly foud at a statoary pot of the orgal objectve
3 Equalty Costrats Lagrage Multplers fucto. If f( ) 0 the s s because the feasble rego defed by the equalty costrats cludes the ucostraed u of the fucto. The use of ths ethod ca be cleared up by cosderg a eaple. Cosder the followg sple proble R wth oe equalty costrat: f() 5 -- ( + ) Geoetrcally, the proble s to fd the pot of shortest dstace fro the org 5 legth + f( ) legth λ λ Fgure 7. Sple equalty costraed eaple. Method : Ellato of a varable The frst ethod we try s to ellate oe of the varables. Therefore, solvg for ters of we have 5 whch whe substtuted back to the objectve fucto, gves us a ew objectve fucto of just oe varable ˆ f( ) -- ( + ( 5) ) ˆ f ( 5) 0 3
4 Systes Optzato Solvg for gves us, ; that s (, ). Ths s a very sple eaple because the equalty costrat was such that oe of the decso varables could be easly ellated. We wll ow see how the Lagaga ethod ca be used to solve the sae proble. Method : Lagraga We frst costruct the Lagraga as L(, λ) ad the set the gradet of ths fucto to zero: -- ( + ) + λ( 5) L + λ 0 L λ 0 L λ 5 0 Ths s a set of 3 equatos 3 ukows. We frst solve the frst two for the Lagrage ultpler, λ, ad the substtute to the thrd, gvg 4λ λ 5 0 λ Oce we have the Lagrage ultpler, we ca easly solve for the reag varables (, λ ) (,, ) Ths Lagraga, L, s a quadratc fucto whch ca be wrtte atr for as L 00 5 λ + -- λ λ fro whch we see that the Hessa atr s gve by A ad sce A 5, A s ot postve defte or se-defte. Therefore, the soluto (, λ ) s ot a u of L(, λ) but s a u of the costraed fucto f( ). We ow 4
5 check Equalty Costrats Lagrage Multplers A to see f t s postve defte. Sce A, A s ot a egatve defte or egatve se-defte. Therefore the soluto (,λ ) s a saddle pot of the Lagraga fucto. 7. Quadratc Objectve Fuctos wth Lear Equalty Costrats We ow cosder the specalzed proble where the objectve fucto s a postve defte quadratc fucto varables, gve by the atr equato C R, ad there est lear equalty costrats d. For a soluto to est we ust have <. Thus we have: ze f( ) -- T A + b T subject to: C d where A s a postve defte atr, ad C s a atr wth rak C The Lagraga s easly fored as L(, λ) -- T A+ b T + λ T [ C d]. where λ s a colu vector of Lagrage ultplers. The ecessary codtos for a etreu are ow wrtte as: whch are a syste of L(, λ ) A + b + C T λ 0 λ L(,λ ) C d 0 + equatos ad + ukows. These ca be succctly wrtte as: AC T C O λ b d If we defe a ew atr M R ( + ) ( + ) such that M AC T C O the we ca wrte the soluto as 5
6 Systes Optzato λ M b d If M s osgular, the M ests, ad the soluto (, λ ) ests. Alteratvely, we ca develop a soluto by frst sovg for ters of λ fro the frst set of equatos. Therefore we have ( A )( b + C T λ ) A b A C T λ ad substtutg ths to the secod set of equatos C( A b A C T λ ) d. Ths s easly solved for the Lagrage ultplers as λ ( CA C T ) ( d + CA b). Ths s substtuted back to the soluto A b A C T λ whch gves A b A C T (( CA C T ) ( d + CA b) ) Eaple: As a eaple, cosder the zato of the learly costraed postve defte quadratc fucto ze subject to: f( ) I atr otato, the objectve fucto becoes f( ) -- T I A I ad the costrats are wrtte as 6
7 Equalty Costrats Lagrage Multplers C d C 3 d We ca ow for the Lagraga as L(, λ) -- T + λ T [ C d] wth the ecessary codtos for a etreu gve by I C T C O λ λ λ We could vert the atr M order to solve for the λ T I C T C O varables. O the other had, we ca proceed as we dd prevously by frst solvg for ters of λ fro the frst set of equatos. Thus we have + C T λ 0 C T λ whch ca be substtuted to the secod set of equatos Ths ca be solved for λ as C( C T λ ) d For our partcular proble, we have d λ ( CC T ) 7
8 Systes Optzato CC T Ivertg ths we have ( CC T ) Therefore we have the Lagrage ultplers λ Fro the secod equato we have the soluto we were lookg for C T λ ad the u of the fucto s f( ) Eaple: Ecoocal Dspatch of Power Cosder the proble of ecoocally dspatchg power fro power plats to loads over a trassso etwork. We wsh to detere the ost ecoocal way to geerate power so that the dead s satsfed. Let P deote the power producted by plat, F ( P ) be the cost of geeratg power P the plat, ad let the fucto L( P, P ) L( P) represet the power lost due to trassso over the lossy trassso les. The total dead for power s deoted as D. The objectve s to ze the cost of producg power ad at the sae te eet all the dead. Thus we have ze P F ( P ) subject to the dead that all the dead be et whch ca be wrtte as 8
9 Equalty Costrats Lagrage Multplers where P P L( P) D s the total power produced. We ca ow for the Lagraga as: L( P, λ) F ( P ) + λ P L( P) D The ecessary codtos ca be wrtte as L P df dp λ λ L P 0,,, L P λ L( P) D 0 (7.6) (7.7) Fro the frst of these we have df L λ λl dp P where we defe the loss factor as L L P Therefore, we have ---- df λ L dp ad λ s detered by solvg (7.6) ad (7.7) sultaeously. 7.4 Ecooc Iterpretato of Lagrage Multplers Cosder ze subject to: F( ) g ( ) b,,, ad thk of the values b as soe avalable resource. Now we wsh to vestgate how the soluto (,λ ) vares as the b vary. We frst wrte the Lagraga as 9
10 Systes Optzato L(, λ) F() + λ T [ g ( ) b] ad thkg of ths as a fucto of b we ca wrte t ore eplctly as ad dfferetatg ths wth respect to b L( b) F( b ( )) + λ T ( b) [ gb ( ( )) b] L b j F F λ T g j + ( b) j b λ j b + j T λ b b ( )[ gb ( ( )) b] F j g k j λ k + λ j b k λ j b + [ g k ( ) b k ] b j j j b F j k j k k g k λ k + λ k + [ g k b k ] j b k λ At (, λ ) the ecessary codtos for a etreu gve: ad F + k λ k g k j L (, λ ) 0 j ( ) 0 g k b k ad therefore sce F( ) L(, λ ) L F (, λ ) λ, b b, Therefore, the Lagrage ultpler λ gves the rate of chage of the objectve fucto wth respect to the resource b, soetes called shadow prces or sestvty coeffcets. 8 Eaple: optal allocato of a scarce resource betwee two processes. Say the total aout of a resource s b. Process gves us a retur based o the aout of resource b used of F( ), whle process gves us a retur of F( ). aze F F( ) + F ( ) 0
11 Equalty Costrats Lagrage Multplers b subject to: + b process process F ( ) F ( ) As a partcular eaple, cosder the Lagraga ad the ecessary codtos are solvg we have Fgure 7. Two processes wth a sgle resource. F ( ) 50 ( ) F ( ) 50 ( ) L(, λ) 00 ( ) ( ) + λ[ + b] L ( ) + λ 0 L ( ) + λ 0 L λ + b 0 ( ) ( ) 0 + b 0 b b b λ b 4 The sestvty ca ow be calculated as
12 Systes Optzato λ F 4 b b therefore f b < 4 ths ples that creasg b wll crease the retur; f b > 4 the creasg b eas a decreasg retur; f b 4 the λ 0 whch eas that the solutos to the costraed ad ucostraed probles are the sae. b F + F costat creasg F creasg F + b b 3 b 4 Fgure 7.3 Cotour plot of F the eaple wth lear equalty costrat. We ca get a clearer vew of what s gog o by vewg the cotour plot show Fgure 7.3. The crcles represet level curves for F F( ) + F ( ). The straght les represet the sgle equalty costrat for dfferet values of b. We see that whe b 4 the costrat le passes through cetre of crcles, that s at pot (, ). Here λ 0 ad the costraed ad ucostraed probles have the sae soluto. Whe b < 4 creasg b results a creasg value for the au of F + F ( λ ) 4 b > 0 whe b > 4 creasg b results a decreasg value for the au of F + F ( λ) 4 b < 0
13 Systes Optzato What do the Lagrage ultplers tell us about the purchase of ore resource b? We kow that the rate of chage of the proft s λ wth respect to creasg the resource b. If the cost of b s y [$/ut of b] the as log as λ > y we should purchase ore b. For ths reaso, λ s soetes called the argal retur. I the preset eaple λ 4 b ( geeral λ wll deped o b ). If the cost of buyg ore b s y $/ut, the we should buy eough to brg b 3 at whch pot y λ. Icreasg b further wll cost ore tha the creased proft t wll produce. 3
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