A Damped Guass-Newton Method for the Generalized Linear Complementarity Problem

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1 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 A Dmed Guss-Newto Method for the Geerlzed Ler Comlemetrty Problem Huju L Houchu Zhou Dertmet of Mthemtcs Ly Uversty L Shdog 765 P R Ch School of Mthemtcl Scece Shdog Norml Uversty J Shdog 5 P R Ch e-ml: lhuju8987 {t} 6com Abstrct I ths er we cosder the geerlzed ler comlemetrty roblem (GLCP over ffe subsce To ths ed we frst reformulte the GLCP s system of osmooth equto v the Fscher fucto Bsed o ths reformulto the fmous dmed Guss-Newto (DGN lgorthm s emloyed for obtg ts soluto d we show tht the DGN lgorthm s qudrtclly coverget wthout odegeerte soluto Some umercl exermets of the lgorthm re lso reorted ths er Keywords- GLCP; DGN lgorthm; qudrtcl covergece I INTRODUCTION The geerlzed ler comlemetrty roblem deoted by the GLCP s to fd vector ( R x y such tht T Mx Ny x y ( x y where M ( m N R re two gve mtrces d l ml m Qz q z R ( Q R q R s ffe m subsce R The GLCP s secl cse of the exteded ler comlemetrty (XLCP whch ws frstly troduced by Mgsr d Pg ([]The geerlzed comlemetrty roblem lys sgfct role ecoomcs equlbrum roblems ocooertve gmes trffc ssgmet roblems egeerg d oerto reserch d of course otmzto roblems ([] For the GLCP my effectve methods hve bee roosed recet yers ([3] Zhg et l ([4] reformulte the GLCP s ucostred smooth otmzto roblem v Fscher fucto d desg Newto-tye lgorthm for solvg t DSBrt d MDBrt ([5] develo double descrto method to fd ll ts solutos d show the geerl roblem s NP-hrd roblem Dfferet from the lgorthms lsted bove ths er we equvletly reformulte the GLCP s system of osmooth equtos v the Fscher fucto Bsed o ths reformulto we roose dmed Guss- Newto lgorthm to solve ths system We show tht the lgorthm s qudrtclly coverget uder mlder hyotheses We ed ths secto wth some ottos used ths er The er roduct of vectors x y R s deoted by x T y Let deote -orm of vectors Euclde scethe trsosed Jcob ( x s deoted by F(x colum vector F of vector-vlued fucto F (x For smlct we use ( x z for T T T T ( x y z For R D dg ( deotes the dgol mtrx whch the -th dgol elemet s II EQUIALENT STATEMENTS OF GLCP I ths secto we wll gve some equvlet sttemets reltve to the soluto of the GLCP Frst the followg result s strghtforwrd Theorem f there exsts ( x y s soluto of the GLCP f d oly l z R such tht Mx Ny Qz q x y ( x y To roose qudrtclly coverget lgorthm for the soluto of the GLCP we ow formulte the GLCP s system of equtos v the Fscher fucto ([6] : R R defed by ( b b b for b R A bsc roerty of ths fucto s tht ( b b b For rbtrry vectors fucto s follows b R we defe vector-vlued wwwjctcom

2 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 ( b ( b ( b ( b ( b b b Obvousl Combg ths cocluso wth Theorem we c estblsh the followg equvlet formulto of the GLCP Theorem oly f there exsts ( x y s soluto of the GLCP f d l z R such tht Mx Ny Qz q ( x y From the lyss bove we defe vector-vlued fucto l m : R R d rel-vlued fucto l : R R s follows: f Mx Ny Qz q ( x z : (3 ( x y f ( x z : ( x z ( x z ( x z (4 Combg ( wth (4 the followg cocluso s obvous Theorem 3 f f ( x y z III ( ( x y s soluto of the GLCP f d oly SOME DEFINITIONS AND BASIC RESULTS I ths secto we frst revew some deftos whch wll be used the sequel m For loclly Lschtz mg H : R R we let H(x deote the Clres geerlzed Jcob of H (x t x R whch c be exressed s the covex hull of the set ([7] where BH(x B H x R H x ( m lm ( x x H (x s dfferetble t x for ll The followg deftos re due to Q d Su ([8] Defto 3 A loclly Lschtz cotuous vector vlued fucto x R exsts for y m : R R s sd to be semsmooth t f the lmt h R lm H ( xth h h t h Defto 3 The fucto H strogly semsmooth t x f H s semsmooth t x d for y H ( x h h t holds tht m : R R s sd to be H( x h H( x h O( h Next we dscuss the dfferetl roertes of d f ( x z s ot defed by (3 d (4 The fucto dfferetble everywhere wth resect to l ( x z R R R However t s loclly Lschtz cotuous vector vlued fucto d there hs oemty geerlzed Jcob the sese of Clre ([9] I the followg we gve roch to clculte elemet of ( x z From Proosto 3 [] we gve the followg result Proosto 3 For x y R choose v R such tht v for y dex wth x d y Let W ( D D b where x y b ; fx y x y x y v b fx y v v W ( x y W B The or more recsel ( x y Chgg v we wll obt dfferet elemet of v f B( x y I our code we choose to set x y otherwse Thus elemet ( x z c be clculted s M N D D b Q where D d D b re defed Proosto 3 It s esly see tht whe m l s squre We summrze the dfferetl roertes of d f the followg lemm d ts roof c be foud [] Lemm 3 For the vector-vlued fucto d relvlued fucto f defed by (3 d (4 the followg sttemets hold: ( s strogly semsmooth (b f s cotuously dfferetble d ts grdet t l ot ( x z R R R s gve by f ( x z ( x z ( x z where s rbtrry elemet belogg to wwwjctcom 683

3 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 I STATIONARY POINT AND NONSINGULARITY CONDITIONS To estblsh qudrtc covergece rte of our lgorthm roosed the ext secto we eed to study the codtos uder whch every elemet of the geerlzed Jcob s full row r t soluto ot of the equto Frst we gve the eeded defto ([] Defto 4 Gve two mtrces tht codto M N m R we sy M N hs the row P -roerty f t stsfes the ( M N R ( M ( N ( M ( N Theorem 4 Suose tht M N hve the row P - roerty d r[ M N] m the for y s of full row r Moreover whe m l s osgulrty Proof: Assume tht s ot of full row r The there s m ozero vector ( u v R such tht ( u v e M u D v N u D v Q u (5 Ths mles tht ( M u ( N u ( Dv ( Dbv b v By b d (6 we hve b (6 ( M u ( N u (7 Suose tht ( M u( N u by r [ M N] m we get u Sce ( uv d u we hve v Wthout loss of geerlt we ssume tht v from (5 u we obt Dv Dv d b whch cotrdcts tht b d b defed Proosto 3 So ( M u( N u (8 By (8 d the row P -roerty of M N we hve tht there ( M u ( N u exsts such tht whch s cotrdcto to (7 So s of full row r Obvousl whe m l s osgulrty By Theorem 3 we ow tht ot ( x y s soluto of the GLCP f d oly f f ( x y z or equvletl ( x y z s globl mmzer wth zero objectve fucto vlue of the ucostred otmzto roblem m R l f ( It s ecessry to estblsh codtos whch gurtees tht every sttory ot of (9 solves the GLCP The followg theorem gves sutble codto Theorem 4 Let ( x y z s sttory ot of (9 f M N hs the row P -roerty d r [ M N] m the ( y Proof: Sce x s soluto of the GLCP (9 s sttory ot of (9 the f ( e ( ( hve s of full row r so ( ( x y s soluto of the GLCP By Theorem 4 we ALGORITHM AND CONERGENCE Moreover I ths secto we formlly stte dmed Guss-Newto (DGN lgorthm whch s smlr to the lgorthm [] For coveece let ( x y z the sequel DGN Algorthm l R d be gve Ste Let Ste If f ( sto; otherwse go to ste3 Ste3 Choose elemet ( set (( where f ( I f ( Ste4 Let be the lrgest elemet the set { / } such tht Ste5 Set f ( f ( f ( Lemm 5 For y ( x z R f ( The gve ( I f ( go to Ste l Suose tht the drecto gve by s scet drecto for f ( t s tht f ( Proof: Obvousl there exst costts d such tht r Thus for y r ( r r r R r R ( r r ( I r ( r Sce f ( ( the d ( wwwjctcom 684

4 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 By ( I f ( r ( we get we hve If we let f ( ( I ( It follows tht ( drecto for f ( f d tht s scet Accordg to Lemm 5 t s esy to sy tht descet drecto of f ( t s d the DGN Algorthm s well defed Obvousl f f ( the s sttory ot of roblem (9 Thus ( x y s soluto of the GLCP uder sutble codtos Theorem 5 Let { } be the sequece determed by the DGN Algorthm the ether { } termtes t sttory ot of f ( or else every ccumulto ot of { } t exsts s sttory ot of f ( Proof: The frst sserto s obvous we rove oly the secod Let l R be ccumulto ot of { } f e there exsts fte subsequece of { } coverges to Wthout loss of geerlt we ssume tht { } coverges to Sce the subdfferetl s uer semcotuous the sequece { } s bouded Wth loss of geerlt we my ssume tht sequece { ( } { f( } f( {( ( { } Sce the f s decresg d bouded from below { f ( } { the we hve Therefore ( } I} {( f ( I ( { } Suose tht ( ( f ( f ( I} I f the ( f ( ( ( I d ostve defte Thus f ( Let such tht f ( (( I f ( be the lrgest elemet the set { / } f ( f ( f ( By the cotuty of f for suffcetly lrge we hve f ( f ( f ( s From the stesze rule of we ow tht f ( f ( f ( f ( f ( f ( f ( Tg the lmt o both sde of ( we get f ( f ( f ( But t s mossble sce ( f ( Therefore f ( e s sttory ot of f The ext result follows mmedtely from Theorem 4 d Theorem 5 Theorem 5 Let { } be the sequece determed by the DGN Algorthm ( x y z be ccumulto ot of { } Suose tht M N hs the row P -roerty d r[ M N] m the ( x y s soluto of GLCP Next we rove the qudrtc covergece of DGN Algorthm Theorem 53 Let { } be the sequece determed by the DGN Algorthm { } Suose tht r be ccumulto ot of { } M N hs the row P -roerty d [ M N] m the we hve the sequece { } coverges to { } qudrtclly Proof: Sce s ccumulto ot of { } exsts fte subsequece of { } coverges to there Wthout loss of geerlt we ssume tht { } coverges to By Theorem 5 we hve ( x y s soluto of GLCP so ( we obt ( f By the roof of Theorem 5 { } { f ( } f ( { } The there exst costts such tht for ll suffcetly lrge d for y ( ( ( Moreover there exsts costt such tht (( I (3 By Lemm 3 we hve ( s strogly semsmooth d by the defto of strog semsmoothess we hve wwwjctcom 685

5 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 ( ( So there exsts 3 such tht ( ( ( ( ( 3 (4 Smlrly to the roof of Theorem 3 [] we c show tht for suffcetly lrge d Thus (( I( (( I( (( I( f ( ( ( ( f ( [ f ( ( ( ] ( [( ( ( ( ( ( ] ( ( [ ( ( ( ] ( Sce the cotuty of f d f ( costt 4 such tht ( f ( f ( 4 (5 there exsts f (6 By ( (6 we hve ( ( ( where 3 4 s costt Therefore we hve { } coverges to I COMPUTATIONAL EXPERIMENTS qudrtclly I the followg we wll mlemet the DGN Algorthm Mtlb d ru t o Petum I comuter Exmle 6 (Murty 988 vrbles M N We te Q q ( x ( s our strtg ot The soluto s x ( The umercl results for ths test roblem c be foud Tble For ths roblem Hrer d Pg ([3] used the dmed- Newto method (DNA d Zhg et l ([4] used the Newto-tye method (NTA The results for the bove two methods d severl vlues of the dmesos re summrzed Tble From Tble d Tble we c coclude tht our lgorthm excels the other two methods lsted bove TABLE Numercl results of our lgorthm for Exmle 6 Dmeso Iterum y ( TABLE Numercl results by DNA NTA Dmeso DNA terum >3 NTA terum Exmle 6 Ths exmle s LCP used by Noor ([4] vrbles M N 4 Q q ( Tble 3 lst the results for ths exmle wth tl ot y N 5 q for dfferet dmesos d rmeter Comred wth the results of Tble 4 [3] we c coclude tht our lgorthm excels methods [3] TABLE 3 Numercl results of our lgorthm for Exmle 6 Dmeso 5 8 Iterum f ( ACKNOWLEDGMENT The uthors would le to th the referees for ther very helful suggestos whch gretly mroved ths er Ths wor ws suorted by the Nturl Scece Foudto Ch (Grt No 76 the Projects for Reformto of Chese Uverstes Logstcs Techg d Reserch (JZW65 d the tol college studets ovto d etrereeursh trg rogrm (457 wwwjctcom 686

6 Itertol Jourl of Comuter d Iformto Techology (ISSN: olume Issue 4 July 3 REFERENCES [] O L Mgsr d J S Pg "The exteded ler comlemetrty roblem" ol [] M C Ferrs d J S Pg " Egeerg d ecoomc lctos of comlemetrty roblems" vol39( [3] F Fcche J S Pg Fte-Dmesol vrtol equlty d com-lemetrty roblems Srger New Yor 3 [4] X Z Zhg F M M d Y J Wg "A Newto tye lgorthm for geerlzed ler comlemetrty roblem over olyhedrl coe" vol [5] D S Brt d M D Brt "The exted ler comlemetrty roblem" vol [6] A Fsher "A secl Newto-tye otmzto method" vol [7] L Q "Covergece lyss of some lgorthms for solvg osmooth equtos" vol [8] L Q d J Su "A osmooth verso of Newto s method" vol [9] F H Clre "Otmzto d osmooth lyss" NewYor NY 983 F Fcche d J Sores "A ew mert fucto for oler comlemetrty roblems d relted lgorthm"vol [] F Fcche d J Sores "A ew mert fucto for oler comlemetrty roblems d relted lgorthm" vol [] R W Cottle J S Pg d R E Stoe "The ler comlemetrty roblem" New Yor NY 99 [] P K Subrm d N H Xu "Covergece lyss of Guss- Newto methods for the comlemetrty roblem" vol [3] P T Hrer d J S Pg "A dmed-newto method for the ler comle-metrty roblem" vol [4] M A Noor Y J Wg d N H Xu "Projecto tertve schemes for geerl vrtol equltes" vol3(34 wwwjctcom 687

MATRIX AND VECTOR NORMS

MATRIX AND VECTOR NORMS Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the