A quadratic convergence method for the management equilibrium model

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1 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 A quadraic covergece ehod for he aagee equilibriu odel Jiayi Zhag Feixia School Liyi Uiversiy Feixia Shadog PRChia Absrac i his paper we sudy a class of ehods for solvig he aagee equilibriu odel We firs give a esiae of he error boud for he odel ad he based o he esiae of he error boud propose a ehod for solvig he odel We prove ha our algorih is quadraically coverge wihou he requiree of exisece of a o-degeerae soluio Keywords Maagee equilibriu odel; esiaio of error boud; algorih; quadraic covergece I INRODUCION he aagee equilibriu odel sees a vecor ( x y R such ha x y x y Mx Ny Qz q 0 0( 0 ( Where M N R l Q R q R ad here l exiss z R he odel origiaed fro equilibriu probles i ecooic aagee egieerig ec Applicaios of copleeariy probles fro he field of ecooics iclude geeral Walrasia equilibriu spaial price equilibria ivaria capial soc are equilibriu opial soppig ad gae-heoreic odels I egieerig he copleeariy probles also plays a sigifica role i coac echaics probles srucural echaics probles obsacle probles aheaical physics Elasohydrodyaic lubricaio probles raffic equilibriu probles(such as a pahbased forulaio proble a ulicoodiy forulaio proble ewor desig problesec [] For exaple he equilibriu of supply ad dead i a ecooic syse is ofe depiced as a copleeary odel bewee wo decisio variables As aoher exaple he ypical Walras Law of copeiio equilibriu i ecooic rasacios ca also be covered o copleeary odel bewee price ad excess dead [3] Recely ay effecive ehods have bee proposed o [ 6] solve ( he basic idea of hese ehods is o cover ( io a ucosraied or a siply cosraied opiizaio proble As we ow if he Jacobia arix a a soluio o ( is o-sigular he i is guaraeed ha he Leveberg- Marquard (L-M algorih is quadraically coverge [56] Laely Yaashia ad Fuushia have proved ha he codiio for he local error boud o hold is weaer ha he o-sigulariy of he Jacobia arix [7] his oivaes he esablishe of a error boud for ( he esablishe of LCP error boud has bee exesively sudied (see lieraure review [8] For exaple Magasaria ad Re have give a error boud uder he R0 - arix codiio [9] Clearly ( is a geeralizaio of LCP which props wheher or o he LCP error boud ca be geeralized o ( For his reaso we focus o he esablishe of a error boud for ( desig a sooh algorih for solvig ( usig he error boud ad aalyze he covergece of he algorih as well as he rae of covergece I secio we give priarily a equivale coversio of ( I secio 3 usig a ew residual fucio we esablish a error boud for ( uder ore geeral codiios I secio based o he esablished error boud we propose a sooh algorih for solvig ( ad prove ha he give algorih is quadraically coverge wihou he requiree of exisece of a o-degeerae soluio Copared wih he covergece of algorihs i [56] he codiio is weaer Now we give soe oaios he ier produc of vecors x y R is wrie as x y Le be he Euclidea or For ease of preseaio we wrie ( x y z for colu vecor ( x y z ad use dis( for he shores disace fro vecor o a closed covex se II EQUIVALEN CONVERSION OF HE MANAGEMEN EQUILIBRIUM MODEL We give i his secio a equivale coversio of ( l For coveiece le ( x y z R he ( ca be covered equivalely o he followig proble : Fid such ha A B 0 0 ( A ( B 0 ( M N Q q 0 ( Where A ( I00 B (0 I0 Le be he se of soluios of( ad assue ha i is oepy wwwijacsahesaiorg 97 P a g e

2 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 We have he followig coclusio heore Vecor ( x y R is a soluio o ( l if ad oly if here exiss z R such ha ( x y z is a soluio o ( III ESIMAION OF HE ERROR BOUND OF HE MANAGEMEN EQUILIBRIUM MODEL his secio aily esablishes he error boud for he aagee equilibriu odel Firs we give soe relaed resuls he defiiio of projecio operaor ad i relaed properies heore For a give posiive cosa here exiss a cosa 0 such ha dis ( r( where r( i{ A B l { R ( M N Q q} Proof Assue ha he heore does o hold he here exiss a sequece { }such ha for ay posiive ieger we have ha is dis where ad ( r( 0 r( 0 dis( (3 Sice sequece { } boudedad r( is coiuous ogeher wih(3we have ( r 0 I addiio sequece { } has a coverge subsequece { i } Le i ( where We have he followig coclusio Where is a posiive cosa i i r( ( i i O he oher hadfro (3we have r( r( i i i i i dis( 0( his coradics wih (hece he heore is proved is ( We give i he followig he error boud esablished by Hoffa [0] Lea For a polyhedral coe P { xr D x d D x d} l l where DR DR d R d R exiss a cosa c 0 such ha here dis( x P c[ D x d ( D x d ] x R Now we also give he defiiio of projecio operaor ad is relaed properies [] For a oepy closed covex se S R he orhogoal projecio fro vecor x R oo S is PS ( x : argi{ y x y S} ad i has he followig propery Lea For ay vecors u v R we have Ps( u Ps( v u v Usig heore Lea ad Lea we have he ai resul heore For ay posiive cosa here exiss a cosa 0 such ha dis M N Q q r ( ( ( ( Where r( i{ A B Proof For ay vecor l R here exiss such ha dis( Fro Lea here exiss a cosac 0 such ha dis( c ( M N Q q Furherore r( r( i{ A B} i{ A B} ( A P ( A B [ A P ( A B ] R A( ( P ( A B P ( A B R A( ( A B ( A B A( B B ( A B dis( Where he secod iequaliy is based o Lea Cobied wih he above forula we have R R r( r( ( A B dis( (5 Fro (5 ad heore we have wwwijacsahesaiorg 98 P a g e

3 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 dis( dis dis dis r ( ( ( ( dis( [ r( ( A B dis( ] [ ( A B ] dis( r( [ ( A B ] c ( M N Q q r( [ ( M N Q q r( ] Where ax{[ ( A B ] c } I he followig we usig Fischer fucio ([] o esablish aoher error boud Defie : R R ad ( a b a b a b a b R I has he followig propery: ( a b 0 a 0 b 0 ab 0 I addiio seg [3] gives he followig coclusio Lea 3 [ ] i( a b ( a b ( i( ab For ay vecors a b R defie a vecor-valued fucio ( a b ( ( a b ( a b ( a b Based o his appig ( ca be covered io he followig equaio ( A B ( : 0 ( M N Q q Clearlyusig Lea 3 ad heore i is easy o have he followig resul heore 3 For ay give posiive cosa here exiss a cosa 3 0 such ha dis ( 3 ( As fucio ( x is o sooh le : R R deoe sooh Fisher-Bureiser fucio ( a b a b a b Where 0 is a sooh paraeer For ease of preseaio le ( x y ( ( x y ( x y Where x ( x x y ( y y Ad p( a b ( a b We defie appig F R l : (0 R (0 ha is ( A B F( ( M N Q q Le f ( F( F( F( Obviously F( 0 ( 0 is a soluio o herefore we cosruc a sooh ehod o solve F( 0 ad assue ha he se of soluios o F( 0 is Firs we give he followig properies of p( a b Lea Fucio p( a b properies: [5] has he followig a O R (0 fucio p( a b is coiuously differeiable ad srogly sei-sooh ha is p( a a b b p( a b V ( a b O ( a b ( a b R [0 Where V p( a a b b ad p is he Clare geeralized gradie of p b ( a b R (0 we have ( a b p( a b ( a b ( a b Based o Lea we have he followig resul heore Fucio F( has he followig properies: l a O R (0 fucio F( is coiuously differeiable locally Lipschiz coiuous ad srogly sei-sooh ha is here exis cosas L 0 L 0 b 0 such ha F( F( L ( (6 F( F( H ( l L ( ( R (0 H F( (7 wwwijacsahesaiorg 99 P a g e

4 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 ( N(0 b {( ( b 0} Where F( is he Clare geeralized gradie of F( b For ( 0 here exiss a eighbourhood N(( 0 b {( ( ( 0 b 0} Ad a cosa c 0 for ay We have ( N(( 0 b dis c F (8 (( ( Proof he resul of (i follows fro Lea direcly (ii For ay here exiss a cosa b3 0 such ha Le Where dis ( ( N( b { b } dis( where Fro Lea (ii we have ( ( ( ( ( A B ( : ( M N Q q for ay ( N(( 0 b {( ( ( 0 b} We have dis(( ( ( 0 ( ( ( ( ( ( ( ( ( ( F( where c ( IV ALGORIHM AND CONVERGENCE I his secio we give a sooh ad coverge algorih for solvig ( ad usig he error boud esablished i secio prove he quadraic covergece of he give sooh algorih wihou he codiio of exisece of a o-degeerae soluio Algorih 3 Sep : Choose paraeers (0 0 ad 0 0 l 0 iiial value ( R ad ( Le 0 Sep : Sop if f( ;oherwise ur o Sep 3 Sep 3: Choose he Jacobia arix H of F( ad le d ( be he soluio o he followig sric quadraic prograig Where i ( d s ( d + ( d F( H d d F( Sep : Le : : : ur o Sep I he followig covergece aalysis assue ha Algorih 3 geeraes a ifiie sequece We have he followig resul heore 3 Assue ha Algorih 3 geeraes a sequece{( } If he iiial value is close sufficiely o {( 0} which is a soluio o F( 0 he {dis(( } (9 coverges quadraically o 0 ie sequece { } coverges quadraically o Proof Le : ( : ( 0 For ay iy 0 defie B l ( (0 R ( : ( ( 0 I he followig we prove he heore i hree seps Firs we prove he followig resul If B ( / he wwwijacsahesaiorg 00 P a g e

5 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 d c dis( (0 F( H d c3dis ( ( Where c 0 c3 0 are cosas Le he closes poi i o be ha is dis( Le d As o(9we have ( d is he globally opial soluio ( d ( d ( (3 Sice Hece we have B ( / we have B ( Fro he defiiio of (8 ad( F( c dis( = c ( Usig ( - ( ad ( 7 ogeher wih he defiiio of ( d we ow ha d [/ u ] ( d [/ u ] ( [/ u ][ F( H ( ] [/ u ] F( F( H ( [/ u ] L L / c L / c c dis( / c L Where c L / c Fro he defiiio of ( d we ow he (0 holds F( H d ( d (5 I addiio fro (3(7ad he defiiio of ( d we have ( d ( F( F( H ( L Fro (6we also have F( F( F( L ogeher wih(5 ad(6we have F( H d ( d L L L L ( L 3 c3dis( c Wherec L L 3 (6 Nex for ay aural uber if ( B / here exiss c 0 such ha dis( cdis ( (7 I fac sice B ( / ad d ogeher wih(8we have ha is F( F( H d F( F( H d d L d F( d L d d F( H d (8 Usig (8(8(0ad(we ow dis( X c F( c F d ( c L d c ( d F H c L c dis( c c dis( 3 ( c L c c c dis( 3 c dis( c Where cl c cc 3 wwwijacsahesaiorg 0 P a g e

6 (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 Las we prove ha he codiio of(7holds ha is for a posiive cosa i /[( c ]/( c whe iiial value 0 B ( for ay aural uber we have B ( / We prove he above resul by aheaical iducio Whe 0 fro he way is chose we ow 0 / ad he B ( / Now assue ha 0 B ( / for 0 We prove i he followig B ( / d d d d d d 0 d cdis 0 0 ( where fro (0 he las iequaliy holds I addiio sice 0 ogeher wih(7 we have B ( / dis( c dis( 0 Hece dis( c dis( ( c c dis 0 ( c c c dis 0 c c Fro he above forula ad he way is chose we ow ha /( c /[( c] Ad c dis 0 ( 0 0 c c c c c c 0 0 ( c / Now heore 3 is proved NOE: heore 3shows ha he give sooh algorih has he propery of quadraic covergece wihou he codiio of exisece of a o-degeerae soluio his is a ew resul V CONCLUSIONS I his paper we propose a algorih for solvig he aagee equilibriu odel Uder wihou he requiree of odegeerae soluio we also show ha he algorih is quadraic covergece based o error boud esiaio isead of he osigular assupio jus as was doe i [56] his coclusio ca be viewed as exesio of previously ow resul i [5 6] How o use he algorih o solve he pracical aagee based o he copuer his is a opic for fuure research ACKNOWLEDGMEN he auhors wish o give heir sicere has o he edior ad he aoyous referees for heir valuable suggesios ad helpful coes which iproved he preseaio of he paper REFERENCES [] FERRIS M C PANG J S Egieerig ad ecooic applicaios of copleeariy probles[j] Sociey for idusrial ad applied aheaics (: [] COLE R W PANG J S SONE R E he liear copleeariy proble[m] New Yor: Acadeic press 99 [3] WALRAS L Elees of Pure Ecooics [M] Lodo : Alle ad Uwi 95 [] FACCHINEI F PANG J S Fiie-Diesioal Variaioal Iequaliies ad Copleeariy Probles[M] New Yor: Spriger 003 [5] REN Q J SUN H C A Quadraically Coverge Algorih for he Geeralized Liear Copleeariy Proble[J] Ieraioal Maheaical Foru : [6] SUN H A o-sooh quadraically coverge algorih for solvig horizoal liear copleeariy probles [J] Joural of Sichua Noral Uiversiy 007 5: [7] YAMASHIA N FUKUSHIMA M O he rae of covergece of he Leveberg-Marquard ehod[j] Copuig [Suppl] 00 5: 39-9 [8] PANG J S Error bouds i aheaical prograig [J] Mah Prograig : [9] MANGASARIAN O L REN J New iproved error boud for he liear copleeariy proble[j] Mah Prograig 99 66: - 55 [0] HOFFMAN A J o he approxiae soluios of liear iequaliies[j] J Res Naioal Bureau of Sadards 95 9: [] ZARANONELLO E H Projecios o Covex Ses i Hilber Space ad Specral heory Coribuios o Noliear Fucioal Aalysis New Yor: Acadeic Press 97 [] FISCHER A A special Newo-ype opiizaio ehod[j] Opi 99 : 69-8 [3] SENG P Growh behavior of a class of eri fucio for he oliear copleeariy proble[j] J Opi heory Appl : 7-37 [] ENGELKE S KONZOW C Iproved soohig ype ehod for he soluio of liear progras[j] Prepri Isiue of Applied Maheaics Uiversiy of Haburg Haburg March 000 [5] QI L SUN D Soohig fucio ad soohig Newo ehod for copleariy probles ad variaioal iequaliy probles[j] Repro Uiversiy of New Souh Wales Sydey Ausralia 998 wwwijacsahesaiorg 0 P a g e

EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar

Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded