A nonsmooth Levenberg-Marquardt method for generalized complementarity problem

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1 ISSN Egla UK Joural of Iformato a Computg Scece Vol. 7 No. 4 0 pp A osmooth Leveberg-Marquart metho for geeralze complemetarty problem Shou-qag Du College of Mathematcs Qgao Uversty Qgao 6607 Cha Receve May 8 0 accepte October 0 0 Abstract. A ew metho for the soluto of the geeralze complemetarly problem s trouce. he metho s base o a o smooth equato reformulato of the geeralze complemetarly problem a o a o smooth Leveberg-Marquart metho for ts soluto. he metho s show to be globally coverget. Numercal results are also gve. Keywors: Nosmooth Leveberg-Marquart metho; geeralze complemetarty problem; global covergece.. Itroucto Complemetarty theory s a brach of the mathematcal sceces wth a we rage of applcatos ustry physcal regoal a egeerg sceces. I ths paper we coser the followg geeralze complemetarty problem F 0 G 0 F G = 0 where F G : R R are ay two cotuously fferetable fuctos. hs problem s eote GCP F G.see[-]. Several problems arsg fferet fels such as game theory mathematcal programmg mechacs a geometry have the same mathematcal form whch may be state as.. A. covers some relate problems such as f F = x the. reuces to the olear complemetarty problem. I the past years several vestgators have bee cocere wth both the theoretcal a computatoal aspects of the above problem. Several mportat results have bee establshesee [-0]. I ths paper we coser a osmooth Leveberg-Marquart metho wth Golste le search for geeralze complemetarty problem. hs paper s orgaze as follows. I the ext secto we trouce the osmooth Leveberg-Marquart metho a the global covergece of the metho. Fally umercal expermetal results are presete.. New Leveberg-Marquart metho a ts covergece I ths secto we escrbe a osmooth Leveberg-Marquart metho for geeralze complemetarty problem. I paper [] C.Kazow M.Fuushma have stue a ucostrae mmzato reformulato of.. he mert fucto s base o the fucto ϕ a b = a + b a b. he approach presete ths paper s smlar but we use a fferet mert fucto whch s base o the followg fucto m{ a b} where "m" eotes the compoetwse mmum operator. Whe x satsfe m{ F G } = 0 M m{ F G } = 0 x solves. hroughout ths secto we eote Publshe by Worl Acaemc Press Worl Acaemc Uo

2 68 Shou-qag Du: A osmooth Leveberg-Marquart metho for geeralze complemetarty problem h = m{ F G } x R = L H = h L h x R hus the equatos ca be brefly rewrtte as H x = h x L h x = 0 3 whch s osmooth equatos. For solvg the systems of equatos we tae as a tool stea of the Clare geeralze Jacoba B-fferetal a b-fferetal. We gve the followg H for H 3 H = { h L h x R } 4 where h = F f F < G h = F or h = G f F = G h = G f F > G. I what follows we use 4 as a tool stea of the Clare geeralze Jacoba a b-fferetal osmooth Leveberg-Marquart metho. Proposto. Suppose that H a H are efe by 3 a by 4 a all V H are osgular. he there exsts a scalar ξ > 0 such that V ξ V Hx. V ϑ V H x N x ε hols for some costats ϑ > 0 ε > 0 a N x ε s a eghbor of x. By the cotuously fferetable property of F a G. the above Proposto. ca be easly obtae. Deote the correspog mert fucto as ψ x = H. We assume that the above mert fucto s cotuously fferetable. Now we gve the followg osmooth Leveberg-Marquart metho wth Golste le search for geeralze complemetarty problem.. Nosmooth Leveberg-Marquart metho wth Golste le search Step 0. Gve a starg vector x 0 R ρ > 0 p > σ 0 ε 0. Step. If ψ x ε stop. Step. Select a elemet V H x f a approxmate soluto R of the system V V + λ I = V H x 5 where λ 0 s Leveberg-Marquart parameter. If the coto s ot satsfe set = V H x. Step 3. F α by Golste le search Set x p x ρ 6 ψ ψ x σα ψ x ψ x + α + = x + α let : = + a go to Step. ψ x + σ α ψ x ψ x + α I what follows as usual aalyzg the behavor of algorthms we assume that the above metho prouces a fte sequece of pots. Base upo the above metho we gve the followg global covergece result about osmooth Leveberg-Marquart metho wth Golste le search for solvg geeralze complemetarty problem. he ma proof of the followg theorem s smlar to heorem JIC emal for cotrbuto: etor@jc.org.u

3 Joural of Iformato a Computg Scece Vol. 7 0 No. 4 pp [4]. But the H 5 a the le search 7 8 whch s use for the soluto of α s fferg to heorem [4]. heorem. Suppose that the sequece { λ } s boue. he each accumulato pot of the sequece x geerate by the above metho s a statoary pot of ψ. Proof Assume that { x } K x. If there are ftely may K such that = ψ x the the asserto follows mmeately from Proposto.9 a Proposto.6 [0]. Hece we ca assume wthout loss of geeralty that f { x } K s a coverget subsequece of { x } the s always gve by 5. We show that for every coverget subsequece {x } K for whch lm ψ x 0 9 there hols a K lm sup < 0 K lm sup ψ x > 0. K I the followg we assume that x x. Suppose that x s ot a statoary pot of ψ. By 5 we have ψ x = V V + λ I V V + λ I So V ψ x V + λ I Note that the eomator the above equalty s ozero otherwse by. we have ψ x = 0. x woul be a statoary pot a the algorthm woul have stoppe. By assumpto λ I M < + a Proposto. there exsts a costat > 0 such that V V + λ I from the above equalty we get ψ x 3 Formula 0 ow realy follows from the fact that we are assumg that the recto satsfes 6 wth p > whle the graet ψ x s boue o the coverget sequece { x }. If s ot satsfe there exsts a subsequece } of {x } hs mples by 6 that ogether wth 3 mples { x K K lm ψ x = 0. K lm = 0. K lm ψ x = 0. K cotractg 9. he sequece { } s uformly graet relate to { x } accorg to the efto gve [0] a the asserto of the theorem also follows from Proposto 0 a Proposto 7 [0]. We complete the proof.. JIC emal for subscrpto: publshg@wau.org.u

4 70 Shou-qag Du: A osmooth Leveberg-Marquart metho for geeralze complemetarty problem. Remar. Suppose that has a oempty soluto set x R s a soluto of f a oly f ψ x = Numercal results I ths secto orer to show the performace of the above osmooth Leveberg-Marquart metho wth Golste le search we preset some umercal results for the osmooth Leveberg-Marquart metho wth Golste le search. he results cate that the metho wor qut well practce. We coe the algorthms Matlab 7.0. Example 3. We coser the geeralze complemetarty problem. where the fuctos F x x = x x 3x G x x = 4x 0 x + + Both F a G are R R cotuously fferetable fuctos. We use osmooth Leveberg-Marquart metho wth Golste le search to compute Example 3.. Results for Example 3. wth tal pot x 0 = 000 are presete able 3.. able 3. Step ψ e e e e e e e e e e e e e Refereces [] [] [3] [4] C.Kazow M.Fuushma. Equvalece of the geeralze complemetarly problem to fferetable ucostrae mmzato. Joural of Optmzato heory a Applcatos H.Jag M.Fuushma L.Q D Su. A trust rego metho for solvg geeralze complemetarty problems. SIAM Joural o Optmzato C.F.Ma J.ag. he quaratc covergece of a smoothg Leveberg -Marquart metho for olear complemetarty problem. Apple Mathematcs a Computato F.Facche C.Kazow. A osmooth exact Newto metho for the soluto of large-scale olear complemetarty problems. Mathematcal Programmg JIC emal for cotrbuto: etor@jc.org.u

5 Joural of Iformato a Computg Scece Vol. 7 0 No. 4 pp [5] N.Yamashta M.Fuushma. Mofe Newto methos for solvg a semsmooth reformulato of mootoe complemetarty problems. Mathematcal Programmg [6] L.Q P.seg. O almost smooth fuctos a pecewse smooth fuctos. Nolear Aalyss [7] E.G.Brg J.M.Martez. Structure mmal-memory exact quas-newto metho a secat precotoers for augmete Lagraga optmzato. Computatoal Optmzato a Applcatos [8] A.Fscher V.Jeyaumar D..Luc. Soluto pot characterzatos a covergece aalyss of a escet algorthm or osmooth cotuous complemetarty problems. Joural of Optmzato heory a Applcatos [9] B.Che X.Che C.Kazow. A pealze Fscher-Burmester NCP-fucto. Mathematcal Programmg [0] D.P.Bertseas. Costrae optmzato a lagrage multpler metho. Acaemc New Yor NY 98. JIC emal for subscrpto: publshg@wau.org.u