A Min-Max Algorithm for Solving the Linear Complementarity Problem

Transcription

1 Joural of Mathematcal Sceces ad Applcatos, 2013, Vol 1, No 1, 6-11 Avalable ole at Scece ad Educato Publshg DOI: /jmsa A M-Ma Algorthm for Solvg the Lear Complemetarty Problem Youssef ELFOUAYENI 1,2,*, Mohamed KHALADI 1,2 1 UMI UMMISCO, IRD - UPMC, Pars, Frace 2 MPD Laboratory, UCAM, Marrakech, Maroc *Correspodg author: youssef_foutaye@yahoofr Receved December 13, 2012; Revsed March 20, 2013; Accepted March 23, 2013 Abstract he Lear Complemetarty Problem LCP M, q s to fd a vector q 0 IR satsfyg 0, M ad M+q=0, where M as a matr ad q as a vector, are gve data I ths paper we show that the lear complemetarty problem s completely equvalet to fdg the fed pot of the map = ma 0, I-M-q; to fd a appromato soluto to the secod problem, we propose a algorthm startg from ay terval vector X ad geeratg a sequece of the terval vector X k=1 whch coverges to the eact soluto of our lear complemetarty problem We close our paper wth some eamples whch llustrate our theoretcal results Keywords: lear complemetarty problem, m-ma algorthm, fed pot, Brouwer theorem, terval vector, closed bouded cove set 1 Itroducto he Complemetarty Problem oted CP s a classcal problem from optmzato theory of fdg IR such that f f where f, a cotuous operator from IR to tself, s gve data he cotrat f s called the complemetarty codto sce for ay, 1, f > 0 f that = f however, ad vce versa It may be the case hs problem becomes preset the subject of much research mportat because t arses may areas ad t cludes mportat felds, we cte for eample the lear programmg LP, the olear programmg NLP, the cove quadratc programmg ad the varatoal equaltes problems I the case that the fucto f s a olear IR to tself, so t s called a cotuous operator from NoLear Complemetarty Problem assocated wth the fucto f ad oted NLCP Numerous methods ad algorthms est to solve olear complemetarty problems such as the homotopy methods of Merrll [23] ad several other authors see for eample Eaves et al [3,5], Sagal [24], usg a reformulato of the NLCP f due to Magasara [22] whch the zero fdg problem ca be made as s mooth as desred, Watso [26] appled the homotopy or cotuato method of Chow, Mallet-Paret ad Yorke [1] to solve the problem Istead of reformulatg the NLCP f as a zero fdg problem, other authors adjusted smplcal fed pot algorth ms to solve the NLCP f drectly, see eg Fsher ad Gould [17], Garca [18], Kojma [19] or Lüth [21] ad recetly our modest work Elfoutaye ad Khalad [14], I ths paper we have gve a ew method for solvg ths problem whch coverges very rapdly relatve to most of the estg methods ad does ot requre a lot of arthmetc operatos to coverge For ths we have showed that solvg the CP f s equvalet to solvg F where F s a fucto from IR to tself defed by F = f After that we have bult a sequece of smooth fuctos whch s uformly coverget to the fucto F k C F ad we have showed that a appromato of the soluto of the CP f s obtaed by solvg F k for a parameter k large eough For solvg the system of olear equatos F k we have used the Gauss-Sedel-He algorthm he umercal results obtaed ths paper are very favorable ad showed that our method works well for the problems tested I the case that the fucto f s affe, e, ts accurate form as below f = q M, where q s a IR ad M s a real square matr of order elemet of, so t s called a Lear Complemetarty Problem

2 Joural of Mathematcal Sceces ad Applcatos 7 assocated wth the matr M ad the vector q ad oted LCP o solve ths problem, there are several methods ad algorthms, we cte for eample Lemke [20] frst preseted a soluto for ths problem Hs deas were later eploted by Scarf [25] hs work o fed pot algorth ms he relatoshp betwee the LCP q, M ad the fed pot problem s well descrbed by Eaves ad Scarf[6] ad by Eaves ad Lemke [4] Cottle ad Datzg s prcpal pvot method[2] ad recetly our modest works Elfoutaye ad Khalad [7,8] I the frst oe we have bult a teror pot method to solve a lear co mplemetarty problem LCP M, q for some matr M ad q IR ; the covergece of ths method requres L s the legth of a bary codg of the put data of the problem LCP M, q hs teror pot method s globally effcet ad has a good terato complety but t has the problem of fdg a strctly feasble startg pot I the secod oe we have gve a globally coverget hybrd method whch s based o vector dvsos ad the secat method for solvg the ; we have gve ths paper some LCP M, q o umber of teratos where L umercal smulatos to llustrate our theoretcal results, ad to show that ths method ca solve effcetly largescale lear complemetarty problems We have to ote that our paper Elfoutaye ad Khalad [15] we have gve a geeral characterzato of a lear complemetarty problem Furthermore, through ths paper, we ca provde the soluto f t ests of ths problem a straghtforward maer ad accordg to the data Precsely, we have demostrated that LCP M, q has a soluto f ad oly f there s a set X such that the system of lear equatos M[ X ] q[ X ] has a oegatve soluto 0 ad M[ X, X ] q[ X ] 0 ; ths soluto s the gve by f X, ad f X = where N = We have to ote that as oe of the remarks we would lke to pot out that we ca fd may sets X, but a uque soluto ad cotrary, we ca fd a set X but may solutos we must thk of the vertblty of the submatr each case As a fal remark we lke to stress that the LCP has a uque soluto f ad oly f there s a uque set X such that the system of lear equatos M[ X ] q[ X ] has a uque oegatve soluto ad 1 M[ X, X] M[ X] q[ X ] q[ X ] 0 I ths paper we show that the lear complemetarty problem s completely equvalet to fdg the fed pot of the map = ma0, I M q ; to fd a appromato soluto to the secod problem, we propose a algorthm startg from ay terval vector X ad X k geeratg a sequece of the terval vector = 1, whch coverges to the eact soluto of our lear complemetarty problem We close our paper wth some eamples whch llustrate our theoretcal results he paper s orgazed as follows I secto 2 we brefly gve some deftos ad otatos to be used through the paper I secto 3 we wrte LCP the equvalet form of fdg a soluto of a fed pot of the fucto g = ma0, f I secto 4 we gve some umercal eamples ad we gve a cocluso secto5 2 Prelmares I ths secto, we summarze some basc propertes ad related deftos whch be used the followg dscusso I partcular, IR deotes the space of real dmesoal vectors ad IR := { IR : 0, = 1 } the oegatve orthat of IR Let, y IR, y the Eucldea orm ad vector For e 1,,1 s ther er product; s s mamum orm he := s the vector of oes IR IR ad k a oegatve teger, refers to the vector obtaed after k teratos; for 1, refers to the th elemet of, ad refers to the elemet of the vector obtaed after k teratos Let, y IR, the epresso y respectvely < y meag that y respectvely < y for each = 1 he traspose of a vector s deoted by super scrpt, such as the traspose of the vector s gve by X = [ a1, b1 ],,[ a, b ] s a -dmesoal terval vector whch a = a 1,, ad b X = [ a, b] = b 1,, b are two vectors IR a th ; oted by: [IR ] deotes the set of all terval vector IR Let X be a terval vector [ IR ], X X for each = 1,, mea g that Let h be a fucto from IR to [IR ] the h X := [ mh ; mah ] IR ] X X m h X := mh IR ; X ma h X := mah IR X For ay X = [ a, b ] a terval vector [IR ], we defe the fucto from [ IR ] X := b a to IR by:

3 8 Joural of Mathematcal Sceces ad Applcatos he a k lm k otato = X k lm = k lm b = k ad meag that Recall that the spectrum S of the matr A s the set of ts values ad ts spectral radus s gve by A := sup{ such that S} 3 Equvalet Reformulato of CP ad Algorthm Our frst objectve ths secto s to show that f s a soluto of CP the s a fed pot of the map g := ma0, f 2 ad vse versa Proposto 1 Solvg the complemetarty problem CP f s equvalet to fdg the soluto of g = Proof Let the set I := {1 : be a soluto of CP ad let s cosder } ad the complemetarty of I defed by I := {1 : 0} = {1 : f } the we have for each I g ad for each the = ma0, f = ma0, f = I g = ma0, f = ma0, = s a fed pot of the fucto g Now let be a fed pot of the fucto g, the we have 0 Now f I the we have ad, therefore f 0 e, f 0 else f I the we have f I the two cases we have 0, f 0 ad f, therefore g s a soluto of CP hs cocludes the proof Now we propose a algorthm geeratg a sequece k=1, X of the terval vector [IR ] to the fed pot of the fucto g ad tedg For ths, we choce X = [ a, b ] a terval X vector [IR ] so large to esure that We defe the terval vector [IR ] at terato k by X := [ a, b ] such that k a k b 1 1 := ma a := m b,ma0, a,ma0, b m ma f X f X It s easy to show that f X s a soluto of g =, the g X Now we show that Proposto 2 If there s a the set {1,,} such that ma0, b m f X < a or b < ma0, a ma f X the the soluto of 2 s ot estg X Remark If the soluto of 2 s ot estg X, we redefe the algorthm wth a ew X IR] larger tha the older oe 3 4 Proof Assume the cotrary, e, we suppose there s a soluto for each 1 we have thus X of the 2, the g X a b ma0, a ma f X ma0, b m f X k ma0, b m f X a b ma0, a ma f X ths cotradcts 4 Now we show that Proposto 3 If ma0, a ma0, b ma m f X f X a b, e, the there s a soluto of 2 X Proof o prove that let s X = [ a, b ], the we have bouded cove set of the g X ad X s a closed IR, sce the fucto g s a cotuous fucto from X to tself the from Brouwer theorem, we kow that there s a fed pot X satsfed = g Now we show that Lemma For each X terval vector [IR ] have g X X f X we

4 Joural of Mathematcal Sceces ad Applcatos 9 If the Proof For each = 1,, we have g X := ma0, b ma0, a b m f X 0 a ma f X 0 m f X ma f X g X = b m f X a ma f X = X f X If the b m f X 0 a ma f X 0 g X = b m f X If the b m f X a ma f X = X f X b m f X 0 a ma f X 0 g X b a ma f X m f X ad f the = b m f X a ma f X = X f X b m f X 0 a ma f X 0 ma f X a b m f X ths cotradcts the fact that est at least a teger j {1,, } such that m f X < ma f X j hs completes the proof of the lemma Now we provde a theorem to prove the covergece of the algorthm the case where the fucto f s lear, e, q f = M q ; wth M s matr ad IR heorem Let M be a matr satsfy that, f there s a soluto of 2 I M < 1 the X k = lm k Proof Usg the fact 3 we have k1 X g X, 5 ad by prevous lemma t follows that j X 1 X k X f X I M X Further, the matr M satsfed I M < 1 the 1 X k 0 6 O the other had f there s a soluto 0 X the g X 1 X X that of 2, ad from 3 we have ; f we use the smple prcple, we ca deduct herefore from 6 we have X k lm k = Wth the above deas, we suggest the followg algorthm for solvg the lear complemetarty problem Algorthm Italzato k X = [ a, b ] a terval vector [IR ] tolerace ; Iteratve step Step 1 Compute Step 2 Compute m f X := m f X ma f X := ma f X ; k 1 a := ma a,ma0, a ma f X k 1 b := m b,ma0, b m f X Step 3 If there s a the set {1,,} such that k ma0, b m f X < a or b < ma0, a ma f X the the soluto s ot estg ; ad therefore we redefe the algorthm wth a ew X IR] larger tha the older oe ad go to step 1 Else go to step 4 k1 Step 4 If X < the we obta the soluto k1 X ad termate the algorthm X Otherwse k ad retur to step 1 4 Numercal tests I ths secto, we provde umercal eamples to demostrate the effcecy of our algorthm o test the effcecy of our proposed algorthm, we coducted the umercal epermets o some test problems I the followg, we wll mplemet our algorthm Matlab 714 ad ru t o a persoal computer wth a 25 GHZ CPU processor ad 512 MB memory We

5 10 Joural of Mathematcal Sceces ad Applcatos k 1 6 stop the teratos f the codto X 10 s satsfed Eample 1 Let us cosder the followg lear complemetarty problem LCP M, q, fd a vector satsfyg M q 0, 0 ad M q where M = ad 4 3 q = 4 2 he eact soluto of ths problem s 5 5 =,0,, I M = 5 < We have to ote that Whe lookg for a appromato wth s sgfcat dgts, we obta that, our algorthm requres CPU tme s he test results of ths eample are summarzed able 1 able 1 Numercal results for the frst eample wth k=20 X X a b a b b a b a b a b a b a b a b a b he able 1 shows that the umercal results usg a m-ma algorthm to solve ths lear complemetarty problem Eample 2 Cosder the followg class of lear complemetarty problems: For a gve teger, fd a vector IR satsfyg where = 1 M q, Mz q 0 ad 0 m = where s the Kroecker s delta j j/ ad j f j ad q q 1 = 1 1 I M = < q = where We have to ote that 1 hs eample s used by Elfoutaye ad Khalad [14] he eact soluto of ths problem s =,,, 1 2 Whe lookg for a appromato wth s sgfcat dgts, we obta that, our algorthm requres: CPU tme, s, for =3 CPU tme, s, for =5 CPU tme, s, for =7 he test results of ths eample are summarzed able 2 able 2 Numercal results for the secod eample for =3 k X X * [ , ] [ , ] 07 [ , ] [ , ] [ , ] [ , ] for =5 K X X * [ , ] [ , ] [ , ] [ , ] 13 [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] for =7 k X X * [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] 27 [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] [ , ] he able 2 shows that the umercal results usg a m-ma algorthm to solve ths lear complemetarty problem 5 Cocluso I ths paper we have demostrated that, o the oe had, solvg the lear complemetarty problem LCP M, q s equvalet to fdg the fed pot of the system = ma0, I M q ; o the other had, we have descrbed a algorthm for fdg a appromato soluto to the fed pot of the secod problem; ths algorthm startg from ay terval vector X ad geeratg a sequece of the terval vector X satsfyg where s the k=1, lm X k k = eact soluto of lear complemetarty problem he umercal results dcate that our algorthm works relably ad effcetly Refereces [1] S N Chow, J Mallet-Paret, J A Yorke, Fdg zeros of maps: homotopy methods that are costructve wth probablty oe, Math Comp, [2] R W Cottle, G B Datzg, A lfe mathematcal programmg, Math Program, [3] B C Eaves, Homotopes for the computatoal of fed pots, Math Program, [4] B C Eaves, C E Lemke, Equvalece of LCP ad PLS, Math Oper Res, [5] B C Eaves, R Sagal, Homotopes for computatoal of fed pots o ubouded regos, Math Program, [6] B C Eaves, H Scarf, he soluto of systems of pecewse lear equatos, Math Oper Res, [7] Y Elfoutaye, M Khalad, A New Iteror Pot Method for Lear Complemetarty Problem, Appl Math Sc, [8] Y ELFoutaye, M Khalad, Usg vector dvsos solvg the lear complemetarty problem, J Comput Appl Math,

6 Joural of Mathematcal Sceces ad Applcatos 11 [9] Y ELFoutaye, M Khalad, A ZEGZOUI, Proft mamzato of fsherme eplotg two fsh speces competto, submtted for publcato [10] Y ELFoutaye, M Khalad, A Zegzout, A geeralzed Nash equlbrum for a boecoomc porblem of fshg, Studa Iformatca Uversals-HERMANN, [11] Y ELFoutaye, M Khalad, A bo-ecoomc model of fshery where prces deped harvest, J Adv Model Optm, [12] Y ELFoutaye, M Khalad, A geeralzed bo-ecoomc model for competg multple-fsh populatos where prces deped o harvest, J Adv Model Optm, [13] Y ELFoutaye, M Khalad, Prey Swtchg: How to mamze the speces s well beg, J Adv Model Optm, [14] Y ELFoutaye, M Khalad, Gauss-Sedel-He method for solvg a complemetarty problems, submtted for publcato [15] Y ELFoutaye, M Khalad, Geeral Characterzato of a Lear Complemetarty Problem, submtted for publcato [16] Y ELFoutaye, Modélsato et étude mathématque et formatque d u modèle boécoomque d eplotato d espèces mares e compétto, thèse Uversté Cad Ayyad, Marrakech Maroc [17] M L Fsher, P J Gould, A smplcal algorthm for the olear complemetarty problem, Math Program, [18] C B Garca, he complemetarty problem ad ts applcato, Ph D hess, Resselaer Polytechc Isttute, roy, NY 1973 [19] M Kojma, Computatoal methods for solvg the olear complemetarty problem, Keo Egeerg Reports 27, Keo Uversty, Yokohama, Japa 1974 [20] C E Lemke, Bmatr equlbrum pots ad mathematcal programmg, Maag Sc, [21] H J Lüth, A smplcal appromato of a soluto for the olear complemetarty problem, Math Program, [22] O L Magasara, Equvalece of the Complemetarty Problem to a System of Nolear Equatos, SIAM J Appl Math, [23] O H Merrll, Applcatos ad Etesos of a Algorthm that Computes Fed Pots of Certa No-Empty, Cove, Upper Sem-Cotuous Pot to set Mappgs, Dept of Idustral Egeerg, Uv of Mchga echcal Report No [24] R Sagal, O the covergece rate of algorthms for solvg equatos that are based o methods of complemetarty pvotg, Math Oper Res, [25] H Scarf, he appromato of fed pots of a cotuous mappg, SIAM J Appl Math, [26] L Watso, Solvg the Nolear Complemetarty Problem by Homotopy Method, SIAM J Cotrol Optm