A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problems *

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1 ISSN , England, UK Journal of Information and Computing Science Vol 5, No,, pp A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problem * Lipu Zhang and Yinghong Xu Department of Mathematic, Zhejiang A&F Unierity, Linan, 33, China Department of Mathematic, Zhejiang SCI-ECH Unierity, Hangzhou, 38, China (Receied October,, accepted October 8, ) Abtract In thi paper, we gie a full-newton tep primal-dual interior-point algorithm for monotone horizontal linear complementarity problem he earching direction i obtained by modification of the claic Newton direction, and which alo enjoy the quadratically conergent property in the mall neighborhood of central path he compleity bound i deried, which i n log n Keyword: horizontal linear complementarity problem, interior-point algorithm, full-newton tep, compleity bound Introduction where n A monotone horizontal linear complementarity problem (LCP) i to find a pair, R uch that uw, R M N q,,, () m m n q R and M, N R, moreoer M and N hae the column monotonic property, ie, for any n Mu Nw u w () he formulation () include linear and cone quadratic programming problem epreed by their optimality condition in their uual format Propertie of thi formulation are decribed in [], where the rank r [ M, N] n ha been proed under the monotonic hypothei here are a ariety of olution approache for LCP which hae been tudied inteniely Among them, the interior-point method (IPM) gained much attention than other method Due to the cloe connection between LCP and linear and cone quadratic programming problem, ome IPM for linear and cone quadratic programming problem hae been etended to LCP For intance, Gonzaga et al [,3] tudied the larget tep path following algorithm for LCP and howed that the fat conergence of the implified larget tep path following algorithm Huang et al [] propoed a high-order feaible IPM for LCP with n log iteration Monteiro et al [5] tudied the limiting behaior of the deriatie of certain trajectorie aociated with the monotone LCP Zhang [6] preented a cla of infeaible IPM for LCP and howed that the algorithm ha n log under ome mild aumption Some other releant reference can be found in [,7,8] In thi paper, we gie a full-newton tep IPM for LCP, the algorithm ue a modified Newton direction, which enjoy the nice property of quadratically conergent in the mall neighborhood of central path We * he reearch i upported by Foundation of Zhejiang Educational Committee (NoY9858) and Foundation of Zhejiang SCI- ECH Unierity (No73683) zhanglipu@hueducn Publihed by World Academic Pre, World Academic Union

2 36 Lipu Zhang, et al: A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problem derie the compleity bound for the algorithm, and the compleity reult i the bet-known for LCP he paper i organized a follow: In Section, the baic concept of IPM are gien, which include central path and the claic Newton direction In ection 3, we gie a caled erion of the claic Newton direction, and from which we gie a modified Newton direction he generic algorithm i decribed in ection In ection 5, the propertie of full-newton tep are analyzed, which include the etimation of the upper bound for dual gap and the increae of the proimity after one full-newton tep, the decreae of proimity after the parameter update i alo gien in thi ection At the end of thi ection, we gie a compleity reult for the full-newton tep IPM Section 6 gie a imple numerical eample Section 7 end the paper with a concluion Some notation ued throughout the paper are a follow and denote the -norm and -norm n of a ector repectiely For any,,, n R, min denote the mallet alue of the component of and denote the componentwie (or Hadamard) product of the ector and Preliminary We aume the following hypothee hold: the eitence of an interior feaible olution and the eitence of a trictly complementarity optimal olution he central path he baic idea of the IPM i to replace the econd equation in () by the parameterized equation e, with hu we conider the ytem M N q, e,, (3) he parameterized ytem (3) ha a unique olution for each hi olution i denoted a, and i called the -center of LCP he et of -center (with running through all poitie real number) gie a homotopy path, which i called the central path of LCP If, then the limit of the central path eit and ince the limit point atify the complementarity condition, the limit yield an optimal olution for LCP, ee [6,9] he claic Newton direction he earch direction ued in all primal-dual IPM were computed from the linear ytem M N q e () Neglecting the quadratic term in the left-hand ide epreion of the econd equation, we obtain the o-called claic Newton direction and MN (5) e he unique olution of the ytem (5) i guaranteed by Lemma in [9] 3 New earch direction o decribe the idea underlying thi paper, we need to conider a caled erion of the ytem (5) that define the earch direction 3 A caled-newton direction Now we introduce the caled ector and the caled earch direction According to (6), the ytem (5) can be rewritten a d and d according to and d, d (6) JIC for contribution: editor@jicorguk

3 Journal of Information and Computing Science, Vol 5 () No, pp where M MV X he earch direction Md Nd d d, N NV S, V diag(), X diag( ) and S diag() (8) d and d are obtained by oling (7), o 3 A modified Newton direction Rearrange the econd equation in (7), we obtain d d e taking quare root at both ide the equation, one ha d d e and can be computed ia (6) Uing aylor erie at, which gie the following equation d d e rearrange the aboe equation and ubtitute the econd equation in (7), one obtain the new Newton ytem Md Nd (9) d d e Once ytem (9) i oled, and can be computed ia (6) too It hould be mentioned that the idea of equialent algebraic tranformation aboe wa alo propoed by [] here, the power tranformation t t wa focued on pace Moreoer, we define a proimity meaure to the central path by, ; e () Let u introduce the notation then we hae p d d, q d d p q p q d, d and dd We compare the norm of p and q by the following lemma Lemma 3One ha q p (7) p q () Proof: By the monotonicity property, ee (), one ha MN d d () hu the reult follow Generic Primal-Dual IPM for LCP q e d d e d d d d p, We inetigate a full-newton tep algorithm uing the modified Newton direction It i aumed that we JIC for ubcription: publihing@wauorguk

4 38 Lipu Zhang, et al: A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problem are gien a poitie primal-dual pair, and uch that the ene of the proimity meaure, ; In the algorithm and, denote the modified Newton tep, a defined before Generic primal-dual IPM for LCP Input: A threhold parameter ; an accuracy parameter ; a fied barrier update parameter : a trictly feaible, and / n uch that, ; begin end : ; : ; : while do begin : ; : end : 5 Compleity analyi i cloe to the -center in In thi ection, we derie the compleity bound for the IPM baed on the modified Newton direction 5 Feaibility condition Lemma5 If, and denote and, then the iterate, i trictly feaible Proof: For each, let introduce the notation and, hen we hae by (6), we obtain d d dd Furthermore, from () we get p q p Uing the econd equation of (9) we find that p p ee e and thi relation lead to e p q (3) JIC for contribution: editor@jicorguk

5 Journal of Information and Computing Science, Vol 5 () No, pp Eidently, the inequality Since By Lemma 3, one ha i atified if p q p q p q p q p q p p Hence, for each, we hae Conequently, the ign of the continuou function of, and yield and remain the ame for eery hi complete the proof Hence and 5 Dual gap In the following lemma, we analyze the effect of the full-newton tep on the dual gap Lemma 5Let and hen we hae n () (5) Proof: Obere that making the ubtitution in (3) that equation become q e (6) and uing thi equation we obtain hi implie the lemma eq q e e e n n 53 Quadratic conergence We firt etimate the increae of the proimity after one full-newton tep heorem 53Let, ; and, If hen Hence, the full-newton tep i quadratically conergent Proof We deduce from Lemma 5 that the full-newton tep i trictly feaible, thu and Obere that by (6), we hae hu q e (7) JIC for ubcription: publihing@wauorguk

6 3 Lipu Zhang, et al: A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problem Furthermore, (7) and (8) lead to q q p p min (8) e q e min e, the lat inequality follow from the fact that Conequently, we hae q q p, and thi implie the lemma 5 Proimity change after one iteration After the full-newton tep, a -update will arie the change of -centre We aume that i reduced by the factor in each iterate Lemma 5Let, ; and, where We hae Proof From (7) and (8) we deduce, ; e, ; n e e e q q e n min n which complete the proof, 55 Fiing the parameter We want to find a update parameter and a threhold parameter hu, after each iterate of the algorithm the property, ; i maintained, and hence the algorithm i well defined By Lemma 5, it uffice if n he left ide of the inequality (9) i monotonically increaing according to, it certainly uffice if n (9) () JIC for contribution: editor@jicorguk

7 Journal of Information and Computing Science, Vol 5 () No, pp At thi tate, if we et and aume that n, it uffice if / n () that the inequality () certainly etablih hu the full-newton tep interior-point algorithm well defined for LCP 56 Compleity bound Lemma 55If the barrier parameter ha the initial alue and i repeatedly multiplied by, then after at mot n log iteration, we hae Proof At the initial point, one ha n, after one iterate, by Lemma 5, the dual gap thu, after k So, it uffice if aking logarithm gie Since iterate, the dual gap atifie n, k k k n k n n klog log log log log It certainly uffice if k log nlog log, thi gie n k log, thi complete the proof he following theorem hold triially heorem 56Setting and / n primal-dual IPM for LCP ha the compleity bound n log n Proof Subtitute () in Lemma 55, the reult follow 6 A imple numerical eperiment In general, though there eit, the initial dual gap i, with n, the full-newton tep, for the LCP problem i trictly feaible, we don't know the JIC for ubcription: publihing@wauorguk

8 3 Lipu Zhang, et al: A Full-Newton Step Primal-Dual Interior Point Algorithm for Linear Complementarity Problem alue of, hu we hould modify the ytem (5) a follow We conider the following eample: MN qmn () and N E M, q Without lo of generality, we chooe e a the initial point Setting and / n After 9 iterate, an optimal olution of the eample i gien by 8, and * 69,,,,76, 7 Concluion *,33,9,7,,69 In thi paper, we gae a full-newton tep IPM for LCP, the method ha the quadratically conergent property in the mall neighborhood of central path he compleity bound i the bet-known reult for LCP Although the imple numerical eample how that the algorithm approimate the optimal olution after finite number of iteration, from a practical perpectie they are not efficient hi i becaue they alway perform according to their wort-cae theoretical compleity bound Our further reearch include to find an IPM for LCP with damped-newton tep and to get the ame compleity bound a that with full-newton tep 8 Reference [] JE Bonnan and CC Gonzaga Conergence of interior point algorithm for the monotone linear complementarity problem Math Oper Re (): -5 [] C Gonzaga, J Bonnan Fat conergence of the implified larget tep path following algorithm Math Programming 76()(997), pp95-5 [3] C Gonzaga he larget tep path following algorithm for monotone linear complementarity problem Math Programming 997, 76(): [] ZH Huang Polynomiality of high-order feaible interior point method for oling the horizontal linear complementarity problem Journal of Sytem Science and Mathematical Science, (): 3-38 [5] R Monteiro, uchiya Limiting behaior of the deriatie of certain trajectorie aociated with a monotone horizontal linear complementarity problem Math Oper Re 996, (): [6] Y Zhang On the conergence of a cla of infeaible interior-point method for the horizontal linear complementarity problem SIAM J Optim 99, (): 8-7 [7] S Bellaia, B Morini Global conergence enhancement of claical line earch interior point method for MCP JIC for contribution: editor@jicorguk

9 Journal of Information and Computing Science, Vol 5 () No, pp J Comput Appl Math 3, 5(): 7-99 * [8] GQWang, YQBai Polynomial interior-point algorithm for P horizontal linear complementarity problem J Comput Appl Math 9, 33(): 8-63 [9] M Kojima, N Megiddo, Noma, A Yohie A Unified Approach to Interior Point Algorithm for Linear Complementarity Problem Lecture Note in Computer Science New York: Springer-Verlag, 99, 538 [] Zolt Daray A weighted-path-following method for linear optimization Studia Uni Babe-Bolyai Informatica, XLVII, JIC for ubcription: publihing@wauorguk