A New Second-Order Corrector Interior-Point Algorithm for P (κ)-lcp

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Filomat 3:0 07, 6379 639 https://doi.org/0.9/fil70379k Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat A New Secod-Order Corrector Iterior-Poit Algorithm for P κ-lcp B.

1 Filomat 3:0 07, Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: A New Secod-Order Corrector Iterior-Poit Algorithm for P κ-lcp B. Kheirfam a, M. Chitsaz a a Departmet of Applied Mathematics, Azarbaija Shahid Madai Uiversity, Tabriz, Ira Abstract. I this paper, we propose a secod-order corrector iterior-poit algorithm for solvig P κ- liear complemetarity problems. The method geerates a sequece of iterates i a wide eighborhood of the cetral path itroduced by Ai ad Zhag. I each iteratio, the method computes a corrector directio i additio to the Ai-Zhag directio, i a attempt to improve performace. The algorithm does ot deped o the hadicap κ of the problem, so that it ca be used for ay P κ liear complemetarity problems. It is show that the iteratio complexity boud of the algorithm is O + κ 3 L. Some umerical results are provided to illustrate the performace of the algorithm.. Itroductio The P κ-liear complemetarity problem LCP requires the computatio of a vector pair x, s R satisfyig Mx + s = q, xs = 0, x, s 0, where q R ad M R is a P κ-matrix. The class of P -matrices was itroduced by Kojima et al. [4] ad it cotais may types of matrices ecoutered i practical applicatios. Let κ be a oegative umber. A matrix M is called a P κ-matrix iff it satisfies the followig coditio: + 4κ x i Mx i + x i Mx i 0, x R, i I + i I where I + = {i : x i Mx i 0} ad I = {i : x i Mx i < 0} are two idex sets. The class of all P κ-matrices is deoted by P κ, ad the class P is defied by P = κ 0 P κ, i.e., M is a P -matrix iff M P κ for some κ 0. Obviously, P 0 is the class of positive semidefiite matrices. LCPs have may applicatios, e.g., liear ad quadratic programmig, fidig a Nash-equilibrium i bimatrix games ad calculatig the iterval hull of liear systems of iterval equatios [, 4]. There are a variety of solutio approaches for LCPs which have bee studied itesively. Amog them, the iterior-poit methods IPMs gaied much attetio tha other methods. IPMs ot oly have polyomial complexity but also they are the most effective methods for solvig large scale optimizatio problems. Examples of IPMs that are reliable both i theory ad i practice iclude the primal-dual path-followig 00 Mathematics Subject Classificatio. Primary 90C5; Secodary 90C33 Keywords. Iterior-poit methods, liear complemetarity problem, P κ matrix, wide eighborhood, polyomial complexity Received: 06 April 06; Revised: December 06; Accepted: December 07 Commuicated by Predrag Staimirović addresses: b.kheirfam@azaruiv.edu B. Kheirfam, Chitsaz@azaruiv.edu M. Chitsaz

2 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, methods of Kojima et al. [5] ad sice the may other algorithms have bee developed based o the primaldual strategy. Mizuo, Todd ad Ye [0] proposed the MTY predictor-corrector algorithm. It was the first algorithm for liear optimizatio LO that had both polyomial complexity ad superliear covergece. More precisely, it has O L iteratio complexity ad the duality gap of the sequece geerated by the MTY algorithm coverges to zero quadratically [7]. I 995 [9] Miao exteded the MTY predictor-corrector method for P κ-lcp. His algorithm has O+κ L iteratio complexity ad is quadratically coverget for odegeerate problems. However, the costat κ is explicitly used i the costructio of the algorithm, which implies that the algorithm ca ot be used for sufficiet liear complemetarity problems. Potra ad Sheg [3] geeralized the MTY predictor-corrector method for sufficiet complemetarity problems. Although the algorithms of [3] do ot deped o the costat κ, their computatioal complexity does: if the problem is a P κ-lcp they termiate i at most O + κ L iteratios. The proposed algorithm i [3] has Q-order i odegeerate case, ad.5 i the degeerate case. Predictor-corrector algorithms with higher order of covergece for degeerate sufficiet LCPs were give i [5]. These algorithms have O + κ L iteratio complexity for P κ LCPs [6]. Although some of the above metioed algorithms have optimal iteratio complexity, i practice those which use wide eighborhood perform better. This is oe of the paradoxes of the iterior-poit methods because algorithms which use large eighborhoods of the cetral path are usually more difficult to aalyze ad, i geeral, their computatioal complexity is worse tha the correspodig oe for algorithms usig smaller eighborhoods. Potra ad Liu [] preseted two predictor-corrector methods for sufficiet liear complemetarity problems based o the N wide eighborhood. The first algorithm depeds o the hadicap κ while the secod does ot. Both algorithms are superliearly coverget eve for degeerate problems ad have O + κ +/m L iteratio complexity. I the ext paper, the authors preseted a corrector-predictor algorithm actig i the wide eighborhood N of the cetral path that does ot deped o the hadicap κ of the problem, has O + κ L iteratio complexity, ad is superliearly coverget eve for geeral sufficiet liear complemetarity problems. Cosiderig other wide eighborhoods that are differet from the classical N eighborhood could be a choice for improvig the complexity of iterior-poit methods. I [] Ai ad Zhag itroduced a ew wide eighborhood Ñ τ α of the cetral path. Their algorithm decomposes the classical Newto directio ito two orthogoal directios usig differet step-legth for each of them. Based o Ai ad Zhag idea, Liu ad Liu [6] proposed a primal-dual secod order corrector IPM for liear programmig LP. The mai differece betwee the method i [] ad [6] lies i that at each iteratio, the latter method computes a corrector directio i additio to the Ai-Zhag directio. Later o, the authors geeralized the proposed algorithm i [6] to semidefiite optimizatio SDO [7]. Recetly, Potra [] preseted three iterior-poit algorithms for sufficiet horizotal liear complemetarity problems HLCP actig i a wide eighborhood of the cetral path proposed by Ai ad Zhag. Motivated by the above metioed works, we preset a secod-order corrector iterior-poit algorithm for P κ-lcp based o the Ai-Zhag wide eighborhood, ad we derive the complexity boud for our algorithm. Numerical results show that our algorithm is promisig. The outlie of this paper is as follows. I sectio, we itroduce P κ-lcp ad review some basic cocepts for IPMs for solvig LCPs, such as the cetral path. I sectio 3, we state ad prove some techical lemmas ad the, based o these results, we establish the iteratio complexity boud of the proposed algorithm. Numerical results are preseted i sectio 4. Fially, some coclusios ad remarks are give i sectio 5. The followig otatios are used through the paper. R deotes the -dimesioal Euclidea space. All the vectors are colum vectors ad e deotes the vector with all compoets equal to oe. For ay vectors x ad s, xs deotes compoetwise product Hadamard product of vectors x ad s, ad so is true for other operatios, e.g., if x R +, the x deotes the vector with compoet x i ad xs with compoet xi s i.the positive ad egative parts of a vector v R are defied by v + = max{v, 0} ad v = mi{v, 0}, so that v + 0, v 0 ad v = v + + v.

3 . The Cetral Path ad Wide Neighborhood B. Kheirfam, M. Chitsaz / Filomat 3:0 07, The cocept of the cetral path plays a critical role i the developmet of IPMs. Kojima et al. [4] first proved the existece ad uiqueess of the cetral path for P κ-lcp. Throughout the paper, we assume that P κ-lcp satisfies the iterior-poit coditio IPC, i.e., there exists a pair x 0, s 0 > 0 such that s 0 = Mx 0 + q, which implies the existece of a solutio for P κ-lcp [4]. The set of feasible iterior poits is deoted by F 0 := { x, s R : s = Mx + q, x, s > 0 }. The basic idea of IPMs is to replace the secod equatio i, the so-called complemetarity coditio for P κ-lcp, by the relaxed equatio xs = µe, with µ > 0. Thus, we cosider the system Mx + s = q, xs = µe, x, s 0. Sice M is a P κ-matrix ad the IPC holds, the system has a uique solutio for each µ > 0 cf. Lemma 4.3 i [4]. This solutio is deoted as xµ, sµ ad is called the µ-ceter of P κ-lcp. The set of µ-ceters with all µ > 0 gives the cetral path of P κ-lcp, i.e., C := { x, s F 0 : xs = µe }. It has bee show that the limit of the cetral path as µ goes to zero exists ad yields a solutio for P κ-lcp Theorem 4.4 i [4]. Applyig Newto s method to for a give feasible poit x, s gives the followig liear system of equatios M x s = 0, x s + s x = µe xs. 3 Sice M is P κ-matrix, the system 3 uiquely defies x, s for ay x > 0 ad s > 0. At each iteratio, the method would choose a target o the cetral path ad apply the Newto method to move closer to the target, while cofiig the iterate to stay withi a certai eighborhood of the cetral path. As usual accordig, a eighborhood of the cetral path, the so-called small eighborhood, is defied as N β := { x, s F 0 : xs µe βµ }, where β 0, is a give costat ad µ = xt s. Alteratively, the so-called wide eighborhood is defied as follows: N α := { x, s F 0 : xs αµe }, where 0 < α <. I this paper, we will work with the followig eighborhood cosidered i []: Ñ τ α = { x, s F 0 : xs τµe ατµ }. 4 It is clear that xs τµe = 0, for all x, s N τ, ad that for ay x, s Ñ τ α we have xs τµe ατµ ad x i s i τµ, which imply Therefore, 0 x is i τµ α, ad or equivaletly x is i ατµ. 5 N τ Ñ τ α N ατ α, τ 0,. 6 Sice N τ is a wide eighborhood, so is Ñ τ α. We ote that a iterior poit method from [, 6] use the eighborhood Ñ τ α oly for α 0, ], while i this paper we will costruct a iterior poit method based o the eighborhood Ñ τ α for ay value of α 0,.

4 3. A Large Update Path Followig Algorithm B. Kheirfam, M. Chitsaz / Filomat 3:0 07, I this sectio, we preset a large update path followig method for solvig P k-lcp. Our algorithm geeralizes the large update path followig method proposed i [6] for P k-lcp. Similar to Ai ad Zhag [] ad Liu et al. [6], we decompose the Newto directio 3, from xs to the target o the cetral path τµe large update, ito two separate parts accordig to the positive ad egative parts of τµe xs. The, we solve the followig two systems: ad M x s = 0, s x + x s = τµe xs, M x + s + = 0, s x + + x s + = τµe xs +. 7 Based o the directio from 7, we compute the corrector directio by M x c sc = 0, s x c + x sc = x s. 9 Fially, the ew iterate is give by xθ, sθ := x, s + xθ, sθ := x, s + θ x, s + θ x +, s + + θ xc, s c, 0 where θ = θ, θ, 0 θ, θ is the step legth vector. To get the best step legth for three of the directios, we expect to solve the followig subproblem mi θ [0,] [0,] s.t. µθ xθ, sθ Ñ τ α. We are ow i the positio to describe our secod-order corrector algorithm for P k-lcps. Secod Order Corrector Al orithm Iput : accuracy parameter ɛ > 0; eighborhood parameter α, 0 < α < ; ceterig parameter τ, 0 < τ 0.5; a iitial poit x, s Ñ τ α, µ 0 = x 0 T s 0 /; set k := 0. begi while x k T s k > ɛ do compute the directios x k, sk by 7 ad xk +, sk + by. compute the directios x c,k, sc,k by 9. Fid a step legth vector θ k = θ k, θk givig a sufficiret reductio of µ k ad assurig xθ k, sθ k Ñ τ α. let x k+, s k+ := xθ k, sθ k, µ k+ = x k+ T s k+ /. set k := k +. ed Figure : The algorithm Here, we give two techical lemmas that will be used durig the aalysis. Their proofs are the same to the proofs of the lemmas 3. ad 3.4 i [6].

5 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, Lemma 3.. Let x, s F 0, x, s be the solutio of 7. Note I + = {i I : x i s i > 0} ad I = I\I + The x i s i x is i 4, i I +. Lemma 3.. Suppose that x, s F 0 ad a + xs 0. Let u, v be the solutio of Mu v = 0, su + xv = a. If x + t 0 us + t 0 v > 0 for some 0 < t 0, the x + tus + tv > 0 for all 0 t t 0. It is easy to verify that xθ, sθ as defied i 0 satisfies the followig system: M xθ sθ = 0, s xθ + x sθ = θ τµe xs + θ τµe xs + θ x s. The term a + xs is sometimes called the target to be tracked, ad it is aturally oegative for most IPMs. I particular, for Algorithm i Fig., this property is give i the ext lemma. Lemma 3.3. Let x, s Ñ τ α ad let xθ, sθ be defied by 0. The xθ, sθ Ñ τ α if ad oly if µθ = xθt sθ > 0 ad xθsθ τµθe ατµθ. Proof. Let us assume that µθ > 0 ad xθsθ τµθe ατµθ. Sice M xθ sθ = 0, we obtai sθ = Mxθ + q. I order to complete the proof of lemma, it suffices to show that xθ > 0 ad sθ > 0. From 5 we deduce that xθsθ > 0. Accordig to Lemma 3., it is sufficiet to prove that s xθ + x sθ + xs 0. From 3 it follows that the left-had side of this iequality is equal to xs + θ τµe xs + θ τµe xs + θ x s. 4 We cosider the followig cases: Case : Let i I. I this case, either x i s i τµ or x i s i τµ, the relatio 4 becomes positive. Case : Let i I + ad x i s i τµ. I this case, the ith compoet of 4 becomes x i s i + θ τµ x i s i θ x i s i θ θ xi s i + θ τµ > 0. 4 Case 3: Let i I + ad x i s i τµ. I this case the ith compoet of 4 reduces to x i s i + θ τµ x i s i θ x i s i θ θ xi s i + θ τµ > 0. 4 Note that i case ad case 3 the first iequalities follow by Lemma 3.. These complete the proof. It follows that the optimizatio problem is equivalet to mi θ [0,] [0,] µθ s.t. µθ > 0, xθsθ τµθe ατµθ. If θ + = θ +, θ+ is the solutio of the above miimizatio problem, the accordig to lemma 3.3 the poit xθ +, sθ + belogs to the wide eighborhood Ñ τ α ad the process ca be repeated. Lemma 3.4. cf. Lemma i [3] If LCP is P k, the for ay x, s R ++ ad ay a R the liear system has a uique solutio u, v for which the followig estimates hold uv + κ ã, κ ã ut v 4 ã, where ã = xs / a. 3 5

6 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, Lemma 3.5. The solutios of 7 ad 3 satisfy the followig iequalities: x s + κ µ, κµ x T s µ 4, 6 xθ sθ µ + κ θ + θ α τ α + θ + κ, 7 ατ xθ T sθ κµ θ + θ α τ α + θ ατ + κ, xθ T sθ µ 4 θ + θ α τ α + θ ατ + κ. 9 Proof. For ay x, s R++ there holds xs / τµe xs = xs / xs τµe + xs/ = µ. By cosiderig u, v = x, s ad a = τµe xs i system, usig Lemma 3.4 ad the above iequality, we obtai ad x s + κ xs / τµe xs + κ µ, κµ κ xs / τµe xs xt s 4 xs / τµe xs µ 4. The above iequalities prove 6. If x, s Ñ τ α, the accordig to 5 we ca write xs / τµe xs + τµe xs+ ατµ τµα α. Therefore, usig the orthogoality of τµe xs ad τµe xs + we have xs / θ τµe xs + θ τµe xs + = θ xs / τµe xs xs / τµe xs + µθ + τµα θ α. Further, from 5 ad 6 we get + θ xs / x s x s ατµ µ + κ. ατµ Now, comparig system with the system 3 ad cosiderig u, v = xθ, sθ ad a = θ τµe xs + θ τµe xs + θ x s, we have ã = xs / θ τµe xs + θ τµe xs + θ x s xs / θ τµe xs + θ τµe xs + + θ xs / x s θ + τα θ α + θ ατ + κ µ. 0

7 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, The iequalities 7, ad 9 ca be derived from 0 ad Lemma 3.4. complete. I the sequel, we deote The proof of lemma is ψθ := ψθ, θ := θ + τα θ α + θ ατ + κ. I the ext lemma we give upper ad lower bouds for µθ. We first ote that, from 0 ad the secod equatio of 3, we have Moreover, Sice xθsθ = x + xθs + sθ = xs + s xθ + x sθ + xθ sθ µθ = xθt sθ = xs + θ τµe xs + θ τµe xs + θ x s + xθ sθ. = et xθsθ = µ + θ e T τµe xs + θ e T τµe xs + θ x T s + xθt sθ. 3 τµ = e T xs τµe = e T xs τµe + + e T xs τµe, 4 by applyig the Cauchy-Schwartz iequality ad the defiitio of the eighborhood Ñ τ α, we get e T xs τµe xs τµe ταµ. 5 It follows from 4 ad 5 that τµ e T xs τµe + τ + τα µ. 6 Lemma 3.6. For ay x, s Ñ τ α ad θ = θ, θ with 0 θ θ, we have µθ τ + τα θ + ταθ + κθ + ψθ µ, 7 4 µθ θ τ 4 θ κψθ µ. Proof. By usig 3, the equality τµe xs = xs τµe +, 4, 5 ad 6 we obtai e T xs τµe + e T xs τµe µθ = µ θ θ τµ e T xs τµe = µ θ θ x T s + xθt sθ e T xs τµe θ θ x T s = µ θ τµ θ θ et xs τµe θ µ θ τµ + θ θ ταµ θ x T s + xθt sθ x T s + xθt sθ = µ θ τ + τα ταµ µ + θ θ x T s + xθt sθ τ + τα θ + ταθ + κθ + ψθ µ, 4 + xθt sθ 9

8 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, where the last iequality derives from 6 ad 9, which proves the iequality 7. Similarly, we get µθ = µ θ τµ θ θ et xs τµe θ x T s + xθt sθ µ θ τµ θ x T s + xθt sθ θ τ 4 θ κψθ µ, where the last iequality holds due to 6 ad. This completes the proof of. To follow the cetral path, we eed to make sure that the iterates remai i the prescribed eighborhood of the cetral path. So i the ext lemma we give a upper boud for -orm of xθsθ τµθe. Lemma 3.7. For ay x, s Ñ τ α ad θ = θ, θ with we have ταθ τ + ατ θ θ, 30 xθsθ τµθe θ τα + θ + κ + + κ ψθ µ. Proof. By subtractig ad addig τµe to the right-had side of we obtai xθsθ = τµe + xs τµe θ xs τµe + θ xs τµe θ x s + xθ sθ = τµe + θ xs τµe + + θ xs τµe θ x s + xθ sθ. From the above equality ad 9 we deduce that xθsθ τµθe = θ xs τµe + + θ xs τµe + τθ e T xs τµe + + xθ sθ + τθ O the other had, by 4, 5 ad 30, we have e T xs τµe e + τθ e θ x s x T s e τ xθt sθ e. θ e T xs τµe + + θ e T xs τµe e T xs τµe = θ + θ θ et xs τµe τµθ θ θ ταµ = τ + τα µθ ταµ θ 0. Therefore, we obtai the followig iequality τ x xθsθ τµθe θ xs τµe + θ T s τ xθ T sθ e x s + xθ sθ e,

9 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, which implies, by v u v u ad u u, xθsθ τµθe θ xs τµe + θ x s τ xt s e + xθ sθ τ xθt sθ e θ ταµ + θ x s + xθ sθ θ τα + θ + κ + + κ ψθ µ, where the last iequality follows from 6 ad 7, ad the proof of lemma is complete. Lemma 3.. If 0 < τ, the the poit xθ, sθ defied i 0 belogs to Ñ τ α for ay θ = θ, θ satisfyig ταθ θ = τ,, 0 < θ θ α τ 4 := + 5 κ κ. 3 Proof. We first ote that the first equatio i 3 implies that τθ = ταθ. 3 Now, by usig ad the first equatio of 3, we obtai ψθ α τ + α α + α α τ + κ τθ α + κ α α + α τ = + α τ α τ τ α α + α τ + α + κ = α τ τθ τθ τα + + κ θ α τ 4, 33 where the last iequality follows from τ < ad α α. It follows from ad 33 that µθ µ θ τ κτα + + κ θ 4 θ α τ 4 + κτα + κ θ 4 α τ 4 4 κτα 5 κ + 4 κ 4 5 > 0, κ + 4

10 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, where the secod iequality follows from the iequality of 3 ad θ ad the third iequality follows from 3. Now, by usig Lemma 3.7, 33 ad we deduce that xθsθ τµθ ταµθ = θ τα + θ + κ + + κ θ τα + + κ θ α τ 4 τα θ τ 4 θ κ α τ 4 + κ + τα τα + + κ θ + + κ + τακ 4 α τ κ τ κ α τ 4 θ + 5 κ κ 4 τ + θ α τ 4 α τ κ + τα + ταθ µ ταθ µ < 0, µ τα + + κ θ µ θ τθ τα µ τα ταθ µ τα ταθ µ where the secod iequality follows from the first equatio of 3 ad the followig iequalities + κ + τα κ + + 6, + κ + τακ 5 4 κ Thus uder the hypothesis of our lemma, we showed that xθsθ τµθe ατµθ ad µθ > 0. This completes the proof of lemma accordig to Lemma 3.3. I the remider of this paper, we will use the otatios α τ θ τα θ := + 5 κ κ, θ := τ, θ := θ, θ. 34 Theorem 3.9. If LCP is P κ, the the Algorithm is well defied, produces a sequece of poits x k, s k belogig to the eighborhood Ñ τ α, ad cα, τ µ k+ 5 κ κ µ k, k = 0,,... 35

11 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, where cα, τ = τα α τ 4. 4 Proof. The first part of the theorem follows from the defiitio of Algorithm ad Lemma 3.. Let us deote φθ = φθ, θ = + τ + τα θ + ταθ τα + κ θ + κθ +, 4 α τ 4 so that accordig to 7 we have µθ φθµ. It is easy to verify that ταθ φ τ, θ = τ + τα ταθ τ + ταθ + ταθ + ταθ κ + τ + ταθ τα With the otatio from 34, it follows that µθ + µ θ κ τ + κ τ κ 4 α τ κ 4 α τ κ 4 α τ 4 ταθ ταθ ταθ κ τα α τ τα 5 κ κ ταθ ταθ = cα, τ 5 κ κ µ. ταθ τα 4 5 κ + 4 ταθ This completes the proof. Corollary 3.0. Uder the hypothesis of Theorem 3.9, Algorithm produces a poit x, s Ñ τ α with µx, s ɛ i at most O + κ 3 L iteratios, where L = L ɛ = log µx 0, s 0. ɛ 4. Numerical Results I this sectio, we preset some umerical results for the test problems LCP, i order to get a feel of how the algorithm might perform i practice. Numerical results were obtaied by usig MATLAB R009a, versio , o a 3-bit system. We choose the parameters α = 0.5 ad τ = Furthermore, we

12 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, take q = e Me to obtai a LCP problem with the startig iterior poit x 0, s 0 = e, e. The algorithm termiates if the relative duality gap Relgap satisfies x T s + x 0 T s 0 0. The set of testig LCP problems are geerated as follows. After oe iputs ay positive iteger, MATLAB geerates a matrix A = rad radomly. The, we take M = A T A ad q = e Me. The umerical results preseted i Table. To test the ifluece from the skewess of matrix M, we also test the LCPs geerated as follows: A = rad, B = rad, M = A T A + B B T ad q = e Me. The umerical results of this set of problems are showed i Table. It turs out that the umber of iteratios Iter. ad CPU time secods is better tha those required whe M is purely positive semidefiite. Our prelimiary implemetatios show that this algorithm is promisig. Table Iter. CPU Relgap E E E E E E E E E- Table Iter. CPU Relgap E E E E E E E E E- 5. Coclusio I this paper, we have exteded the recetly proposed secod-order corrector iterior-poit algorithm of Liu et al. for LP to P κ-lcp ad derived the iteratio boud for the algorithm, amely, O + κ 3 log x0 T s 0. Moreover, we use the eighborhood ɛ Ñ τ α for ay value of α 0,, while i [6] this eighborhood has bee used oly for α 0,. Our algorithm does ot use explicitly the hadicap κ of the problem, ad it ca solve ay LCPs. Furthermore, our prelimiary umerical experimets show that the ew algorithm may also perform well i practice. Refereces [] W. Ai, S. Zhag, A O L iteratio primal-dual path-followig method, based o wide eighborhoods ad large updates, for mootoe LCP, SAIM J. Optim , [] S.C. Bullups, K.G. Murty, Complemetarity problems, J. Comput. Appl. Math , [3] F. Gurtua, C. Petra, F.A. Potra, O. Shevcheko, A. Vacea, Corrector-predictor methods for sufficiet liear complemetarity problems, Comput. Optim. Appl. 4 0, [4] M. Kojima, N. Megiddo, T. Noma, A. Yoshise, A Uified Approach to Iterior Poit Algorithms for Liear Complemetarity Problems, Lecture Notes i Comput. Sci. 53, Spriger-Verlag, Berli 99. [5] M. Kojima, S. Mizuo, A. Yoshise, A primal-dual iterior poit for liear programmig, I: Megiddo, N. ed. Progress i Mathmatical Programmig: Iterior Poit ad Related Methods., pp Spriger, New York 99. [6] C. Liu, H. Liu, A O L iteratio primal-dual secod-order corrector algorithm for liear programmig, Optim. Lett. 5 0, [7] C. Liu, H. Liu, A ew secod order corrector iterior-poit algorithm for semidefiite programmig, Math. Meth. Oper. Res. 75 0, [] X. Liu, F.A. Potra, Corrector-predictor methods for sufficiet liear complemetarity problems i a wide eighborhood of the cetral path, SIAM J. Optim , [9] J.A. Miao, Quadratically coverget Oκ + L iteratio algorithm for the P κ-matrix liear complemetarity problem, Math. Program , [0] S. Mizuo, M.J. Todd, Y. Ye, O adaptive-step primal-dual iterior poit algorithms for liear programmig, Math. Oper. Res ,

13 B. Kheirfam, M. Chitsaz / Filomat 3:0 07, [] F.A. Potra, Iterior poit methods for sufficiet horozatal LCP i a wide eighborhood of the cetral path with best kow iteratio complexity, SIAM J. Optim. 4 04,. [] F.A. Potra, X. Liu, Predictor-corrector methods for sufficiet liear complemetarity problems i a wide eighborhood of the cetral path, Optim. Methods Softw , [3] F.A. Potra, R. Sheg, A large-step ifeasible- iterior-poit method for the P -matrix LCP, SIAM J. Optim , [4] U. Schäfer, A liear complemetarity problem with a P-matrix, SAIM Rev , 9 0. [5] J. Stoer, M. Wechs, S., Mizuo, High order ifeasible iterior-poit methods for solvig sufficiet liear complemetarity problems, Math. OPer. Res. 3 99, 3 6. [6] J. Stoer, M. Wechs, Ifeasible-iterior-poit paths for sufficiet liear complemetarity problems ad their aalyticity, Math. Program. 3 99, [7] Y. Ye, O. Güler, R.A. Tapia, Y. Zhag, A quadratically coverget O L-iteratio algorithm for liear programmig, Math. Program , 5 6.

The Limit of a Sequence (Brief Summary) 1

The Limit of a Sequence (Brief Summary) 1 The Limit of a Sequece (Brief Summary). Defiitio. A real umber L is a it of a sequece of real umbers if every ope iterval cotaiig L cotais all but a fiite umber of terms of the sequece. 2. Claim. A sequece