2 The LCP problem and preliminary discussion

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1 Iteratioal Mathematical Forum, Vol. 6, 011, o. 6, Polyomial Covergece of a Predictor-Corrector Iterior-Poit Algorithm for LCP Feixiag Che College of Mathematics ad Computer Sciece, Chogqig Three Gorges Uiversity, Wazhou, Chogqig, , P.R. Chia cfx00@16.com Abstract We establishe the polyomial covergece of a ew class of pathfollowig methods for liear complemetarity problems (LCP). Namely, we show that the predictor-corrector methods based o the L orm eighborhood. Mathematics Subject Classificatio: 90C33, 65G0, 65G50 Keywords: iterior-poit algorithm; polyomial complexity; path-followig methods; liear complemetarity problems 1 Itroductio LCPs arise i may areas, such as quadratic programmig, bimatrix games, variatioal iequalities, ad ecoomic equilibria problems, ad they have bee the subject of much research iterest. A umber of direct as well as iterative methods have bee proposed for their solutio. The book by Cottle, Pag ad Stoe [1] is a good referece for pivotig methods developed to solve LCPs. Aother importat class of methods used to tackle LCPs are the iterior-poit ad ifeasible-iterior-poit methods, which were first desiged to solve liear programs [,3,4,5]. The LCP problem ad prelimiary discussio Liear complemetarity problems(lcp)determies a vector pair (x, z) satisfyig Mx c = z, x T z =0, (x, z) R + R +, (1)

2 19 Feixiag Che where c R ad M S+. LCP (1) is equivalet to the followig oliear system with oegative costraits. ( ) Mx z c F (x, z) = =0, (x, z) R+ XZe R +. It is straightforward to calculate [ ] F M I (x, z) =. Z X The Jacobia matrix F (x, z) is osigular for all (x, z) > 0. Its proof will be give later. Problem (1) ca be formulated as the miimizatio problem miimize x T z s.t Mx z = c, (x, z) R + R +. The set of feasible solutio ad the iterior feasible solutio of (1) are F = {(x, z) R+ R +,z = Mx c}, F 0 = {(x, z) R ++ R ++,z = Mx c}, respectively. Throughout this paper, we assume that F ad F 0 are ot empty. The path-followig algorithms studied i this paper are based o the followig cetrality measures of a poit for (x, z) R + R + : d(x, z) = Xz μe, where μ = xt z. The path-followig methods are based o the followig cetral path eighborhoods: where η>0 is a give costat. N(η) ={(x, z) F 0, Xz μe ημ}, 3 The Newto directio It is ow easy to see that the Newto directio (Δx, Δz) for system (3) is the solutio of the followig system of liear equatios: { XΔz + ZΔx = ημe Xz MΔx Δz =0. ()

3 Predictor-corrector iterior-poit algorithm for LCP 193 Lemma 3.1 For ay vectors p, q > 0, let the fuctio f(μ) = Pq μe, the μ = pt q is the miimizer of f(μ), where P = diag(p). Moreover, Pq p T q Pq. proof. f(μ) = Pq μ = i=1 (p i q i μ) = μ i=1 (p i q i )μ+ i=1 p i q i. Observig the quadratic equatio, the coclusio is obvious. Lemma 3. Let M S+, vectors x R ++, z R ++, the the matrix [ Z X ] M I is osigular. proof. Coversely, let the matrix above is sigular, the there exists vector (d 1,d ) R 0, satisfies Zd 1 + Xd =0, Md 1 + d =0. This implies X(X 1 Z +M)d 1 =0. Cosiderig the fact that X, ad X 1 Z +M are defiite, we obtai d 1 = 0 ad d = 0. This is a cotradictio with that (d 1,d ) R 0, the we complete the proof. The proof of ext lemma is straightforward ad therefore we omit the details. Lemma 3.3 For the algorithm above, the followig equatios hold (1) x(θ) T z(θ) =(1 θ(1 η))x T z + θ Δx T Δz; () X(θ)z(θ) μ(θ) =(1 θ)(xz xt z e)+θ [ΔXΔz ΔxT Δz e], where x(θ) =x + θδx, z(θ) =z + θδz,μ(θ) =x(θ) T z(θ)/. 4 Predictor-corrector algorithm I this sectio we give the predictor-corrector algorithm which geerates iterates i the followig eighborhood of the cetral path: N(η) ={(x, z) F 0, Xz μe ημ}, where μ x T z/ ad η is a costat such that η (0, 1). ALGORITHM I Let (x 0,z 0 ) N(η), η (0, 1/4), ad set k =0. Repeat util (x k ) T z k ε do 1. Predictor step: Set (x, z) = (x k,z k ) ad compute the solutio (Δx, Δz) of system (4) with η = 0; compute the largest ˆθ so that (x(θ),z(θ)) N(η), θ [0, ˆθ] where x(θ) =x + θδx, z(θ) =z + θδz.

4 194 Feixiag Che. Corrector step: Set (x,z )=(x k + ˆθΔx, z k + ˆθΔz) ad compute the solutio (Δx, Δz ) of system (4) with η = 1; set (x k+1,z k+1 )=(x +Δx,z + Δz ). 3. Let k:=k+1 ad retur to step 1. Ed The followig lemma establishes some importat bouds which play a importat role i the covergece aalysis of the Algorithm I. Lemma 4.1 With the otatios above, the followig equalities hold (1) (x ) T z =(1 ˆθ)x T z + ˆθ Δx T Δz; () (x k+1 ) T z k+1 =(x ) T z +(Δx ) T Δz. proof. The two statemets follow immediately from lemma.4(1) with η = 1 ad θ =1. Lemma 4. With the otatios above, the followig iequalities hold (1) Δx T Δz (x) T z/4; () (Δx ) T Δz (x ) T z /8. proof. Defie D =(X) 1/ (Z) 1/, μ =(x) T z/, Multiplyig system (4) o both sides by (XZ) 1/, we have D 1 Δz + DΔx = (XZ) 1/ (ημe Xz). Takig L -orm squared we obtai D 1 Δz + DΔx +Δx T Δz = (XZ) 1/ (ημe Xz). Usig 4Δx T Δz D 1 Δz + DΔx +Δx T Δz, we ca easily coclude that 4Δx T Δz (XZ) 1/ (ημe Xz) μ. From η = 0, we complete the first part. Whe η = 1, we ca obtai the we complete the proof. 4(Δx ) T Δz (X Z ) 1/ (μ e X z ) (X Z ) 1/ (μ e X z ) 4η μ μ 1 η Lemma 4.3 With the otatios above, the followig iequalities hold, the followig relatio holds: μ k+1 (1 + 1 ˆθ )(1 8 ) μ k. proof. From lemma 3.1() ad lemma 3.(), we obtai (x k+1 ) T z k+1 = (x ) T z +(Δx ) T Δz ( )(x ) T z (1 + 1 ){(1 ˆθ)(x k ) T z k + ˆθ Δx T Δz}. 8 (1 + 1 ˆθ )(1 8 ) (x k ) T z k.

5 Predictor-corrector iterior-poit algorithm for LCP 195 The, we complete the proof. By lemma 3.3, we ca obtai that the improvemet of the objective value depeds o the size of ˆθ, so we wish to boud ˆθ from below. Lemma 4.4 Let vectors p, q R such that p T q 0, the the followig iequality holds Pq 4 p + q. Theorem 4.5 With the otatios above, we let ˆθ = max{θ (0, 1], (x(θ),s(θ)) N(η)}, the (1) ˆθ ΔXΔs/μ /η ; () ΔXΔs/μ. 4 proof. Usig lemma.4, we have the followig iequality: X(θ)z(θ) μ(θ)e (1 θ)(xz μe) + θ ΔXΔz ΔxT Δze (1 θ)(xz μe) + θ ΔXΔz (1 θ)ημ + θ ΔXΔz. We see that for 0 ˆθ ΔXΔs/μ /η ; X(θ)z(θ) μ(θ)e (1 θ)ημ + θ ΔXΔz η(1 θ). This because the quadratic term i θ: ΔXΔz/μ θ +ηθ η 0 for θ betwee zero ad the root η + η +4 ΔXΔs/μ /η = ΔXΔs/μ ΔXΔs/μ /η. Thus, X(θ)z(θ) μ(θ)e η(1 θ)μ = ημ(θ) or (x(θ),s(θ)) N(η) for 0 ˆθ ΔXΔs/μ /η. From D =(X 1/ )S 1/, the we obtai (DΔx) T D 1 Δs =(Δx) T Δs 0 ad D 1 Δs + DΔx =(XS) 1/ ( Xs) with η = 0. Usig lemma 3.4 we obtai We complete the proof. ΔXΔs/μ 4.

6 196 Feixiag Che Theorem 4.6 Suppose that η (0, 1/4). The, every iterate (x k,s k ) geerated by Algorithm-I is i N(η) ad satisfies (x k ) T s k ( )( ) k (x 0 ) T s 0. (3) Moveover, Algorithm-I termiates i at most O( logε 1 ) iteratios. proof. A a argumet similar to the oe used i lemma 3., we have ΔX Δs 4 (X S ) 1/ (μ e X s ) 4 4η μ 1 η ημ The, from lemma.4, X (θ)z (θ) μ (θ)e (1 θ)(x z μ e) + θ ΔX Δz (ΔxT ) Δz e (1 θ)(x z μ e) + θ ΔX Δz (1 θ)ημ + θ ημ ημ ημ (θ). hece, we get X (θ)z (θ) > 0 for every θ (0, 1], whe settig θ =1,we coclude the the first part. Relatio (5) follows from the lemma 3.3 ad lemma 3.5. Refereces [1] R. Cottle, J. S. Pag ad R. E. Stoe, The Liear Complemetarity Problem. SIAM, 009. [] M. Kojima, N. Megiddo, S. Mizuo, A primaldual ifeasible-iterior-poit algorithm for liear programmig, Mathematical Programmig, vol.61, pp.63-80, [3] S. Mizuo, Polyomiality of ifeasible-iterior-poit algorithms for liear programmig. Mathematical Programmig, vol.67, pp , [4] F. A. Potra, A quadratically coverget predictorcorrector method for solvig liear programs from ifeasible startig poits. Mathematical Programmig vol.67, pp , [5] U Schäfer, A liear complemetarity problem with a P-matrix. SIAM review, vol.46, No., pp , 004. Received: Jauary, 011