A new error bound for linear complementarity problems for B-matrices

Transcription

1 Electroic Joural of Liear Algebra Volume 3 Volume 3: (206) Article A ew error boud for liear complemetarity problems for B-matrices Chaoqia Li Yua Uiversity, lichaoqia@yueduc Megtig Ga Shaorog Yag Follow this ad additioal works at: Part of the Numerical Aalysis ad Computatio Commos Recommeded Citatio Li, Chaoqia; Ga, Megtig; ad Yag, Shaorog (206), "A ew error boud for liear complemetarity problems for B-matrices", Electroic Joural of Liear Algebra, Volume 3, pp DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository For more iformatio, please cotact scholcom@uwyoedu

2 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July 206 A NEW ERROR BOUND FOR LINEAR COMPLEMENTARITY PROBLEMS FOR B-MATRICES CHAOQIAN LI, MENGTING GAN, AND SHAORONG YANG Abstract A ew error boud for the liear complemetarity problem is give whe the ivolved matrix is a B-matrix It is show that this boud improves the correspodig result i [M García- Esaola ad JM Peña Error bouds for liear complemetarity problems for B-matrices Appl Math Lett, 22:07 075, 2009] i some cases, ad that it is sharper tha that i [CQ Li ad YT Li Note o error bouds for liear complemetarity problems for B-matrices Appl Math Lett, 57:08 3, 206] Key words Error boud, Liear complemetarity problem, B-Matrix AMS subject classificatios 90C33, 65G50, 65F35 Itroductio A liear complemetarity problem LCP(M, q) tries to fid a vector x R such that () x 0, Mx+q 0, (Mx+q) T x = 0, where M = [m ij ] R ad q R The LCP(M,q) has various applicatios i the Nash equilibrium poit of a bimatrix game, the cotact problem ad the free boudary problem for joural bearig; for details, see [4, 5, 7] It is well-kow that the LCP(M,q) has a uique solutio for ay q R if ad oly if M is a P-matrix [5] Here, a matrix M R is called a P-matrix if all its pricipal miors are positive [6] I [3], Che ad Xiag gave the followig error boud of the LCP(M,q) whe M is a P-matrix: x x d [0,] (I D +DM) r(x), where x is the solutio of the LCP(M,q), r(x) = mi{x,mx + q}, D = diag(d i ) with 0 d i, ad the mi operator r(x) deotes the compoetwise miimum of two vectors If M satisfies certai structure, the some bouds of d [0,] (I D+ DM) ca be derived; for details, see [2, 7, 8, 0, 4] ad refereces therei Received by the editors o March 4, 206 Accepted for publicatio o July 2, 206 Hadlig Editor: Michael Tsatsomeros School ofmathematics ad Statistics, Yua Uiversity, Kumig, Yua, 65009, PRChia (lichaoqia@yueduc) 476

3 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July 206 A New Error Boud for Liear Complemetarity Problems for B-Matrices 477 Whe M is a B-matrix itroduced by Peña i [6] as a subclass of P-matrices, García-Esaola ad Peña i [0] preseted the followig upper boud which is oly related with the etries of M Here a real matrix M = [m ij ] R is called a B-matrix [6] if for each i N = {,2,,}, (2) ( ) m ik > 0, ad m ik > m ij for ay j N ad j i k N k N Theorem [0, Theorem 22] Let M = [m ij ] R be a B-matrix with the form (3) where (4) B + = [b ij ] = M = B + +C, m r + m r + m r + m r +, ad r + i = {0,m ij j i} The (5) d [0,] (I D+DM) mi{β,}, where β = mi {β i} ad β i = b ii b ij i N As show i [5], if the diagoal domiace of B + is weak, ie, β = mi {β i} = mi i N i N b ii b ij is small, the the boud (5) maybe verylargewhe M is ab-matrix, which leadsto that the estimate i (5) is alwaysiaccurate, for details, see [5, 6] To improve the boud (5), Li ad Li [5] gave the followig boud for d [0,] (I D+DM) whe M is a B-matrix Theorem 2 [5, Theorem 4] Let M = [m ij ] R be a B-matrix with the form M = B + +C, where B + = [b ij ] is the matrix of (4) The (6) d [0,] (I D+DM) mi { βi, } + βj b jk,

4 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July CQ Li, MT Ga, ad SR Yag where β i = b ii j=i+ b ij l i (B + ), l k (B + ) = + βj k i b ii j=k, b jk = if i = b ij ad Very recetly, whe M is a weakly chaied diagoally domiat B-matrix, Li ad Li [6] gave a boud for d [0,] (I D + DM) This boud holds true for the case that M is a B-matrix because a B-matrix is a weakly chaied diagoally domiat B-matrix [6] Theorem 3 [6, Corollary ] Let M = [m ij ] R be a B-matrix with the form M = B + +C, where B + = [b ij ] is the matrix of (4) The where β i = b ii d [0,] (I D+DM) j=i+ b ij > 0 ad mi{ β i,} = if i = I this paper, we also focus o the error boud for the LCP(M,q), ad gave a ew boud for d [0,] (I D+DM) whe M is a B-matrix It is show that this boud is more effective to estimate d [0,] (I D + DM) tha that i Theorem, ad sharper tha those i Theorems 2 ad 3, 2 Mai results We first recall some defiitios A matrix A = [a ij ] C is called a strictly diagoallydomiat (SDD) matrix iffor eachi N, a ii > a ij It is well-kow that a SDD matrix is osigular [] A matrix A = [a ij ] is called a Z-matrix if a ij 0 for ay i j, ad a osigular M-matrix if A is a Z-matrix with A beig oegative [] Next, several lemmas which will be used later are give Lemma 2 [8, Theorem 32] Let A = [a ij ] R be a SDD M-matrix The A, a ii ( u i (A)l i (A)) u j (A)l j (A),

5 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July 206 where u i (A) = a ii A New Error Boud for Liear Complemetarity Problems for B-Matrices 479 j=i+ a ij, l k (A) = k i a ii j=k, = if i = u j (A)l j (A) a ij ad (2) ad (22) Lemma 22 [5, Lemma 3] Let γ > 0 ad η 0 The for ay x [0,], x+γx mi{γ, } ηx x+γx η γ Lemma 23 [6, Lemma 5] Let A = [a ij ] R with a ii > The for ay x i [0,], i N, j=i+ x i +a ii x i x i +a ii x i a ij for each i N j=i+ a ij x i a ii a ii j=i+ a ij Theorem 24 Let M = [m ij ] R be a B-matrix with the form M = B + +C, where B + = [b ij ] is the matrix of (4) The (23) d [0,] (I D +DM) where β i is defied i Theorem 2 ad Proof Let M D = I D+DM The mi { βi, } = if i =, M D = I D+DM = I D +D(B + +C) = B + D +C D, where B + D = I D + DB+ = [b ij ] Similarly to the proof of Theorem 22 i [0], we ca obtai that B + D is a SDD M-matrix with positive diagoal elemets ad C D = DC, ad that (24) M D (I +(B + D ) C D ) (B + D ) ( ) (B + D )

6 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July CQ Li, MT Ga, ad SR Yag By Lemma 2, (25) (B + D ) ( ( d i +b ii d i ) ( u i (B + D )l i(b + D )) u j (B + D )l j(b + D ), where u i (B + D ) = b ij d b ij d i i j=k, j=i+, ad l k (B + D d i +b ii d ) = i k i d i +b ii d i By Lemma 22, we ca easily get that for each k N, l k (B + D ) (26) b ij k i b ii = l k(b + ) <, ad that for each i N, (27) j=k, ( d i +b ii d i ) ( u i (B + D )l i(b + D )) = d i +b ii d i Furthermore, by Lemma 23, = mi { b ii mi { βi, } j=i+ j=i+ b ij d i l i (B + D ) b ij l i (B + ), } (28) u i (B + D )l i(b + D ) = d i +b ii d i d i +b ii d i b ii b ij d i l i (B + D ) β i j=i+ By (25), (26), (27) ad (28), we have (29) (B + D ) mi { β, } + i=2 mi { βi, } The coclusio follows from (24) ad (29)

7 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July 206 A New Error Boud for Liear Complemetarity Problems for B-Matrices 48 The comparisos of the bouds i Theorems 2, 3 ad 24 are established as follows Theorem 25 Let M = [m ij ] R be a B-matrix with the form M = B + +C, where B + = [b ij ] is the matrix of (4) Let β i ad β i be defied i Theorems 2 ad 3 respectively The mi { βi, } mi { βi, } + βj mi{ β i,} b jk Proof Note that ad l k (B + ) = (20) k i β i = b ii b ii j=i+ j=k, Meatime, for j =,2,,, (2) + βj b ij, βi = b ii j=i+ b ij l i (B + ) b ij < Hece, for each i N, β i β i ad mi{ β i,} mi{ β i,} b jk + βj By (20) ad (2), we have mi { βi, } + βj (22) Moreover, for j =,2,,, b jk = βj βj + b jk b jj b jk l j (B + )+ = β j + b jk l j (B + ) = mi{ β i,} b jk l j (B + ) b jk =

8 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July CQ Li, MT Ga, ad SR Yag this implies (23) = + + mi { βi, } b jk l j (B + ) b jk, mi { βi, } + βj The coclusio follows from (22)ad (23) Example 26 Cosider the family of B-matrices i [5]: M k = 08 0 k k+ 8 07, b jk where k The M k = B + k +C k, where B + k = k k By Theorem (Theorem 22 i [0]), we have It is obvious that By Theorem 3, we have d [0,] 4 (I D +DM k) 4 = 30(k +) mi{β,} d [0,] 4 (I D +DM k) By Theorem 2, we have 30(k +) + whe k mi{ β i,} d [0,] 4 (I D +DM k) 297(90k+9)(90k+92)+624(00k+0)2 099(90k+9) 2,

9 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July 206 A New Error Boud for Liear Complemetarity Problems for B-Matrices 483 ad 297(90k+9)(90k+92)+624(00k+0) 2 By Theorem 24, we have 099(90k+9) 2 < 52675, for ay k d [0,] 4 (I D +DM k) 297(90k+9)(90k+9)+597(00k+00)2 099(90k+9) 2, ad I particular, whe k =, ad Whe k = 2, ad 297(90k+9)(90k+9)+597(00k+00) 2 099(90k+9) 2 < 297(90k+9)(90k+92)+624(00k+0)2 099(90k+9) 2 297(90k+9)(90k+9)+597(00k+00) 2 099(90k+9) , 297(90k+9)(90k+92)+624(00k+0) 2 099(90k+9) (90k+9)(90k+9)+597(00k+00) 2 099(90k+9) 2 370, 297(90k+9)(90k+92)+624(00k+0) 2 099(90k+9) I these two cases, the bouds i (5) are equal to 60 (k = ) ad 90 (k = 2), respectively This example shows that the boud i Theorem 24 is sharper tha those i Theorems, 2 ad 3 3 Coclusios I this paper, we give a ew boud for d [0,] (I D + DM) whe M is a B-matrix, ad show that it improves the boud of Theorem 22 of [0] i some cases, ad that it is always sharper tha those of Theorem 4 of [5] ad of Corollary of [6]

10 Electroic Joural of Liear Algebra ISSN Volume 3, pp , July CQ Li, MT Ga, ad SR Yag Ackowledgmet The authors are grateful to the referees for their useful ad costructive suggestios This work is supported by Natioal Natural Sciece Foudatios of Chia (36074) ad Natural Sciece Foudatios of Zhejiag Provice (LY4A00007) REFERENCES [] A Berma ad RJ Plemmos Noegative Matrix i the Mathematical Scieces SIAM Publisher, Philadelphia, 994 [2] TT Che, W Li, X Wu, ad S Vog Error bouds for liear complemetarity problems of MB-matrices Numer Algorithms, 70(2):34 356, 205 [3] XJ Che ad SH Xiag Computatio of error bouds for P-matrix liear complemetarity problem Math Program, 06:53 525, 2006 [4] XJ Che ad SH Xiag Perturbatio bouds of P-matrix liear complemetarity problems SIAM J Optim, 8: , 2007 [5] RW Cottle, JS Pag, ad RE Stoe The Liear Complemetarity Problem Academic Press, Sa Diego, 992 [6] JM Peña A class of P-matrices with applicatios to the localizatio of the eigevalues of a real matrix SIAM J Matrix Aal Appl, 22: , 200 [7] PF Dai Error bouds for liear complemetarity problems of DB-matrices Liear Algebra Appl, 434: , 20 [8] PF Dai, YT Li, ad CJ Lu Error bouds for liear complemetarity problems for SBmatrices Numer Algorithms, 6:2 39, 202 [9] PF Dai, CJ Lu, ad YT Li New error bouds for the liear complemetarity problem with a SB-matrix Numer Algorithms, 64(4):74 757, 203 [0] M García-Esaola ad JM Peña Error bouds for liear complemetarity problems for B-matrices Appl Math Lett, 22:07 075, 2009 [] M García-Esaola ad JM Peña Error bouds for the liear complemetarity problem with a Σ-SDD matrix Liear Algebra Appl, 438(3): , 203 [2] M García-Esaola ad JM Peña Error bouds for liear complemetarity problems ivolvig B S -matrices Appl Math Lett, 25: , 202 [3] M García-Esaola ad JM Peña Error bouds for liear complemetarity problems of Nekrasov matrices Numer Algorithms, 67: , 204 [4] M García-Esaola ad JM Peña B-Nekrasov matrices ad error bouds for liear complemetarity problems Numer Algorithms, 72(2): , 206 [5] CQ Li ad YT Li Note o error bouds for liear complemetarity problems for B-matrices Appl Math Lett, 57:08 3, 206 [6] CQ Li ad YT Li Weakly chaied diagoally domiat B-matrices ad error bouds for liear complemetarity problems Numer Algor, DOI: 0007/s [7] KG Murty Liear Complemetarity, Liear ad Noliear Programmig Helderma Verlag, Berli, 988 [8] P Wag A upper boud for A of strictly diagoally domiat M-matrices Liear Algebra Appl, 43:5 57, 2009