Computation of Error Bounds for P-matrix Linear Complementarity Problems

Transcription

1 Mathematical Programmig mauscript No. (will be iserted by the editor) Xiaoju Che Shuhuag Xiag Computatio of Error Bouds for P-matrix Liear Complemetarity Problems Received: date / Accepted: date Abstract We give ew error bouds for the liear complemetarity problem where the ivolved matrix is a P-matrix. Computatio of rigorous error bouds ca be tured ito a P-matrix liear iterval system. Moreover, for the ivolved matrix beig a H-matrix with positive diagoals, a error boud ca be foud by solvig a liear system of equatios, which is sharper tha the Mathias-Pag error boud. Prelimiary umerical results show that the proposed error boud is efficiet for verifyig accuracy of approximate solutios. Keywords accuracy error bouds liear complemetarity problems Mathematics Subject Classificatio (2000) 90C33 65G20 65G50 1 Itroductio The liear complemetarity problem is to fid a vector x R such that Mx + q 0, x 0, x T (Mx + q) = 0, This work is partly supported by a Grat-i-Aid from Japa Society for the Promotio of Sciece. Xiaoju Che Departmet of Mathematical System Sciece, Hirosaki Uiversity, Hirosaki , Japa. Tel.: Fax: che@cc.hirosaki-u.ac.jp Shuhuag Xiag Departmet of Applied Mathematics ad Software, Cetral South Uiversity, Chagsha, Hua , Chia. Tel.: Fax: xiagsh@mail.csu.edu.c

2 2 Xiaoju Che, Shuhuag Xiag or to show that o such vector exists, where M R ad q R. We deote this problem by LCP(M, q) ad its solutio by x. Recall the followig defiitios for a matrix. M is called a P-matrix if max i(mx) i > 0 1 i for all x 0. M is called a M-matrix, if M 1 0 ad M ij 0 (i j) for i, j = 1, 2,...,. M is called a H-matrix, if its compariso matrix is a M-matrix. It is kow that a H-matrix with positive diagoals is a P-matrix. Moreover, M is a P-matrix if ad oly if the LCP(M, q) has a uique solutio x for ay q R. See [4]. It is easy to verify that x solves the LCP(M, q) if ad oly if x solves r(x) := mi(x, Mx + q) = 0, where the mi operator deotes the compoetwise miimum of two vectors. The fuctio r is called the atural residual of the LCP(M, q), ad ofte used i error aalysis. Error bouds for the LCP(M, q) have bee studied extesively, see [3 6,8,9,13]. For M beig a P-matrix, Mathias ad Pag [6] preset the followig error boud for ay x R, where x x 1 + M r(x), (1.1) = { } mi max x i(mx) i. x =1 1 i This error boud is well kow ad widely cited. However, the quatity i (1.1) is ot easy to fid. For M beig a H-matrix with positive diagoals, Mathias ad Pag [6] gave a computable lower boud for, (mi b i )(mi( M 1 b) i ) i i (max( M 1 b) j ) 2 =: c(b), (1.2) j for ay vector b > 0, where M is the compariso matrix of M, that is M ii = M ii Mij = M ij for i j. However, fidig a large value of c(b) is ot easy. For some b, c(b) ca be very small, ad thus the error coefficiet µ(b) := 1 + M c(b) (1.3) ca be very large. See examples i Sectio 3. Iterval methods for validatio of solutio of the LCP(M, q) have bee studied i [1,12]. Whe a umerical validatio coditio for the existece of a

3 Title Suppressed Due to Excessive Legth 3 solutio holds, a umerical error boud is provided. However, the umerical validatio coditio is ot esured to be held at every poit x. I this paper, for M beig a P-matrix, we preset a ew error boud i p (p 1, or p = ) orms, x x p d [0,1] DM) 1 p r(x) p, (1.4) where D =diag(d 1, d 2,..., d ). Moreover, for M beig a H-matrix with positive diagoals, we show d [0,1] DM) 1 p M 1 max(λ, I) p (1.5) where Λ is the diagoal part of M, ad the max operator deotes compoetwise maximum of two matrices. This implies max(λ, I) = diag(max(m 11, 1), max(m 22, 1),..., max(m, 1)). I compariso with the Mathias-Pag error coefficiets (1.1) ad (1.3), we give the followig iequalities. If M is a P-matrix, max d [0,1] (I D+DM) 1 max(1, M ) If M is a H-matrix with positive diagoals, = 1 + M mi(1, M ). (1.6) M 1 max(λ, I) µ(b) M 1 mi(λ, I), for ay b > 0. (1.7) If M is a M-matrix, M 1 max(λ, I) 1 + M I additio, for M beig a M-matrix, the optimal value (I D + D M) 1 1 := M 1 mi(λ, I). (1.8) max d [0,1] (I D + DM) 1 1 ca be foud by solvig a simple covex programmig problem. I Sectio 3, we use some umerical examples to illustrate these error bouds. I particular, for some cases, (1.4),(1.5),(1.6),(1.7), (1.8) hold with equalities, which idicate that they are tight estimates. Prelimiary umerical results show that the ew error bouds are much sharper tha existig error bouds. Notatios: Let N = {1, 2,..., }. Let e deote the vector whose all elemets are 1. The absolute matrix of a matrix B is deoted by B. Let deote the p-orm for p 1 or p =.

4 4 Xiaoju Che, Shuhuag Xiag 2 New error bouds It is ot difficult to fid that for every x, y R, mi(x i, y i ) mi(x i, y i ) = (1 d i )(x i x i ) + d i (y i y i ), i N (2.1) where 0 if y i x i, yi x i 1 if y d i = i x i, yi x i mi(x i, y i ) mi(x i, y i ) + x i x i y i yi + x i x otherwise. i Moreover, we have d i [0, 1]. Hece puttig y = Mx + q ad y = Mx + q i (2.1), we obtai r(x) = (I D + DM)(x x ), (2.2) where D is a diagoal matrix whose diagoal elemets are d = (d 1, d 2,..., d ) [0, 1]. It is kow that M is a P-matrix if ad oly if I D +DM is osigular for ay diagoal matrix D =diag(d) with 0 d i 1 [7]. This together with (2.2) yields upper ad lower error bouds, r(x) max I D + DM x x max d [0,1] (I D+DM) 1 r(x). (2.3) d [0,1] Moreover, it is ot difficult to verify that if M is a P-matrix ad D =diag(d) with d [0, 1], we have max x i((i D + DM)x) i > 0, for all x 0, 1 i that is, (I D + DM) is a P-matrix. Therefore, computatio of rigorous error bouds ca be tured ito optimizatio problems over a P-matrix iterval set, which is related to liear P-matrix iterval systems. The liear iterval system has bee studied itesively ad some highly efficiet umerical methods have bee developed, see [10, 12] for refereces. I the rest part of this sectio, we give some simple upper bouds for d [0,1] DM) 1. Lemma 2.1 If M is a M-matrix, the I D + DM is a M-matrix for d [0, 1]. Proof From I 27 of Theorem 2.3, Chap. 6 i [2], there is u > 0 such that Mu > 0. It is easy to verify that (I D + DM)u > 0. Applyig the theorem agai, we fid that I D + DM is a M-matrix. Theorem 2.1 Suppose that M is a H-matrix with positive diagoals. The we have d [0,1] DM) 1 M 1 max(λ, I). (2.4)

5 Title Suppressed Due to Excessive Legth 5 Proof Let M = Λ B. We ca write (I D + DM) 1 = (I (I D + DΛ) 1 DB) 1 (I D + DΛ) 1. (2.5) We first prove (2.4) for M beig a M-matrix. Note that B 0 with zero diagoal etries, ad for ay d [0, 1], Lemma 2.1 esures that I D+DM ad (I (I D + DΛ) 1 DB) are M-matrices. For each ith diagoal elemet of the diagoal matrix (I D + DΛ) 1 D, we cosider the fuctio φ(t) = t 1 t + tm ii, for t [0, 1]. It is easy to verify that φ(t) 0 ad mootoically icreasig for t [0, 1]. Hece, we have Λ 1 (I D + DΛ) 1 D 0, for d [0, 1]. Sice B is oegative, we get Λ 1 B (I D + DΛ) 1 DB 0, for d [0, 1]. By Theorem 5.2, Chap. 7 ad Corollary 1.5, Chap. 2 i [2], the spectral radius satisfies 1 > ρ(λ 1 B) ρ((i D + DΛ) 1 DB), for d [0, 1]. Therefore, we fid that (I (I D + DΛ) 1 DB) 1 = I + (I D + DΛ) 1 DB + + ((I D + DΛ) 1 DB) k + I + Λ 1 B + + (Λ 1 B) k + = (I Λ 1 B) 1 = (Λ B) 1 Λ = M 1 Λ. Now for each ith diagoal elemet of the diagoal matrix (I D + DΛ) 1, we cosider the fuctio ψ(t) = 1 1 t + tm ii. For t [0, 1], ψ(t) > 0, ad ψ (t) 0 if M ii < 1 otherwise ψ (t) 0. Hece, we obtai { max ψ(t) = 1/Mii if M ii < 1 t [0,1] 1 otherwise. This implies (I D + DΛ) 1 max(λ 1, I), for d [0, 1]. (2.6)

6 6 Xiaoju Che, Shuhuag Xiag Therefore, the upper boud (2.4) for M beig a M-matrix ca be derived by (2.6), (2.5) ad that for all d [0, 1], (I (I D + DΛ) 1 DB) 1 ad (I D + DΛ) 1 are oegative ad (I (I D + DΛ) 1 DB) 1 (I D + DΛ) 1 M 1 Λ max(λ 1, I) = M 1 max(λ, I). Now we show (2.4) for M beig a H-matrix with positive diagoals. Sice for ay matrix A, ρ(a) ρ( A ), we have that for all d [0, 1], ρ((i D + DΛ) 1 DB) ρ((i D + DΛ) 1 D B ) ρ(λ 1 B ) < 1. Therefore, we have (I (I D + DΛ) 1 DB) 1 = I + (I D + DΛ) 1 DB + + ((I D + DΛ) 1 DB) k + I + (I D + DΛ) 1 D B + + ((I D + DΛ) 1 D B ) k + I + Λ 1 B + + (Λ 1 B ) k + = (I Λ 1 B ) 1 = (Λ B ) 1 Λ = M 1 Λ. This together with (2.5) ad (2.6) gives (I D + DM) 1 M 1 Λ max(λ 1, I) = M 1 max(λ, I). Remark 1. Sice M 1 max(λ, I) 0, we have ad M 1 max(λ, I) = M 1 max(λ, I)e M 1 max(λ, I) 1 = (e T M 1 max(λ, I)) T. The upper error boud i (2.4) with or 1 ca be computed by solvig a liear system of equatios mi(λ 1, I) Mx = e or M T mi(λ 1, I)x = e. Theorem 2.2 Suppose that M is a M-matrix. Let V = {v M T v e, v 0} ad f(v) = max 1 i (e + v M T v) i. The we have d [0,1] DM) 1 1 = max f(v). (2.7) v V Proof From Lemma 2.1, we have that This implies that (I D + DM) 1 0, for all d [0, 1]. (I D +DM) 1 1 = (e T (I D +DM) 1 ) T = (I D +M T D) 1 e.

7 Title Suppressed Due to Excessive Legth 7 Therefore, max d [0,1] (I D + DM) 1 1 = max max u s.t. 1 i u i u Du + M T Du = e 0 d e. (2.8) Let v = Du, the from u 0 ad d [0, 1], we have 0 v u = v M T v + e. This implies v V. Hece we obtai d [0,1] DM) 1 1 max f(v). v V Coversely, suppose that v is a maximum solutio of f(v) i V. We set u = v M T v + e ad { vi /u d i = i if u i 0 0 otherwise, for all i N. The d [0, 1] ad u Du + M T Du = e. This implies that u is a feasible poit of the maximizatio problem (2.8). Thus, d [0,1] DM) 1 1 max f(v). v V Furthermore, the feasible set V is covex ad bouded, ad the objective fuctio f is covex. Thus, max f(v) always has a optimal value. The proof v V is completed. Now we show that error bouds give i this paper are sharper tha the Mathias-Pag error bouds. Theorem 2.3 If M is a P-matrix, the for ay x R, the followig iequalities hold. 1 r(x) (Mathias-Pag [6]) 1 + M 1 max(1, M ) r(x) (Cottle-Pag-Stoe [4]) 1 = max d [0,1] I D + DM r(x) x x d [0,1] DM) 1 r(x) max(1, M ) r(x) = 1 + M r(x) mi(1, M ) r(x) 1 + M r(x) (Mathias-Pag [6])).

8 8 Xiaoju Che, Shuhuag Xiag Proof The first iequality is obvious. For the ext equality, we set D =diag(d 1,..., d ) to be a optimal poit such that I D + D M = max I D + DM. d [0,1] From M ii > 0, we have I D + D M = max 1 i 1 d i + d i M ii + d i = max 1 i 1 d i + d i M ij j=1 =: 1 d i 0 + d i 0 M i0j. j=1 M ij Hece the value d i 0 must be a boudary poit of [0, 1]. Moreover, it is easy to fid { I D + D M if M M = > 1 1 otherwise which implies max(1, M ) = j=1 j i max I D + DM. (2.9) d [0,1] The secod ad third iequalities follows from (2.3). For the fourth iequality, we first prove that for ay osigular diagoal matrix D =diag(d) with d (0, 1], (I D + DM) 1 max(1, M ). (2.10) Let H = (I D +DM) 1 ad i 0 be the idex such that j=1 H i 0j = (I D+DM) 1. Defie y = (I D+DM) 1 p, where p = (sg(h i01),,sg(h i0)) T. The p = (I D + DM)y, My = D 1 p + y D 1 y, ad (I D + DM) 1 = y. Furthermore, by the defiitio of, we have 0 < y 2 max i Let j be the idex such that y j ( pj (i) If y j 1, the we have ( pi y i (My) i = max y i + y i y i i d i d i + y d j y ) j j d j y 2 My j My M y. ). ( pi = max y i + y i d i y ) i. i d i

9 Title Suppressed Due to Excessive Legth 9 This implies (I D + DM) 1 y M. (ii) If y j > 1, the p j + d j y j y j > 0 ad p d j > y j d j y j 0. Thus j p j = 1 ad d j > 1 y 1 j. Hece, we obtai 0 < p j + d j y j y j d j 1. This implies 0 < (My) j 1. Thus y 2 y j y ad (I D + DM) 1 1. (iii) If y j < 1, the p j + d j y j y j < 0 ad p d j < y j d j y j 0. Thus j p j = 1 ad d j 1 + y 1 j. Similarly, we obtai 0 > p j + d j y j y j d j 1. This implies 1 (My) j < 0. Thus y 2 y j y ad (I D + DM) 1 1. Combiig the three cases, we claim that (2.10) holds for ay osigular matrix D=diag(d) with d (0, 1]. Now we cosider d [0, 1]. Let d ɛ = mi(d + ɛe, e), where ɛ (0, 1]. The, we have (I D + DM) 1 = lim (I D ɛ + D ɛ M) 1 max(1, M ). ɛ 0 Sice D is arbitrarily chose, we obtai the fourth iequality. The ext equality ad iequality are trivial. Theorem 2.4 If M is a H-matrix with positive diagoals, the for ay x, b R, b > 0, the followig iequalities hold. x x d [0,1] DM) 1 r(x) M 1 max(λ, I) r(x) (µ(b) M 1 mi(λ, I) ) r(x) µ(b) r(x) (Mathias-Pag [6]). I additio, if M is a M-matrix, the for ay x R, the followig iequalities hold. x x M 1 max(λ, I) r(x)

10 10 Xiaoju Che, Shuhuag Xiag ( 1 + M M 1 mi(λ, I) ) r(x) 1 + M r(x) (Mathias-Pag [6]). Proof We first cosider that M is a H-matrix with positive diagoals. The first ad secod iequalities follow (2.3) ad Theorem 2.1. Now we show the third iequality. For ay b R, b > 0, let b 0 = mi b i. The b b 0 e, ad M 1 b 1 i M 1 b 0 e = b 0 M 1 e. Moreover, for every j N (( M 1 b) j ) 2 ( M 1 b) j b 0 ( M 1 e) j ( mi 1 i ( M 1 b) i )( mi 1 i b i)( M 1 e) j. Hece from M 1 e = M 1, we obtai (max( M 1 b) j ) 2 ( mi ( M 1 b) i )( mi b i) M 1. j 1 i 1 i Therefore, from the followig iequalities 1 + M I + M I + Λ max(λ, I) + mi(λ, I) we fid µ(b) = 1 + M c(b) M 1 (1 + M ) M 1 ( max(λ, I) + mi(λ, I) ) M 1 max(λ, I) + M 1 mi(λ, I). Now, we cosider that M is a M-matrix. Let M 1 w = max M 1 y = M 1. From the defiitio of y =1, we have M 1 2 max 1 i (M 1 w) i (MM 1 w) i M 1. By the similar argumet above, we fid 1 + M M 1 (1+ M ) M 1 max(λ, I) + M 1 mi(λ, I). Applyig Theorem 2.1, we obtai the followig relative error bouds Corollary 2.1 Suppose M is a H-matrix with positive diagoals. For ay x R, we have r(x) (1 + M ) M 1 max(λ, I) ( q) + x x x M M 1 max(λ, I) r(x). ( q) +

11 Title Suppressed Due to Excessive Legth 11 Proof. Set x = 0 i (2.3). From Theorem 2.1, we get x M 1 max(λ, I) r(0) = M 1 max(λ, I) ( q) +. Moreover, from Mx + q 0, we deduce ( q) + (Mx ) + Mx. This implies ( q) + Mx, ad ( q) + M x. Combiig (2.3) with the bouds for x x ad x, we obtai the desired error bouds. Remark 2. Note that for M beig a H-matrix with positive diagoals, x solves LCP(Λ 1 M, Λ 1 q) if ad oly if x solves LCP(M, q). Let r(x) = mi(λ 1 Mx + Λ 1 q, x). From (2.3) ad we have I D + DΛ 1 M I D(I Λ 1 M) Λ 1 M, for every x R. Moreover, from with p = 1 or p =, we obtai r(x) Λ 1 M x x M 1 Λ r(x) Λ 1 M p = Λ 1 M p, r(x) p cod p (Λ 1 M) ( q) x x p + p x cod p(λ 1 M) r(x) p, (2.11) p ( q) + p for p = 1 or p =. 3 Numerical examples I this sectio, we first use examples to illustrate error bouds derived i the last sectio. Next we report umerical results obtaied by usig Matlab 6.1 o a IBM PC. Example 3.1 I [12], Schäfer cosidered a applicatio of P-matrix liear complemetarity problems, which arises from computig iterval eclosure of the solutio set of a iterval liear system [10]. The followig P-matrix is from [12] ( ) 1 4 M =. 5 7 This matrix is ot a H-matrix. It is ot difficult to fid that 1 + 6d 2 + 4d 1 d [0,1] DM) 1 = max = 5, 2 d [0,1] d d 1 d 2

12 12 Xiaoju Che, Shuhuag Xiag ad 1 + M 13 mi(m ii ) = 13. Example 3.2 Cosider the followig H-matrix with positive diagoals. (Example i [4]) ( ) 1 t M =, 0 1 where t 1. It is easy to show that 1/t 2. Hece the error boud (1.1) has 1 + M t 2 (2 + t ) = O(t 3 ). For b = e, we have ad c(b) = µ(b) = 1 + M c(b) (mi b i )(mi( M 1 b) i ) i i (max( M 1 b) j ) 2 = 1/(1 + t ) 2 j = (1 + t ) 2 (2 + t ) = O(t 3 ). The error coefficiets give i the last sectio satisfy for p = 1, max d [0,1] (I D(I M)) 1 p = max (1+d 1 t ) = M 1 max(i, Λ) p = 1+ t. 2 d 1 [0,1] Hece, the ew error bouds are much smaller tha the Mathias-Pag error bouds, especially whe t. Moreover, we ca show that the ew error bouds are tight. Let t = 1 ad q = (1, 1) T. The the LCP(M, q) has a uique solutio x = (0, 1) T. For x = (4, 3) T, x x = 4, M 1 max(i, Λ) = 2, r(x) = 2. Hece (1.4) ad (1.5) hold with equality. For M = I, (1.6),(1.7) with b = e ad (1.8) hold with equality. Now we report some umerical results to compare these error coefficiets. Example 3.3 Let M be a tri-diagoal matrix b + α si( 1 ) c a b + α si( 2 ) c M = c a b + α si(1) For b = 2, a = c = 1, α = 0, the LCP(M, q) with various q i a iterval vector arises from the fiite differece method for free boudary problems [11].

13 Title Suppressed Due to Excessive Legth 13 Table 1 Example 3.3, = 400, κ 1 = max d [0,1] (I D + DM) 1 1 α a b c κ 1 M 1 max(λ, I) µ(e) E E E E E E E E E E E E4 Ackowledgemets The authors are grateful to the associate editor ad two aoymous referees for their helpful commets. Refereces 1. Alefeld, G.E., Che, X., Potra, F.A.: Numerical validatio of solutios of liear complemetarity problems, Numer. Math. 83, 1-23(1999) 2. Berma, A., Plemmos, R.J.: Noegative Matrix i the Mathematical Scieces, SIAM Publisher, Philadelphia(1994) 3. Che, B.: Error bouds for R 0-type ad mootoe oliear complemetarity problems, J. Optim. Theory Appl. 108, (2001) 4. Cottle, R.W., Pag, J.-S., Stoe, R.E.: The Liear Complemetarity Problem, Academic Press, Bosto, MA(1992) 5. Ferris, M.C., Magasaria, O.L.: Error bouds ad strog upper semicotiuity for mootoe affie variatioal iequalities, A. Oper. Res. 47, (1993) 6. Mathias, R., Pag, J.-S.: Error bouds for the liear complemetarity problem with a P-matrix, Liear Algebra Appl. 132, (1990) 7. Gabriel, S.A., Moré, J.J.: Smoothig of mixed complemetarity problems. I: M.C.Ferris ad J.-S.Pag (ed.) Complemetarity ad Variatioal Problems: State of the Art, SIAM Publicatios, Philadelphia, PA(1997) 8. Magasaria, O.L., Re, J.: New improved error bouds for the liear complemetarity problem, Math. Programmig 66, (1994) 9. Pag, J.-S.: Error bouds i mathematical programmig, Math. Programmig 79, (1997) 10. Roh, J.: Systems of liear iterval equatios, Liear Algebra Appl. 126, 39-78(1989) 11. Schäfer, U.: A eclosure method for free boudary problems based o a liear complemetarity problem with iterval data, Numer. Fuc. Aal. Optim. 22, (2001) 12. Schäfer, U.: A liear complemetarity problem with a P-matrix, SIAM Review 46, (2004) 13. Xiu, N., Zhag, J.: A characteristic quatity of P-matrices, App. Math. Lett. 15, 41-46(2002)