Error Bounds for P-matrix Linear Complementarity Problems and Their Applications

Transcription

1 Error Bounds for P-matrix Linear Complementarity Problems and Their Applications Xiaojun Chen The Hong Kong Polytechnic University NPA2008, Beijing, 7-9 April

2 Outline Computational Global Error Bounds for P-matrix LCP Math. Programming 2006 (with S. Xiang) Perturbation Error Bounds for P-matrix LCP SIAM J. Optimization 2007 (with S. Xiang) Non-Lipschitzian NCP Math. Comp (with G. Alefeld) Extended Vertical LCP Comp. Optim. Appl (with C. Zhang & N. Xiu)

3 Part I 2 Linear Complementarity Problem to find a vector x R n such that Mx+ q 0, x 0, x T (Mx+ q) =0, where M R n n and q R n. We denote this problem by LCP(M,q) and its solution by x. M is called P-matrix, if max x i(mx) i > 0 for all x 6= 0; 1 i n M-matrix, ifm 1 0, M ij 0(i6= j) for i, j =1, 2,...,n; H-matrix, if its comparison matrix is an M-matrix.

4 3 Natural Residual: r(x) :=min(x, Mx + q) Global error bound if there exists a constant τ such that kx x k τkr(x)k, x R n. Question: How to compute τ?

5 Mathias-Pang Error Bound(1990) M is a P-matrix kx x k 1+kMk kr(x)k, c(m) for any x R n, where ½ ¾ c(m) = min max x i(mx) i. kxk =1 1 i n 4 M is an H-matrix with positive diagonals c(m) (min i b i )(min i ( M 1 b) i ) (max j ( M 1 b) j ) 2 =: c(m,b), for any vector b>0, where M is the comparison matrix of M, thatis M ii = M ii Mij = M ij for i 6= j. µ(b, M) := 1+kMk c(m,b) 1+kMk c(m)

6 New Error Bound M is a P-matrix, kx x k p max d [0,1] n k(i D + DM) 1 k p kr(x)k p, where D =diag(d 1,d 2,...,d n ). M is an H-matrix with positive diagonals, max d [0,1] n k(i D + DM) 1 k p k M 1 max(λ,i)k p M is an M-matrix, max k(i D + d [0,1] DM) 1 k 1 =max f(v). n v V V = {v M T v e, v 0} 5 f(v) = max 1 i n (e + v + M T v) i.

7 Comparison M is a P-matrix, 1 kr(x)k (Mathias-Pang) 1+kMk 1 max(1, kmk ) kr(x)k (Cottle-Pang-Stone) 1 = max d [0,1] nki D + DMk kr(x)k 6 kx x k max d [0,1] n k(i D + DM) 1 k kr(x)k max(1, kmk ) kr(x)k c(m) = 1+kMk c(m) kr(x)k min(1, kmk ) c(m) kr(x)k 1+kMk c(m) kr(x)k (Mathias-Pang).

8 M is an H-matrix with positive diagonals 7 kx x k max d [0,1] n k(i D + DM) 1 k kr(x)k k M 1 max(λ,i)k kr(x)k (µ(m,b) k M 1 min(λ,i)k )kr(x)k µ(m,b)kr(x)k (Mathias-Pang).

9 M is an M-matrix 8 kx x k km 1 max(λ,i)k kr(x)k ( 1+kMk c(m) 1+kMk c(m) km 1 min(λ,i)k)kr(x)k kr(x)k (Mathias-Pang)

10 Numerical Examples Example 1.(P -matrix)schäfer (2004) Ã! 1 4 M = M is a P-matrix but not an H-matrix. New Error Bound max k(i D + DM) 1 k =5, d [0,1] 2 Mathias-Pang Error Bound 1+kMk c(m) 13.

11 Example 2.(H-matrix)Cottle(1992) M = Ã 1 t 0 1!, where t New Error bound max d [0,1] 2 k(i D(I M)) 1 k p Mathias-Pang Error Bound = max (1 + d 1 t ) d 1 [0,1] = k M 1 max(i,λ)k p =1+ t, p =1,. 1+kMk c(m) t 2 (2 + t ) =O(t 3 ).

12 Example 3. (M matrix) b + α sin( 1 n ) c a b+ α sin( 2 n ) c M = c a b+ α sin(1) 11 Table 1. α a b c κ 1 km 1 max(λ,i)k µ(m,e) e e e7 n e e e6 n e e e e e e4

13 12 n =400, κ 1 = max d [0,1] n k(i D + DM) 1 k 1 (min e i )(min(m 1 e) i ) i i (max(m 1 e) j ) 2 =: c(m,e), j µ(m,e) := 1+kMk c(m,e)

14 Part II 13 Perturbation Error Bounds for LCP x is the solution of LCP(M,q) x is the solution of LCP(M + 4M,q + 4q) Question: kx x k?, kx x k kx k?

15 Cottle-Pang-Stone (1992) 14 M is a P-matrix. The following statements hold: (i) for any two vectors q and p in R n, kx(m,q) x(m,p)k c(m) 1 kq pk, where c(m) = min kxk =1 ½ ¾ max x i(mx) i 1 i n. (ii) for each vector q R n, there exists a neighborhood U of the pair (M,q) andaconstantc 0 > 0suchthatfor any (A, b), (B,p) U, A, B are P-matrices and kx(a, b) x(b,p)k c 0 (ka Bk + kb pk ).

16 Remark The above constant c(m) isdifficult to compute, and c 0 is not specified. It is hard to use this result for verifying accuracy of a computed solution of the LCP when the data (M,q) contain errors. 15

17 New Perturbation Error Bounds M is a P-matrix, β p (M) = max d [0,1] n k(i D + DM) 1 Dk p, where D =diag(d 1,d 2,...,d n ). Using the constant β p (M), we give perturbation bounds for M being a P-matrix as follows. kx(m,q) x(m,p)k β p (M)kq pk, 16 and kx(a, b) x(b,p)k β p(m) 2 k( p) + kka Bk (1 η) 2 kx(m,q) x(a, b)k kx(m,q)k 2² 1 η β p(m)kmk + β p(m)kb pk 1 η, for A, B M := {A β p (M)kM Ak η < 1}, and kq bk ²k( q) + k.

18 If M is a P-matrix, then for k k, β (M) 1 c(m). M is an H-matrix with positive diagonals, 17 β p (M) k M 1 k p M is an M-matrix, β p (M) =km 1 k p. M is a symmetric positive definite matrix, β 2 (M) =km 1 k 2.

19 M is a positive definite matrix 18 kx(m,q) x(m,p)k 2 k( M + M T 2 ) 1 k 2 kq pk 2, kx(a, b) x(b,p)k 2 k(m+m T 2 ) 1 k 2 2k( p) + k 2 ka Bk 2 (1 η) 2 and + k(m+m T 2 ) 1 k 2 kb pk 2, 1 η kx(m,q) x(a, b)k 2 2²kMk 2 kx(m,q)k 2 1 η k(m + M T 2 ) 1 k 2 for A, B M := {A k( M+M T 2 ) 1 k 2 km Ak 2 η < 1}.

20 Example 4 19 M = Ã 1 t 0 t!, where t 1. New Error bound β (M) = max d [0,1] 2 k(i D + DM) 1 Dk =2 Mathias-Pang Error Bound 1 c(m) t (t ).

21 Relative Perturbation Bounds for LCP 20 Linear systems Suppose Ax = b, A R n n, 0 6= b R n (A + 4A)y = b + 4b, 4A R n n, 4b R n with k4ak ²kAk and k4bk ²kbk. If ²κ(A) =η < 1 and A is nonsingular, then A + 4A is nonsingular and ky xk kxk 2² 1 η κ(a).

22 P-Matrix LCP Suppose min(x, Mx + q) = 0 21 M R n n, 0 6= ( q) + R n min(y, (M + 4M)y + q + 4q) =0 4M R n n, 4q R n. with k4mk ²kMk and k4qk ² max(k( q) + k, kqk kmx+ qk). If M is a P-matrix and ²β(M)kMk = η < 1, then M + 4M is a P-matrix and ky xk kxk 2² 1 η β(m)kmk.

23 M is an H-matrix with positive diagonals, ²κ ( M) =η < 1, and 22 k4mk ²k Mk and k4qk ² max(k( q) + k, kqk kmx+ qk ) then M + 4M is an H-matrix with positive diagonals and ky xk 2² kxk 1 η κ ( M).

24 M is a symmetric positive definite matrix, ²κ 2 (M) =η < 1, and 23 k4mk 2 ²kMk 2 and k4qk 2 ² max(k( q) + k 2, kqk 2 kmx+ qk 2 ), then M + 4M is a P-matrix and ky xk 2 kxk 2 2² 1 η κ 2(M).

25 M is a positive definite matrix, and and ²κ 2 ( M + M T 2 )=η < 1 k4mk 2 ²k M + M T k4qk 2 ² max(k( q) + k 2, kqk 2 kmx+ qk 2 ) km+m T k 2 2kMk 2, then M + 4M is a positive matrix, and 2 k 2 kx yk 2 2² kxk 2 1 η κ 2( M + M T 2 ). 24

26 Remark The above inequalities indicate that the constant β(m)kmk is a measure of sensitivity of the solution x(m,q) ofthelcp(m,q). Moreover, it is interesting to see that the measure is expressed in the terms of the condition number of M, thatis, 25 κ p (M) :=km 1 k p kmk p = β p (M)kMk p for M being an M-matrix with p 1 and a symmetric positive definite matrix with p = 2. Hence, it makes connection between perturbation bounds of the LCP and perturbation bounds of the systems of linear equations in the Newton-type methods for solving the LCP.

27 Newton-type method 26 (I D k + D k M)(x x k )= r(x k ), (1) or à M I D k I D k!ã x x k y y k! = F (x k,y k ), (2) where D k is a diagonal matrix whose diagonal elements are in [0, 1]. Sensitivity of (??) and(??) will effect implementation of the methods and reliability of the computed solution.

28 Proposition For any diagonal matrix D=diag(d) with0 d i 1, i =1, 2,...,n, the following inequalities hold and κ p à κ à M I D M I D I D! I D! κ (I D + DM) 1 2 κ p(i D + DM), p K p (M) := max k(i D + d [0,1] DM) 1 k p ki D + DMk p. n Ã! 1! ˆK p (M) := max k M I M I k kã k. d [0,1] n D I D D I D ˆK (M) K (M).

29 Example 5 Let M = ai(a 1), ˆK (M) κ à and = k à =(1+a)k =(1+a) 2! M I I 0! à ai I k k I 0 Ã! 0 I I ai ai I I 0 k! 1 k K (M) = max d [0,1] n k(i D + DM) 1 k ki D + DMk max 0 ξ 1 (1 + aξ ξ) min 0 ξ 1 (1 + aξ ξ) = a. 28 For large a, ˆK (M) K (M) a 2 + a + 1 is very large.

30 Connection with Sensitivity of LCP(M,q) 29 M being an M-matrix with kmk 1 κ (M) K (M) κ (M)k max(λ,i)k. The condition number κ (M) is a measure of sensitivity of the solution of the system of linear equations for the worst case. Note that we have shown that κ (M) isa measure of sensitivity of the solution of LCP. Hence we may suggest that if Λ is not large, then the LCP is well-conditioned if and only if the system of linear equations (??) ateachstepofthenewtonmethodis well-conditioned.

31 Numerical Examples 30 kr(x)k macheps =10 16 Example 6 (Free boundary problem for journal bearings(bierlein, 1975). m ij = h 3 i+ 1 2 h 3 i 1 2 h 3 i 1 2 q i = δ(h i+ 1 2 δ = 2, ² =0.8 and n+1 h i 1 2 =, j = i +1, + h 3, j = i, i+ 1 2, j = i 1, 0, otherwise i, j =1,,n h i 1 ), i =1, 2,,n. 2 1+² cos(π(i 1 2 )δ) π, i =1, 2,,n+1.

32 Let 4M = ² M , 4q = ² qe. 31 Table 2. Perturbation bounds of Example 6 (µ = max(m ii )+1) n ² M ² q κ (M) µκ (M) k4xk bound e e e e e e e e e e e e e e e-7 1.0e e e e-7-1.0e e e

33 Example 7 (Ahn,1983). We consider a tridiagonal H-matrix M = , and q = 4e Notice that M is well-conditioned for any n. From our analysis, LCP(M,q) is not sensitive to small changes in data. Let 4M and 4q be defined in Example 6.

34 Table 3. Perturbation analysis of Example 7 33 β( M) =k M 1 k, ν = max(1, kmk )k M 1 max(λ,i)k n ² M ² q β( M)kMk ν k4xk bound e e e-4 1.0e-3 1.0e e e-3-1.0e-3-1.0e e e e e e-3 1.0e-5 1.0e e e-3-1.0e-5-1.0e e e e e e-5 1.0e-7 1.0e e e-5-1.0e-7-1.0e e e e e e-5 1.0e-7 1.0e e e-5-1.0e-7-1.0e e e-5

35 Applications Non-Lipschitizan NCP F ( x) 0, x 0, x T F( x) = 0 Extended Vertical LCP min( M x + q, M x + q,..., M x + q ) = m m 0 Stochastic LCP M( ω) x + q( ω) 0, x 0, x min E min( M( ω) x + q( ω), x 0 ( M( ω) x T + q( ω)) = 0