Examples of Perturbation Bounds of P-matrix Linear Complementarity Problems 1

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1 Examples of Perturbation Bounds of P-matrix Linear Complementarity Problems Xiaojun Chen Shuhuang Xiang 3 We use three examples to illustrate the perturbation bounds given in [CX] X. Chen and S. Xiang, Perturbation bounds of P-matrix linear complementarity problems, Technical Report, Department of Mathematical Sciences, Hirosaki University, February 006, Revised January 007. We use the semi-smooth Newton method [3] to solve (.8) in [CX] with stop criteria kr(x)k 0 4 and computer precision macheps =0 6. We report numerical results in Table and Table where the fourth column and the fifth column represent the measure β(m)kmk for the LCP and the upper bounds (4.5), (4.6) of K(M) for the system of (.8) in [CX], respectively. An exact error k4xk is computed as follows. First, we find approximation solutions ˆx 0 and ˆx+4x 0 of LCP(M,q)and LCP(M +4M,q+4q), respectively. Next, we define ˆq and ˆq + 4ˆq such that ˆx and ˆx + 4x are exact solutions of LCP(M, ˆq) andlcp(m + 4M, ˆq + 4ˆq), respectively. The perturbation bounds bound = k M k (k4mk kx(m + 4M, ˆq + 4ˆq)k + k4ˆqk ) are based on (.4) and Theorem.5 in [CX]. Example We consider a problem which arises from finite difference approximation of free boundary problems for infinite journal bearings [4]. Here M is a tridiagonal M-matrix whose elements m ij are defined m ij = h 3, j = i +, i+ h 3 i + h 3, j = i, i+ h 3, j = i, i 0, otherwise and the elements of vector q are defined q i = δ(h i+ h i ), i =,,,n. In a common model for the infinitely long cylindrical bearing, i, j =,,,n δ = n + and h i = +² cos(π(i )δ) π, i =,,,n+. Following Cryer[4], we chose ² = 0.8. Reformulating the journal bearing problem to the LCP(M,q) causes truncation error and rounding error. One is interested in perturbation This work is partly supported by a Grant-in-Aid from Japan Society for the Promotion of Science. Department of Mathematical System Science, Hirosaki University, Hirosaki , Japan, chen@cc.hirosaki-u.ac.jp. 3 Department of Applied Mathematics and Software, Central South University, Changsha, Hunan 40083, China, xiangsh@mail.csu.edu.cn

2 error in the solution of the LCP(M,q) caused by small changes in M and q. In this problem, M is ill-conditioned for large n. Let 4M = ² M , 4q = ² q e. (0.) Table. Perturbation bounds of Example β(m) =km k, ν = κ (M)k max(λ,i)k n ² M ² q β(m)kmk ν k4xk bound e e e-3.0e e e-3 -.0e e e-3.06e e e-5.0e-3.06e e e-5 -.0e-3.06e e e-5.068e e e-7.0e-5.068e e e-7 -.0e-5.068e e Example []. We consider a tridiagonal H-matrix 4 4 M = , and q = 4e. 4 4 Notice that M is well-conditioned for any n. From Theorem.8 in [CX], LCP(M,q) is not sensitive to small changes in data. Let 4M and 4q be defined by (0.). Table Perturbation bounds of Example β( M) =k M k, ν =max(, kmk )k M max(λ,i)k n ² M ² q β( M)kMk ν k4xk bound e e e-4.0e-3.0e e e-3 -.0e-3 -.0e e e e e e-3.0e-5.0e e-4.000e-3 -.0e-5 -.0e e-4.000e e e e-5.0e-7.0e e e-5 -.0e-7 -.0e e e e e e-5.0e-7.0e e e-5 -.0e-7 -.0e e e-5

3 Example 3 Linear variational inequalities and complementarity problems have often been used to discuss formulation and solution of traffic equilibrium problems [5, 8]. Here, we use a simple traffic network in [5] to illustrate applications of perturbation bounds of the LCP with a positive definite matrix. This network consists of two nodes: w, w, and five paths: p,p,p 3,p 4,p 5. The two nodes are connected by two two-way streets and one one-way street. The paths p,p,p 3 are directed from w to w,andp 4,p 5 are the returns of p and p, respectively. (See Figure in [5].) The travel demands are d = 0 (from w to w ), d = 0 (from w to w ). Let x i denote the flow on path p i, i =,, 3, 4, 5, and let X d denote the set of x satisfying the travel demands d =(d,d ), that is, X d = {x R 5 x 0,x + x + x 3 = d,x 4 + x 5 = d }. Furthermore, the personal travel cost function is given by where M = c(x) :=Mx+ b , b = By the Wardrop principle, a load pattern x X d is user-optimized if and only if. (Mx + b) T (x x ) 0, for all x X d. (0.) This linear variational inequality problem is equivalent to the following constrained complementarity problem x X d, τ 0, Mx+ b τ 0, x T (Mx+ b τ) =0, (0.3) where τ =(τ, τ, τ, τ, τ ). Here τ and τ depict the minimum transportation cost on w and w, respectively. The optimal solution of this example is x = (0, 90, 0, 70, 50) associated with τ = 550 and τ = 640. In practical applications, the travel cost often contain errors, due to inaccurate data, uncertain weather, etc. Such errors affect travel demand and flow. In order for the optimal solution x to be of practical use, it is very important to have some sensitivity information of solution on the cost. Now we show that such information can be obtained by using perturbation bounds in Theorem. in [CX]. It is easy to verify that M is a positive definite matrix, and x is a solution of LCP(M,q) withq = b τ. Moreover, if x is a solution of LCP(M + 4M,q + 4b), then x satisfies ((M + 4M) x + q + 4b) T (x x) 0, for all x X d, (0.4) where d =( d, d )with d = x + x + x 3 and d = x 4 + x 5. 3

4 Suppose that the matrix and vector in the cost function c(x) haveperturbations M = ² M and 4b = ² b Since k( M + M T ) k 0.4 and k4mk 5k4Mk ² M 5 ² M, by Theorem. in [CX], we obtain that for perturbation travel cost M + 4M and b + 4b with ² M < the corresponding perturbation traffic flow x satisfies k x x k α (M) k( q) + k k4mk + α (M)k4bk 5α (M) k( q) + k ² M + 5α (M) ² b =: 4x bd, where α (M) = ² M and k( q) + k = kτ bk Moreover, the corresponding perturbation demand satisfies k d dk 3k x x k 3k x x k =: 4d bd. Table 3. Perturbation bounds of Example 3 ² M ² q k x x k 4x bd 4d bd 0.0.0e-3.7e-4.534e e-4.0e e-3.0e e- -.0e Note that travel cost functions in many traffic equilibrium problems have the form: c(x) = Mx + b, where M is a positive definite matrix. Analysis in this example can be extended to general cases. Moreover, the perturbation bounds are easy to compute. Remark Theoretical analysis and numerical results show that the new perturbation bounds for the P-matrix LCP are rigorous, which improve previous perturbation bounds significantly. Moreover, β(m)kmk is the first measure closely related to the condition number κ(m) for the sensitivity of the solution of the P-matrix LCP. 4

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