Tikhonov Regularization for Nonlinear Complementarity Problems with Approximative Data

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1 Internationa Mathematica Forum, 5, 2010, no. 56, Tihonov Reguarization for Noninear Compementarity Probems with Approximative Data Nguyen Buong Vietnamese Academy of Science and Technoogy Institute of Information Technoogy 18, Hoang Quoc Viet, q. Cau Giay, Ha Noi, Viet Nam Nguyen Thi Thuy Hoa Hanoi Coege of Home Affairs 36, Xuan La, Tay Ho, Ha Noi, Viet Nam Abstract The purpose of this paper is to study the Tihonov reguarization method for soving genera noninear compementarity probem with approximative data. The stabiity, convergence and convergence rates of the reguarized soutions are considered. Keywords: Tihonov reguarization and continuous function Mathematics Subject Cassification: 47H17 1. Introduction Let g(x) and h(x) be continuous functions from the Eucidian space R n into R m with the scaar product and norm of R n denoted by.,. R n and. R n, respectivey. Consider the foowing genera compementarity probem: find an eement x R n such that g( x) 0,h( x) 0, g( x),h( x) R m =0, (1.1) where the symbo y =(y 1,..., y m ) 0 is meant that y i 0,i=1,..., m. In the case that m = n, g(x) = x, and h(x) = F (x), (1.1) is the cassica compementarity probem x 0,F(x) 0, x, F (x) R n =0 (1.2)

2 2788 Nguyen Buong and Nguyen Thi Thuy Hoa which has attracted much attention due to its various appications. We refer the reader to [1]-[4] for review. Recenty, reformuations of the noninear compementarity probem (NCP) as a minimization probem or a system of equations have drawn much attention [4]. A function which can constitute an equivaent minimization probem for the NCP is caed a merit function. Definition. A function φ : R 2 R is caed NCP-function if The foowing three NCP-functions: φ(a, b) =0 ab =0,a 0,b 0. φ NR (a, b) = min{a, b}, φ MS (a, b) =ab + 1 2α (max{0,a αb}2 a 2 + max{0,b αa} 2 b 2 ),α>1, φ FB (a, b) = a 2 + b 2 a b, have been we studied in the iterature. The merit function based on the function φ NR is caed the natura residua. The function φ MS is nonegative on R 2 and the merit function based on the function φ MS is the impicit Lagrangian proposed by Mangasarian and Soodov [5]. Recenty, Luo and Tseng [6] proposed a cass of new merit functions f : R n R defined by f(x) =ψ 0 ( x, F (x) R n)+ n ψ i ( x i, F i )), i=1 where ψ 0 : R [0, ) and ψ i : R 2 [0, ),i =1,..., n, are continuous functions that vanish on the negative orthant ony. There exist severa methods for the soution of the NCP (1.2), see, e.g., [6]- [10]. A those methods are proposed for soving an equivaent minimization probem or a system of equations. The particuar cass of methods to be considered are the so-caed reguarization methods, which are designed to hande i-posed probems. An i-posed probems may be difficut to sove since sma errors in the computations can ead to a totay wrong soution. Reguarization methods try to circumvent this difficuty by substituting the soution of the origina probem with the soution of a sequence of we-posed (i.e. nicey behaved) probems whose soutions form a trajectory converging to the soution of the origina probem. The Tihonov-reguarization scheme in [11], [12] for (1.2) consists in soving a sequence of compementarity probems x ε 0,F ε (x ε ) 0, x ε,f ε (x ε ) R n =0, (1.3)

3 Tihonov reguarization 2789 where F ε (x) =F (x)+εx, and ε is a sma parameter of reguarization. In [13] the reguarization x ε is defined on the base of H(ε, z) =0 ε =0,x S 0, (1.4) where S 0 denotes the soution set of (1.2), z := (ε, x),h(ε, z) :=(ε, G(ε, z)) T, and G i (ε, x) :=φ(x i,f ε,i (x)),i=1,..., n, where F ε,i is the ith component of F ε. The convegence of the reguarized soutions for (1.3) and (1.4) is ony estabished for the case that F is a P 0 -function. Moreover, the convergence rate of the reguarized soutions is sti an opened question. In [14], on the base of transforming the NCP into a vector optimization probem with equaity constraints, we proposed a new approach to reguarize the NCP (1.1) for the more genera case that F is not a P 0 -function. We estabished the convergence rates of reguarized soution provided that g i (x)h i (x) are differentiabe with Lipschitz-continuous derivatives. To do this, set Ceary, ϕ i (x) = max{0,g i (x)},i=1,..., m, ϕ m+i (x) = max{0,h i (x)},i=1,..., m. S i := {x R n : g i (x) 0} = {x R n : ϕ i (x) =0},i=1,..., m, S i := {x R n : h i (x) 0} = {x R n : ϕ m+i (x) =0},i=1,..., m, the functions ϕ j are aso continuous, nonegative on R n and ϕ j (y) =0 y N S j,j =1,..., 2m := N. Moreover, Evidenty, N S j = {x R n : g(x) 0,h(x) 0}. g( x),h( x) R m = m g i ( x)h i ( x) =0 g i ( x)h i ( x) =0,i=1,..., m. i=1 W consider the functions f i (x) =g i (x)h i (x) and set S 0 = {x R n : f i (x) =0,i=1,..., m}. (1.5) Then, NCP (1.1) is equivaent to find an eement x beonging to N i=0s i or satisfying ϕ j ( x) = min ϕ j(x),j =1,..., N, (1.6) x R n

4 2790 Nguyen Buong and Nguyen Thi Thuy Hoa and x S 0. It means that the soution x of (1.1) beongs to the intersection of the soution sets of the vector optimization probem (1.6) and system of equations (1.5). In this paper, we continue to study the agorithm in [14] for the case that {f i,ϕ j } are given by continuous approximations {fi δ,ϕδ j } such that f i (x) fi δ (x) δ, i =1,..., m, ϕ j (x) ϕ δ j (x) δ, j =1,..., N, x Rn,δ 0. (1.7) It is easy to see that condition (1.7) for f i and its approximations fi δ is satisfied, if g i and h i with approximations gi δ and h δ i,respectivey, satisfy this condition and are bounded. The reguarization method is the foowing unconstrained optimization probem: find an eement x δ α Rn such that Fα δ (xδ α ) = min F δ x R n α (x), α > 0,δ 0, Fα δ (x) = F δ (x) 2 R + N ϕ δ m j (x)+α x x 2 R n, F δ (x) =(f δ 1 (x),..., f δ m(x)) T (1.8) depending on a parameter α>0. As spoen in [14], for each α>0 probem (1.8) possesses a soution denoted by x δ α. 2. Main resuts We have the foowing resuts. Theorem 2.1. Let α,δ 0 so that δ /α 0, as. Then every sequence {x }, where x := x δ α is a soution of (1.8) with α, δ repaced by α,δ, respectivey, has a convergent subsequence. The imit of every convergent subsequence is an x -MNS. If, in addition, the x -MNS x is unique, then im x = x. Proof. From (1.8) we have F δ (x ) 2 R m + N ϕδ j (x )+α x x 2 R n F δ (y) 2 R m + N ϕδ j (y)+α y x 2 R n, (2.1)

5 Tihonov reguarization 2791 for every fixed eement y R n. Consequenty, F δ (x ) 2 R + α x x 2 m R F δ (y) 2 n R m N + [ ϕ δ j (y) ϕ j (y)+ϕ j (y) ϕ j (x ) ] + ϕ j (x ) ϕ δ j (x ) +α y x 2 R n. (2.2) By taing y N j=0 S j, we have F (y) =0, and ϕ j (x ) ϕ j (y) =0,j =1,..., N. Then, from (1.8) and (2.2) it impies or F δ (x ) 2 R m+α x x 2 R n F δ (y) 2 R m+2δ x x 2 R n mδ2 α +2 δ α N N +α y x R n (2.3) + y x R n. (2.4) Since δ,δ /α 0as, then {x } is bounded. Let {x } {x } such that x x as. We sha prove that x is a soution of (1.1). Indeed, from (2.3) and ϕ j (x) 0 x R n,j =1,..., N it foows 0 F (x ) 2 R m mδ2 +2δ N + α y x R n. Tending in the ast inequaity, the continuous property of F gives F (x) =0, i.e., x S 0. Now, we prove that x S 1. For any eement y S 0, from (2.1) we aso obtain Therefore, ϕ δ 1 (x )+ ϕ 1 (x )+ ϕ δ N μ 1 ϕ δ j (x )+α 1 μ 1 x x 2 R n j=2 1 (y)+ N j=2 j=2 μ 1 ϕ δ j (y)+α1 μ 1 y x 2 R n. N μ 1 ϕ δ j (x )+α 1 μ 1 x x 2 R n ϕ 1 (y)+2δ + N j=2 μ 1 ϕ δ j (y)+α1 μ 1 y x 2 R n.

6 2792 Nguyen Buong and Nguyen Thi Thuy Hoa After passing in the ast inequaity we obtain ϕ 1 (x) ϕ 1 (y) y S 0. It means that x S 1. Further, we prove that x S 2. For any y S 0 S 1, again from (2.1) and ϕ 1 (x ) ϕ 1 (y) we can write ϕ δ 2 (x )+ N j=3 μ 2 ϕ δ + ϕ δ 2 (y)+ N j=3 j (x )+α 1 μ 2 x x 2 R 2δ α μ1 n α μ 2 μ 2 ϕ δ j (y)+α1 μ 2 y x 2 R n. By the simiar argument, we obtain x S 2. Now, suppose that we have proved x p 1 j=0 S j, and need to show that x S p. Taing y p j=0 S j, from (2.1) and ϕ j (x ) ϕ j (y) =0,j =1,..., p 1, it is obvious ϕ δ p (x )+ N j=p+1 + ϕ δ p (y)+ μ 2 ϕ δ N j=p+1 j (x )+α 1 μ 2 x x 2 R n 2δ α μp μ 2 ϕ δ j (y)+α1 μ 2 y x 2 R n. p 1 It is means that x S p. Consequenty, x N j=0s j. The x -MNS property of x is foowed from (2.4). Theorem is proved. Theorem 2.4. Assume that the foowing conditions hod: (i) F is differentiabe, (ii) there exists L>0 such that F ( x) F (z) R m L x z R n for z in some neighbouhood of x (iii) there exists ω R m such that x x = F ( x) ω (iv) L ω R m < 1. Then, for the choice α δ, we have x x R n = O(δ 1/4 ). Proof. From (1.5), F (y) =0,ϕ j (y) =0 y N j=0 S j,j =1,..., N, we have F (x ) 2 R m + α x x 2 R n mδ2 +2δ +2α ω, F ( x)( x x ) R m. N

7 Tihonov reguarization 2793 Therefore, Thus, F (x ) 2 R m + α x x 2 R n mδ2 +2δ N 1 +2α ω R m F (x ) R m + α ω R ml x x 2 R n. F (x ) 2 R m + α (1 ω R ml) x x 2 R n mδ2 +2δ Using the impication +2α ω R m F (x ) R m. (a, b, c 0,a 2 ab + c 2 ) a b + c we obtain F (x ) R m = O( δ ). Hence, Theorem is proved now. x x R n = O(δ 1/4 ). N 1 This wor was supported by the Vietnamese Natura Foudation of Science and Technoogy Deveopment. References [1] P.T. Harer and J.-S. Pang, Finite-dimensiona variationa inequaity and noninear compementarity probem: A survey of theory, agorithms and appications, Mathematica Programming, 48 (1990) [2] P.T. Harer, Compementarity probem, in Handboo of Goba Optimization, R. Horst and P. Pardaos, eds, Kuwer Academic Pubishers, Boston, pp , [3] M.C. Ferris and J.-S. Pang, Engineering and economic appications of compementarity probems, SIAM Review, 39 (1997) [4] M. Fuushima, Merit functions for variationa inequaityand compementarity probems, in Noninear Optimization and Appications, G. Di Pio and F. Giannessi eds., Penum Pubishing Corporation, New Yor, pp , 1996.

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