Int. Journa of Math. Anaysis, Vo. 3, 2009, no. 34, 1683-1691 Reguarization for Noninear Compementarity Probems Nguyen Buong a and Nguyen Thi Thuy Hoa b a Vietnamse Academy of Science and Technoogy Institute of Information Technoogy 18, Hoang Quoc Viet, q. Cau Giay, Ha Noi, Vietnam nbuong@ioit.ac.vn b Hanoi Coege of Home Affairs 36, Xuan La, Tay Ho, Ha noi, Vietnam Abstract The purpose of this paper is to study the Tikhonov reguarization method for soving genera noninear compementarity probems. The convergence and convergence rates of the reguarized soutions are considered. Mathematics Subject Cassification: 47H17 Keywords: Tikhonov reguarization and continuous function 1. Introduction Let g(x) and h(x) be two continuous functions from an Eucidian space E n to E m where the scaar product and norm of E n are denoted by.,. E n and. E n, respectivey. Consider the probem: find an eement x E n such that g( x) 0,h( x) 0, g( x),h( x) E 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), a continuous function in E n, (1.1) is the cassica compementarity probem: find an eement x E n satisfying x 0,F(x) 0, x, F (x) E n =0 (1.2) which has attracted much attention due to its various appications. We refer the reader to [3], [8], [11] and [12] for review.
1684 Nguyen Buong and Nguyen Thi Thuy Hoa Recenty, reformuations of the noninear compementarity probem (NCP) as a minimization probem or a system of equations have drawn much attention [8]. A function which can constitute an equivaent minimization probem for the NCP is caed a merit function. More precisey, a merit function is a function whose goba minima on a set X E n are coincident with the soution of the origina NCP. To construct such a function, it is effective to take advantage of the Cartesian structure of the NCP. In particuar, the cass of functions defined beow serves as a convennient too for constructing a merit function [5], [9], [14]. Definition. A function φ : E 2 E 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 E 2 and the merit function based on the function φ MS is the impicit Lagrangian proposed by Mangasarian and Soodov [21]. The function φ FB was first considered by Fischer [6] (and attributed to Burmeister). Recenty, the merit function φ FB has extensivey been studied and has shown to have a number of favorabe properties. Luo and Tseng [20] proposed a cass of new merit functions f : E n E defined by f(x) =ψ 0 ( x, F (x) E n)+ n ψ i ( x i, F i )), i=1 where ψ 0 : E [0, ) and ψ i : E 2 [0, ),i =1,..., n, are continuous functions that vanish on the negative orthant ony. This idea is used in [15] to construct a new merit function. There exist severa methods for the soution of the NCP (1.2), see, e.g., [1], [4], [13], [16]-[19]. 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 for noninear compementarity probems 1685 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 Tikhonov-reguarization scheme in [2], [10] for (1.2) consists in soving a sequence of compementarity probems x ε 0,F(x ε )+εx ε 0, x ε,f ε (x ε ) E n =0, (1.3) where F ε (x) =F (x)+εx, and ε is a sma parameter of reguarization. In [25] 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 estabished ony for the P 0 -function F. Moreover, the convergence rate of the reguarized soutions is sti an opened question. In this paper, on the base of transforming the NCP into a vector optimization probem with equaity constraints, we propose a new approach to reguarize the NCP (1.1) without needing that F is the P 0 -function. We estabish 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 E n : g i (x) 0} = {x E n : ϕ i (x) =0},i=1,..., m, S i := {x E n : h i (x) 0} = {x E n : ϕ m+i (x) =0},i=1,..., m, the functions ϕ j are aso continuous, nonegative on E n and ϕ j (y) =0 y N j=1 S j,j =1,..., 2m := N. Moreover, Evidenty, N j=1 S j = {x E n : g(x) 0,h(x) 0}. g( x),h( x) E m = m g i ( x)h i ( x) =0 g i ( x)h i ( x) =0,i=1,..., m. i=1
1686 Nguyen Buong and Nguyen Thi Thuy Hoa Therefore, we consider the functions f i (x) =g i (x)h i (x) and set S 0 = {x E 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=0 S i or satisfying ϕ j ( x) = min ϕ j(x),j =1,..., N, (1.6) x E n 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). Tikhonov reguarization method for (1.1) is constructed on the base of soving the foowing equaity constraint optimization probem: find an eement x E n such that ϕ j ( x) = min ϕ j (x),j =1,..., N. x S 0 Consider the foowing minimization probem: find an eement x α E n satisfying F α (x α ) = min F α(x), (1.7) x E n where F α (x) is defined by F α (x) = F (x) 2 E + N α μ j ϕ m j (x)+α x x 2 E n, j=1 0 μ 1 <μ j <μ j+1 < 1,j =2,..., N 1, F (x) =(f 1 (x),..., f m (x)) T, and x is some eement in E n. It is we-known [27] that probem (1.7) has a soution x α for each α>0. 2. Main resuts We have the foowing resuts. Theorem 2.1. Let α k 0, as k. Then every sequence {x k }, where x k := x αk is a soution of (1.7) with α repaced by α k, has a convergent subsequence. The imit of every convergent subsequence is an x -minima norm soution (x -MNS). If, in addition, the x -MNS x is unique, then im k xk = x. Proof. From (1.7) it foows F (x k ) 2 E + N α μ j m k ϕ j(x k )+α k x k x 2 E n j=1 F (y) 2 E m + N j=1 α μ j k ϕ j(y)+α k y x 2 E n, (2.1)
Reguarization for noninear compementarity probems 1687 for each fixed eement y E n. Taking y N j=0 S j, we have F (y) =0, and ϕ j (x k ) ϕ j (y) =0,j =1,..., N. Then, from (2.1) it impies x k x E n y x E n. (2.2) Consequenty, {x k } is bounded. Let {x } {x k } be such that x x as. We sha prove that x is a soution of (1.1). Indeed, from (2.1) and ϕ j (x) 0 x E n we obtain 0 F (x ) 2 E m α y x 2 E 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) and ϕ j (x) 0 x E n,j =1,..., N, we can write x x 2 E n N ϕ 1 (y)+ ϕ 1 (x )+α 1 μ 1 j=2 α μ j μ 1 ϕ j (y)+α 1 μ 1 y x 2 E n. After passing in the ast inequaity we obtain ϕ 1 (x) ϕ 1 (y) y S 0. Note that x is a oca minimizer of ϕ 1 on S 0. But, because of S 0 S 1, x aso is a goba minimizer of ϕ 1 on E n. It means that x S 1. Further, we prove that x S 2. For any y S 0 S 1, from (2.1) and ϕ j (x) 0 x E n,j =1,..., N we have x x 2 E n N ϕ 2 (y)+ ϕ 2 (x )+α 1 μ 2 j=3 α μ j μ 2 ϕ j (y)+α 1 μ 2 y x 2 E n. By the simiar argument, we obtain x S 2, and x S j,j =3,..., N 1. Consequenty, x N 1 j=0 S j. In fina, we have to prove that x is a soution of (1.1). As above, for any y N 1 j=0 S j, from (2.1) it deduces that ϕ N (x) ϕ N (y) y j=0 N 1 S j. Hence, ϕ N (x) = min y N 1 j=0 S ϕ j N(y), i.e., x S N. The x -MNS property of x is foowed from (2.2). Theorem is proved now. Theorem 2.2. Assume that the foowing conditions hod: (i) F is differentiabe, (ii) there exists L>0 such that F ( x) F (z) E m L x z E n for z in some neighbouhood of x (iii) there exists ω E m such that x x = F ( x) ω (iv) L ω E m < 1.
1688 Nguyen Buong and Nguyen Thi Thuy Hoa Then, we have x k x E n = O( α k ). Proof. Using (1.7) with x = x we obtain F (x k ) 2 E m + N α μ j k ϕ j(x k )+α k x k x 2 E n j=1 (2.3) α k [ x x 2 E n xk x 2 E + n xk x 2 E n]. Since x N j=0s j, then ϕ j (x k ) ϕ j ( x) =0,j =1,..., N. From this fact and (2.3) we have F (x k ) 2 E m + α k x k x 2 E n 2α k ω, F ( x)( x x k ) E m. (2.4) Note that condition (ii) impies F (x k )=F ( x)(x k x)+r k (2.5) with r k E m 1 2 L xk x 2 En. (2.6) Combining (2.4)-(2.6) eads to Thus, F (x k ) 2 E m + α k x k x 2 E n 2α k ω E m F (x k ) E m + α k ω E ml x k x 2 E n. F (x k ) 2 E m + α k(1 ω E ml) x k x 2 R n 2α k ω E m F (x k ) E m. (2.7) Therefore, F (x k ) E m = O(α k ). Then, from (2.7) it foows x k x 0 E n = O( α k ). Theorem is proved. Remark. In the case that the given functions f i and ϕ j are smooth, then the functiona F α, wi have the same property, if instead of ϕ j in (1.7) we use ϕ 2 j. The reguarized soution x k of (1.7) or (2.9) can be approximated by the conjugate gradient method in [7], [22], [28]-[30], the method of trust region in [23], [24] or others in [26]. This work was supported by Vietnamese Natura Foudation for Science and Technoogy Deveopment 2009.
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