1. Introduction The nonlinear complementarity problem (NCP) is to nd a point x 2 IR n such that hx; F (x)i = ; x 2 IR n + ; F (x) 2 IRn + ; where F is


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1 New NCPFunctions and Their Properties 3 by Christian Kanzow y, Nobuo Yamashita z and Masao Fukushima z y University of Hamburg, Institute of Applied Mathematics, Bundesstrasse 55, D2146 Hamburg, Germany, z Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto 661, Japan. February 7, 1996 (Revised June 25, 1996) Abstract. Recently, Luo and Tseng proposed a class of merit functions for the nonlinear complementarity problem (NCP) and showed that it enjoys several interesting properties under some assumptions. In this paper, adopting a similar idea to Luo and Tseng's, we present new merit functions for the NCP, which can be decomposed into component functions. We show that these merit functions not only share many properties with the one proposed by Luo and Tseng but also enjoy additional favorable properties owing to their decomposable structure. In particular, we present fairly mild conditions under which these merit functions have bounded level sets. Key words: Nonlinear complementarity problem, NCPfunction, merit function, unconstrained optimization reformulation, error bound, bounded level sets. 3 The work of the second and third authors was supported in part by the Scientic Research GrantinAid from the Ministry of Education, Science and Culture, Japan. The work of the second author was also supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
2 1. Introduction The nonlinear complementarity problem (NCP) is to nd a point x 2 IR n such that hx; F (x)i = ; x 2 IR n + ; F (x) 2 IRn + ; where F is a function from IR n into itself, h1; 1i denotes the inner product in IR n, and IR n + := fx 2 IR n j x i ; i = 1;... ; ng [1, 2]. Recently, reformulations of the NCP as a minimization problem or a system of equations have drawn much attention [3]. A function which can constitute an equivalent minimization problem for the NCP is called a merit function. More precisely, a merit function is a function whose global minima on a set X IR n are coincident with the solutions of the original NCP. To construct such a function, it is eective to take advantage of the Cartesian structure of the NCP. In particular, the class of functions dened below serves as a convenient tool for constructing a merit function [4, 5, 6, 7]. Denition 1.1 A function : IR 2! IR is called an NCPfunction if (a; b) = () ab = ; a ; b : By using an NCPfunction, we can construct a merit function g : IR n! IR for the NCP as follows: If is nonnegative on IR 2, then dene the function g by g(x) = otherwise, by g(x) = nx i=1 nx i=1 (x i ; F i (x)); (x i ; F i (x)) 2 : It is easy to see that the NCP can be cast as the unconstrained optimization problem with the objective function g. There are many functions that belong to the class of NCPfunctions. Among others, the following three NCPfunctions have been well studied in the literature. Example 1.1 (i) N R (a; b) = minfa; bg. 1
3 (ii) MS (a; b) = ab (maxf; a bg2 a 2 + maxf; b ag 2 b 2 ); > 1. (iii) F B (a; b) = p a 2 + b 2 a b. The merit function based on the function NR is called the natural residual. The function MS is nonnegative on IR 2 [6] and the merit function based on MS is the implicit Lagrangian proposed by Mangasarian and Solodov [8]. The function F B was rst considered by Fischer [9] (and attributed to Burmeister). Recently, the merit function based on F B has extensively been studied and has shown to have a number of favorable properties. Among the above three NCPfunctions, the following useful relations hold. Proposition 1.1 [1, 11] There exist positive constants c1; c2; c3; c4 such that c1 NR (a; b) 2 F B (a; b) 2 c2 NR (a; b) 2 and c3 NR (a; b) 2 MS (a; b) c4 NR (a; b) 2 for all a; b 2 IR. This proposition indicates that if a merit function based on one of the three NCPfunctions provides an error bound or has bounded level sets, then the same is true for the merit functions based on the other NCPfunctions. To the authors' knowledge, for the merit functions based on these NCPfunctions to provide global error bounds, not only the strong monotonicity (or the uniform P property) but also the Lipschitz continuity of the function F involved in the NCP are required [12, 13]. The question whether or not the merit functions based on these NCPfunctions provide global error bounds without the Lipschitz continuity of F remains unanswered. Recently, Luo and Tseng [14] proposed a class of new merit functions f : IR n! IR dened by nx f(x) = (hx; F (x)i) + i=1 2 i(x i ; F i (x)); (1)
4 where : IR! [; 1) and i : IR 2! [; 1); i = 1;... ; n, are continuous functions that vanish on the negative orthant only. The function f is not based on an NCPfunction in the sense of Denition 1.1. Nevertheless, the function f enjoys several interesting properties under some assumptions as shown in [14]. Among others, f turns out to provide a global error bound under the strong monotonicity of F only, and it is convex on IR n under the assumption that hx; F (x)i and Fi(x); i = 1;... ; n, are all convex functions of x. We may naturally come up with the idea of constructing a new NCPfunction by using a similar approach to Luo and Tseng's. In view of the structure of the function f, we notice that the second term is decomposable into n component functions but the rst term is not. So, for constructing a new P merit function based on an NCPfunction, we may substitute the n decomposable function i=1 (xifi(x)) for the function (hx; F (x)i). In other words, we consider the following class of functions g : IR n! IR: where i : IR 2! IR is dened by nx g(x) = i(xi; Fi(x)); (2) i=1 i(a; b) = (ab) + i(a; b) (3) for each i = 1;... ; n. It can be shown that i is an NCPfunction, and hence g is a merit function for the NCP (see Lemma 2.1 and Theorem 2.1). Of course, the function g is not the same as the merit function f. Nevertheless, we can expect that g not only shares many properties with f but also enjoys additional favorable properties owing to its decomposable structure. A possible cost to pay is that g is in general nonconvex, while f becomes convex when F satises some conditions (see [14, Theorem 2.2]). The following concepts [15, 16] will be useful throughout the paper. (a) F : IR n! IR n is a Pfunction if max 1in x i 6=y i (xi yi)(fi(x) Fi(y)) for all x; y 2 IR n ; x 6= y: 3
5 (b) F : IR n! IR n is a P function if max 1in (x i y i )(F i (x) F i (y)) > for all x; y 2 IR n ; x 6= y: (c) F : IR n! IR n is a uniform P function (with modulus ) if there is a > such that max 1in (x i y i )(F i (x) F i (y)) kx yk 2 for all x; y 2 IR n : (d) (e) An n 2 n matrix M is a P matrix if every principal minor of M is nonnegative. An n 2 n matrix M is a P matrix if every principal minor of M is positive. It is known [16, Theorem 5.8] that if a P function is dierentiable, then its Jacobian matrices are P matrices. The paper is organized as follows. In the next section, we introduce a class of new merit functions. In Section 3 we show that the merit functions provide global error bounds for the NCP under the uniform P property of the function F. In particular, this result does not rely on the Lipschitz continuity of F. In Section 4 we show that the merit functions have bounded level sets under fairly mild assumptions. In Section 5 we consider conditions under which any stationary point of the merit functions turns out to be a solution of the NCP. In Section 6 we present a simple descent method for solving the NCP. We conclude the paper with some nal remarks in Section 7. We shall use the following notation. For a vector x 2 IR n and an index set f1; 2;... ; ng, x represents the vector with elements x i ; i 2. By k 1 k and k 1 k 1, we denote the Euclidean norm and l 1 norm, respectively. We denote the (transposed) Jacobian of F at x by rf (x), i.e., rf (x) is the matrix whose (i; j)th element j i. For a vector t 2 IR m, [t] + denotes the vector whose ith element is maxf; t i g. 2. Unconstrained optimization reformulations In this section, we show that the function i dened by (3) is an NCPfunction, so that the unconstrained optimization problem constituted from this function is equivalent to the NCP. 4
6 To begin with, following [14], we dene a class of functions, to which the functions i that constitute i are supposed to belong. Denition 2.1 The class of continuous functions : IR m! [; 1) such that (t) = () t is denoted by 9 m. (3). The following lemma shows the most fundamental property of the function i dened by Lemma 2.1 For any and i 2 9 2, the function i dened by (3) is a nonnegative NCPfunction. Proof. Since and i are nonnegative by denition, i is nonnegative. Now, we prove that i is an NCPfunction. First, suppose that (a; b) is a vector such that a ; b and ab =. Then, by the denition of 9 m, we have (ab) = and i(a; b) =, which yields i (a; b) =. Next, suppose that i (a; b) =. Then, by the nonnegativity of and i, we must have (ab) = and i(a; b) =. Moreover, by the denition of 9 m, we obtain ab, a and b, from which it follows that ab =. Hence, i is an NCPfunction. 2 The following theorem establishes the equivalence between the NCP and the unconstrained minimization problem min x2ir n g(x); where g is dened by (2). Theorem 2.1 Let i : IR 2! IR be dened by (3) with arbitrary and i for i = 1;... ; n, and let g : IR n! IR be dened by (2). Then g is nonnegative on IR n. Moreover, g(x) = if and only if x is a solution of the NCP. 5
7 Proof. By Lemma 2.1, i are nonnegative NCPfunctions for all i = 1;... ; n. Moreover, g(x) = if and only if i (x i ; F i (x)) = for all i = 1;... ; n. Hence the theorem follows. 2 We provide examples of functions that belong to 9 m ; m = 1; 2. Denition 2.2 For p >, let p : IR! [; 1) be dened by p (t) = maxf; tg p ; and let p I ; p II ; p F B ; p M : IR 2! [; 1) be dened, respectively, by p I(t) = k[t] + k p 1 = (maxf; t 1 g + maxf; t 2 g) p ; p II(t) = k[t] + k p = maxf; t 1 g 2 + maxf; t 2 g 2 p=2 ; p p F B(t) = [ F B (t)] + = max ; p M(t) = maxf; t 1 ; t 2 g p : q t t t 1 + t 2 p ; It is easy to observe that p and p I ; p II ; p F B ; p M Moreover, for p > 1, the functions, p p II and p F B are continuously dierentiable up to (p 1)th order. Using the functions given in Denition 2.2, we present some examples of the NCPfunction i dened by (3). Denition 2.3 For p >, let p I ; p II ; p F B ; p M : IR 2! [; 1) be dened, respectively, by p I(a; b) = (ab) p + I(a; p b); p II(a; b) = (ab) p + II(a; p b); p F B(a; b) = (ab) p + p F B(a; b); p M(a; b) = (ab) p + M(a; p b): 6
8 For p > 1, the functions p II and p F B are continuously dierentiable up to (p 1)th order. Now we display four special cases of g, which correspond to the functions given in Denition 2.3. Denition 2.4 For p >, let g p I ; gp II ; gp F B ; gp M : IRn! [; 1) be dened, respectively, by g p I (x) = nx g p II (x) = nx g p F B (x) = nx g p M (x) = nx p I (x i; F i (x)); i=1 p II (x i; F i (x)); i=1 p F B (x i; F i (x)); i=1 p M (x i; F i (x)): i=1 Note that for p > 1, the functions g p II and gp F B are continously dierentiable up to (p 1)th order, provided that F is continuously dierentiable up to (p 1)th order. 3. Global error bounds In this section, we show that if F is a uniform P function and if and 1;... ; n satisfy suitable assumptions, then the merit function g dened by (2) provides a global error bound for the NCP. To the authors' knowledge, this is the rst time that a global error bound result is established for a merit function under the uniform P function property only. For example, for the squared FischerBurmeister function and the implicit Lagrangian, a similar result is currently known only when F is not only a uniform P function but also a globally Lipschitzcontinuous function [4, 12, 17]. The error bound results established below rely on the following growth assumption on and i ; i = 1;... ; n. Assumption 3.1 The functions and i ; i = 1;... ; n, have growth of order p > in the sense that there exists a positive constant such that, for each i = ; 1;... ; n, 7
9 i satises i(t) k[t] + k p for all t: The next proposition shows that all the functions given in Denition 2.2 have growth of order p. Proposition 3.1 The functions p, p I, p II, p F B and p M have growth of order p. Proof. It is easy to show that the functions Here we only prove that p, p I, p II and p M have growth of order p. 1 F B has growth of order 1. Then it will be easy to verify that p F B has growth of order p for any p >. If t 1 and t 2 are both nonnegative, then we have 1 F B qt (t) = t2 2 + t 1 + t 2 q t t2 2 = k[t] + k: If t 1 and t 2 are both nonpositive, then we have 1 F B (t) = = k[t] +k. Finally, without loss of generality, we suppose t 1 > and t 2 <. Then we have 1 F B (t) = qt t2 2 + t 1 + t 2 t 1 = k[t] + k: The proof is complete. 2 The next result relates the function g to the function gi 1. Recall that the function g1 I : IR n! IR is dened by g 1 I (x) = nx = i=1 nx i=1 1 I (x i; F i (x)) ([x i F i (x)] + + [F i (x)] + + [x i ] + ) : 8
10 Lemma 3.1 Let Assumption 3.1 be satised and the function g be dened by (2). Then, there exists a positive constant such that gi(x) 1 g(x) 1=p for all x 2 IR n : (4) Proof. The inequality can be shown to hold in a way similar to [14, Lemma 3.1]. 2 The inequality (4) shows that if gi 1 provides a global error bound, then g also provides a global error bound. We are now in a position to prove the main theorem of this section. Theorem 3.1 Suppose that F is a uniform P function. Let Assumptions 3.1 be satised and the function g be dened by (2). Then there exists a positive constant such that kx x 3 k 2 g(x) 1=p for all x 2 IR n ; (5) where x 3 is the unique solution of the NCP. Proof. Since F is a uniform P function, there exists a constant > such that kx x 3 k 2 max 1in (x i x 3 i )(F i (x) F i (x 3 )) (6) for all x 2 IR n. Also, since x 3 solves the NCP, we have x 3 i ; F i(x 3 ) and x 3 i F i(x 3 ) = for all i. Therefore we have, for each i and all x 2 IR n, (x i x 3 i )(F i (x) F i (x 3 )) = x i F i (x) x 3 i F i(x) F i (x 3 )x i [x i F i (x)] + + x 3 i [F i (x)] + + F i (x 3 )[x i ] + i ([x i F i (x)] + + [F i (x)] + + [x i ] + ) (7) = i 1 I(x i ; F i (x)); where i := maxf1; x 3 i ; F i(x 3 )g >. Let ^ := max 1in i : Then, combining (6) and (7), we obtain kx x 3 k 2 max 1in i 1 I(x i ; F i (x)) ^ nx i=1 1 I(x i ; F i (x)) = ^ g 1 I(x): The desired inequality (5) then follows from Lemma Proposition 3.1 and Theorem 3.1 indicate that the functions g p I ; gp II ; gp F B and g p M all satisfy the inequality (5) and hence provide global error bounds if F is a uniform P function. 9
11 4. Bounded level sets In this section we consider conditions which guarantee that the level sets L(c) := fx 2 IR n j g(x) cg are bounded for all c : To establish the bounded level set results, we need the following assumption on the functions and i ; i = 1;... ; n; which constitute the merit function g. Assumption 4.1 The functions and i ; i = 1;... ; n, have the property that, for each i = ; 1;... ; n, k[t] + k! 1 implies i(t)! 1. Note that every merit function g given in Denition 2.4 is constituted from functions i satisfying Assumption 4.1, cf. Proposition 3.1. We rst deal with monotone problems. Before stating the result, we note that the particular level set L() is equal to the solution set of the NCP. Hence, if the NCP has an unbounded solution set, the level sets L(c) cannot be bounded for any c since L() L(c) for all c : Since it can happen that the monotone complementarity problem has an unbounded solution set (e.g., take F (x) ), we need an additional assumption, namely that the NCP is strictly feasible, i.e., that there exists a vector ^x > with F (^x) > : Theorem 4.1 Assume that F is monotone and the NCP is strictly feasible. Let g be dened by (2) and Assumption 4.1 be satised. Then the level sets L(c) are bounded for all c : Proof. Assume there exists an unbounded sequence fx k g L(c) for some c. We rst note that there is no index j such that x k j! 1 or F j (x k )! 1 on a subsequence, because otherwise we would have j(x k j ; F j(x k ))! 1 by Assumption 4.1, which in turn would imply g(x k )! 1 on a subsequence. Hence all components of the two sequences fx k g and ff (x k )g are bounded from below. On the other hand, the sequence fx k g is unbounded 1
12 by assumption. Hence there exists an index j such that x k j! 1 on a subsequence. Now let ^x satisfy the strict feasibility condition for the NCP. Since F is monotone, we have hx k ; F (^x)i + h^x; F (x k )i hx k ; F (x k )i + h^x; F (^x)i: (8) Since ^x > and F (^x) > ; it follows immediately from the previous remarks that the lefthand side of (8) tends to innity. Hence hx k ; F (x k )i! 1 on a subsequence. Thus, there is an index l such that x k l F l(x k )! 1 on a subsequence. By Assumption 4.1, this implies that (x k l F l(x k )) and therefore g(x k ) itself tend to innity on a subsequence. This is the desired contradiction to fx k g L(c): 2 Note that this bounded level set result for the monotone NCP is also true for the merit function proposed by Luo and Tseng [14]. This follows immediately from the denition of the function f in (1) and the proof given for Theorem 4.1. However, for the merit functions constituted by the NCPfunctions of Example 1.1, this result does not hold. In fact, consider the simple example with n = 1, where F (x) 1. Then F is monotone and the NCP is strictly feasible. However, for the merit function constituted by the NCPfunction NR, we have L NR (1) := fx 2 IR j minfx; F (x)g 2 1g = [1; 1); which is unbounded. By Proposition 1.1, this also implies that the implicit Lagrangian and the squared FischerBurmeister function, which are constituted from MS and F B respectively, have unbounded level sets for this example. We next present another condition which guarantees the boundedness of the level sets. This result is a straightforward modication of the corresponding result by Luo and Tseng [14, Theorem 4.1]. Theorem 4.2 Let g be dened by (2) and Assumption 4.1 be satised. Suppose that, for any sequence fx k g such that kx k k! 1, [x k ] + =kx k k! and [F (x k )] + =kx k k!, we have lim inf k!1 max i fx k i F i(x k )g kx k k > : 11
13 Then the level sets L(c) are bounded for all c : Proof. The theorem can be proved in a way similar to [14, Theorem 4.1]. 2 We can easily verify that lim inf k!1 hx k ; F (x k )i kx k k > =) lim inf k!1 max i fx k i F i(x k )g kx k k > : So, the assumptions on F in Theorem 4.2 are weaker than those in [14, Theorem 4.1], which hold whenever F is a uniform P function or an R function in the sense of Chen and Harker [18]. In particular, the level sets are bounded if F is an ane mapping such that F (x) = Mx + q with an R matrix M. 5. Stationary points Luo and Tseng [14] gave conditions under which the merit function f dened by (1) is convex. However, the merit function g given by (2) is not necessarily convex under such conditions. Nevertheless, since commonly used minimization methods can only nd a stationary point or a local minimum, it is important to investigate conditions under which any stationary point of g is a solution of the NCP. In this section, we focus our attention to such conditions. In what follows, we suppose the functions i consist of continuously dierentiable functions and i Then i are also continuously dierentiable i (a; i (a; b) = b i(a; b) ; (9) = a i(a; b) : (1) Moreover, if F is continuously dierentiable, then the function g dened by (2) is also continuously dierentiable. We dene the following classes of continuously dierentiable functions, which are given by Luo and Tseng [14]. 12
14 Denition m + denotes the class of continuously dierentiable functions 2 9m such that C.1 ht; r (t)i for all t 2 IR m ; C.2 r (t) for all t 2 IR m ; C.3 r (t) = () t. Moreover, 9 m ++ denotes the class of functions 2 9m + such that C.4 r (t) > for all t 6. The functions the function p, p II and p F B with p > 1 belong to 91 +, 92 + and 92 +, respectively. However, p II with p > 1 does not belong to 92 ++, whereas the function p F B with p > 1 belongs to Note that p I and p M are not dierentiable and hence belong to neither 92 + nor The next lemma shows some important properties of the NCPfunction i constituted by i belonging to the classes of functions dened above. Lemma 5.1 Let the NCPfunction i be dened by (3) with and i Then, i has the following properties. (P.1) i (a; b) = () r i (a; b) = ; i(a; i (a; b) for all (a; b) T 2 IR 2. Moreover, if i , then we have i(a; b) = i(a; b) =. Proof. (P.1) First, suppose that i (a; b) =. Then, since i are nonnegative functions, we have ab ; a b 1 C A : 13
15 Hence, C.3 yields (ab) = and r i(a;b) =. It then follows from (9) and (1) that r i (a; b) =. Next, suppose that r i (a; b) =. Then, by (9) and (1), we obtain and b a (ab) i(a;b) (ab) i(a;b) Let the both sides of (11) and (12) be nonzero. Then, since (11) : (12) (ab) and r i(a;b) by C.2, we must have a > and b > from (11) and (12). It then follows from C.3 that r i(a;b) =, which is a contradiction. Therefore, the both sides of at least one of (11) and (12) are zero. Without loss of generality, we assume the both sides of (11) are zero. Then, by C.3, we obtain ab or b =. It can easily be veried by considering the two cases ab and b = separately that the both sides of (12) are zero. Hence, we have r i(a;b) = and (ab) =. It then follows from C.3 that a, b and ab, which yield a, b and ab =. Since i is an NCPfunction, we obtain i (a; b) =. (P.2) By C.1 and C.2, we have ab (ab) 2 = (ab (ab)) (ab) : (13) We also have from C.2 Furthermore, C.1 and C.2 yield (ab) = From (9) and (1), we i(a; i(a; b) i(a;b) : (14)! i(a;b) (ab) D (a;b) ;r T i(a;b) E : i (a; i (a; b) = ab + (ab) 2 i(a;b) (ab) i(a; b)! i(a;b) ; (16) 14
16 which shows (P.2) by (13)(15). (P.3) Suppose i(a;b) = ab + =. Then, by (16), we have (ab) 2 i(a; i(a; b) (ab) i(a; b) i(a; b) Hence, by (13)(15), every term of the righthand side must be zero. Moreover, C.3 and the rst term being zero yield ab. If either a or b is negative, then the second term is positive by C.4. This is a contradiction. So, we must have a and b. Hence, we obtain ab =, a and b, which yield i (a; b) =. Then, by (P.1) proved earlier, we i(a;b) =. We can prove the converse in a similar way. 2 Note that the functions p II and p F B with p > 1 enjoy properties (P.1) and (P.2). Moreover, the function p F B with p > 1 has property (P.3), whereas the function p II with p > 1 does not. Now, we give a necessary and sucient condition for a stationary point of g to be a solution of the NCP. To this end, we dene the index sets ( P(x) := i i (x i ; F i (x)) > ( ) C(x) i i (x i ; F i (x)) = ( ) N (x) i i (x i ; F i (x)) < R(x) := P(x) [ N (x): For simplicity, we shall denote these sets by P, C, N and R. The point x under consideration will always be clear from the context. Using these index sets, we dene regularity conditions. Similar concepts of regularity have been considered by De Luca et al. [19], Facchinei and Kanzow [2] and Luo and Tseng [14] for dierent merit functions. ; ; ;! : Denition 5.2 A point x 2 IR n is said to be regular if for every nonzero vector z 2 IR n such that z C = ; z P > ; z N < ; 15
17 there exists a vector y 2 IR n such that y P ; y N ; y R 6= and hy; rf (x)zi : Moreover, a point x 2 IR n is said to be strictly regular if for every nonzero vector z 2 IR n such that z C = ; z P > ; z N < ; there exists a vector y 2 IR n such that y C = ; y P ; y N and hy; rf (x)zi > : The rst regularity in Denition 5.2 is similar to that of [19], whereas the second strict regularity is similar to that of [2], where it is simply called regular. We next present a sucient condition for a vector x 2 IR n to be (strictly) regular. Lemma 5.2 Let x 2 IR n : If rf (x) is a Pmatrix (P matrix), then x is a (strictly) regular vector. Proof. We only consider the case where rf (x) is a Pmatrix. The proof is similar for the case where rf (x) is a P matrix, see also [2]. So let z 2 IR n be a nonzero vector with z C = ; z P > and z N < : Since rf (x) is a Pmatrix, there exists an index j 2 R such that z j [rf (x)z] j : Now choose y 2 IR n in such a way that y j = z j and y i = for all i 6= j: Then nx hy; rf (x)zi = y i [rf (x)z] i = y j [rf (x)z] j = z j [rf (x)z] j ; i=1 i.e., x is regular. 2 16
18 Similarly to Lemma 5.2, one can show that if rf (x) is (strictly) semimonotone and N = ;; then x is (strictly) regular, see [15] for the denition of a (strictly) semimonotone matrix. The following theorem shows that the (strict) regularity provides a necessary and sucient condition for a stationary point of g to be a solution of the NCP. Theorem 5.1 Let the NCPfunctions i be dened by (3) with and i (respectively, i ) for all i = 1;... ; n, and let g be dened by (2). Suppose that x is a stationary point of g. Then, x is a solution of the NCP if and only if x is regular (respectively, strictly regular). Proof. By using Lemma 5.1, it can be proved in a way similar to the proof of [19, Theorem 4.2] for the case where i are constituted by and i , and the proof of [2, Theorem 3.3] for the case where i are constituted by and i Theorem 5.1 indicates that a necessary and sucient condition for any stationary point x of g p F B with p > 1 to be a solution of the NCP is that x is a regular point, whereas such a condition for g p II with p > 1 is that x is a strictly regular point. From Lemma 5.2, we obtain the following corollary, which gives sucient conditions under which any stationary point of g solves the NCP. Corollary 5.1 Let the NCPfunctions i be dened by (3) with and i (respectively, i ) for all i = 1;... ; n, and let g be dened by (2). If F is a P function (respectively, rf (x) is a P matrix for any x), then any stationary point of g is a solution of the NCP. 2 Corollary 5.1 indicates that a sucient condition for any stationary point of g p F B with p > 1 to be a solution of the NCP is that F is a P function, whereas such a condition for g p II with p > 1 is that rf (x) is a P matrix. As an application of the previous results, we obtain the following corollary. 17
19 Corollary 5.2 Assume that F is monotone and the NCP is strictly feasible. Then the NCP has a nonempty and bounded solution set. Proof. We consider g 2 F B. Note that, by Theorem 4.1, the level set L(c) of g 2 F B is nonempty and compact whenever c g 2 F B(x) for some x 2 IR n : Hence, by the BolzanoWeierstrass Theorem, the function g 2 F B has a global minimum x 3. In particular, x 3 is a stationary point of g 2 BF. Since F is monotone, rf (x 3 ) is positive semidenite and therefore a P  matrix. Hence g 2 BF (x 3 ) = by Corollary 5.1, i.e., x 3 solves the NCP. The boundedness of the solution set follows from the fact that the level set L() is bounded. 2 We note that the results stated in Corollary 5.2 are known to hold under even weaker conditions [1, Theorem 3.4]. But the above proof is completely dierent from classical proofs and, in particular, quite elementary. 6. Descent methods Similar to, e.g., [5, 21, 14, 17], we present in this section a simple descent method which makes no use of the derivatives of F. This method is therefore applicable in particular to largescale problems. Throughout this section, we will use the following F (x))! T 1(x 1 ; F 1 n (x n ; F n (x)) ;... ; 2 IR n F (x)) As a search direction, we will take the vector! T 1(x 1 ; F 1 n (x n ; F n (x)) ;... ; 2 IR n : d(x) F (x)) : (17) The following lemma shows that this direction is a descent direction of the merit function g with and i ; i = 1;... ; n, if rf (x) is positive semidenite. 18
20 Lemma 6.1 Let the NCPfunctions i be dened by (3) with and i for all i = 1;... ; n, and let g be dened by (2). If rf (x) is positive semidenite, then the direction d(x) given by (17) is a descent direction for 9 at x as long as x is not a solution of the NCP. Proof. Suppose that x is not a solution of the NCP. We recall that the gradient of g can be written as rg(x) = Hence we have hrg(x); d(x)i = F F F (x)) ; + rf (x) + F (x)) F F (x)) ; rf (x) By property (P.2) shown in Lemma 5.1, the rst term is nonpositive. Since rf (x) is positive semidenite, the second term is also nonpositive. Hence we have + : (18) hrg(x); d(x)i : Assume now that hrg(x); d(x)i = : This implies the both terms of the righthand side of (18) vanish. In particular, by the abovementioned property (P.2), we i (x i ; F i i (x i ; F i (x)) = for all i = 1;... ; n: It then follows from properties (P.1) and (P.3) in Lemma 5.1 that i (x i ; F i (x)) = for all i. Hence x solves the NCP, a contradiction to the assumption. Therefore we must have hrg(x); d(x)i < ; i.e., d(x) is a descent direction of g at x: 2 Based on Lemma 6.1, we now present a derivativefree descent algorithm for solving the NCP. Algorithm 6.1 (Descent method) (S.) Choose x 2 IR n ; " ; 2 (; 1=2); 2 (; 1) and set k := : (S.1) If g(x k ) "; stop. 19
21 (S.2) Let d k ; F (x k )) : (S.3) Compute a steplength t k = m k, where m k is the smallest nonnegative integer satisfying the Armijotype condition g(x k + m k d k ) (1 ( m k ) 2 )g(x k ): (S.4) Set x k+1 := x k + t k d k ; k := k + 1 and go to (S.1). We summarize the convergence properties of Algorithm 6.1 in the following theorem. Here we suppose that " = in the algorithm and that the generated sequence is innite. Theorem 6.1 Let the NCPfunctions i be dened by (3) with and i for all i = 1;... ; n, and let g be dened by (2). If F is monotone, then Algorithm 6.1 is welldened, and any accumulation point of the generated sequence fx k g is a solution of the NCP. Moreover, if the NCP is strictly feasible, then there exists at least one accumulation point. Proof. Since F is monotone, rf (x) is positive semidenite for any x. Hence, by Lemma 6.1, d k is a descent direction of g at x k for each k. Then the rst part of the theorem can be proved in exactly the same way as Proposition 11 in [21]. (In [21], Jiang considers a similar algorithm based on the squared FischerBurmeister function.) The last part follows from Theorem 4.1 and the descent property of Algorithm Theorem 6.1 demonstrates in particular that the proposed algorithm is valid for the merit function g constituted by the NCPfunctions p F B. For the merit function g constituted by and i for i = 1;... ; n, it is also possible to show that d k! as k! 1, so ;F (x 3 )) = holds at any accumulation point of the generated sequence. However, unlike the case of i for i = 1;... ; n, this does not necessarily imply that g(x 3 ) =, i.e., x 3 solves the NCP (cf. Lemma 5.1). To cope with this diculty, we may slightly modify the search direction as d k ; F (x k )) ; F (x k )) ;
22 where is a suciently small positive constant. Then, the convergence of the modied algorithm will be proved in a similar way to [17]. We will not go into detail here, however. 7. Final remarks In this paper, we have introduced new merit functions for the nonlinear complementarity problem and proved several desirable properties of these functions, which compare favorably with corresponding properties of other existing merit functions. On the other hand, one of the reasons why the squared FischerBurmeister function has attracted much attention of the researchers is the fact that it is the natural dierentiable merit function for the following equation reformulation of the nonlinear complementarity problem: 8 F B (x) := 1 C F B (x 1 ; F 1 (x)). A = ; F B (x n ; F n (x)) where F B is the function displayed in Example 1.1. Similarly, we can view the merit functions presented in [14] and this paper as natural merit functions of the following two equations, respectively: 8 LT (x) := and 8(x) := (hx; F (x)i) 1(x 1 ; F 1 (x)). n(x n ; F n (x)) (x 1 F 1 (x)). (x n F n (x)) 1(x 1 ; F 1 (x)). n(x n ; F n (x)) 21 1 C A 1 C A = = ;
23 where and i ; i = 1;... ; n. Note that these are generalized nonlinear least squares reformulations of the nonlinear complementarity problem, and that there is no corresponding equation for the implicit Lagrangian by Mangasarian and Solodov [8]. References [1] Harker, P.T., and Pang, J.S., FiniteDimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms and Applications, Mathematical Programming, Vol. 48, pp. 161{22, 199. [2] Pang, J.S., Complementarity Problems, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, MA, pp. 271{338, [3] Fukushima, M., Merit Functions for Variational Inequality and Complementarity Problems, Nonlinear Optimization and Applications, Edited by G. Di Pillo and F. Giannessi, Plenum Publishing Corporation, New York, NY, to appear. [4] Fischer, A., An NCPFunction and Its Use for the Solution of Complementarity Problems, Recent Advances in Nonsmooth Optimization, Edited by D.Z. Du, L. Qi and R.S. Womersley, World Scientic Publishers, Singapore, pp. 88{15, [5] Geiger, C., and Kanzow, C., On the Resolution of Monotone Complementarity Problems, Computational Optimization and Applications, Vol. 5, pp. 155{173, [6] Kanzow, C., Nonlinear Complementarity as Unconstrained Optimization, Journal of Optimization Theory and Applications, Vol. 88, pp. 139{155, [7] Kanzow, C., and Kleinmichel, H., A Class of NewtonType Methods for Equality and Inequality Constrained Optimization, Optimization Methods and Software, Vol. 5, pp. 173{198,
24 [8] Mangasarian, O.L., and Solodov, M.V., Nonlinear Complementarity as Unconstrained and Constrained Minimization, Mathematical Programming, Vol. 62, pp. 277{ 297, [9] Fischer, A., A Special NewtonType Optimization Method, Optimization, Vol. 24, pp. 269{284, [1] Luo, Z.Q., Mangasarian, O.L., Ren, J., and Solodov, M.V., New Error Bounds for the Linear Complementarity Problem, Mathematics of Operations Research, Vol. 19, pp. 88{892, [11] Tseng, P., Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 89, pp. 17{37, [12] Kanzow, C., and Fukushima, M., Equivalence of the Generalized Complementarity Problem to Dierentiable Unconstrained Minimization, Journal of Optimization Theory and Applications, to appear. [13] Pang, J.S., A Posteriori Error Bounds for the LinearlyConstrained Variational Inequality Problem, Mathematics of Operations Research, Vol. 12, pp. 474{484, [14] Luo, Z.Q., and Tseng, P., A New Class of Merit Functions for the Nonlinear Complementarity Problem, Working Paper, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada, [15] Cottle, R.W., Pang, J.S., and Stone, R.E., The Linear Complementarity Problem, Academic Press, New York, NY, [16] More, J.J., and Rheinboldt, W.C., On P  and SFunctions and Related Classes of ndimensional Nonlinear Mappings, Linear Algebra and Its Applications, Vol. 6, pp. 45{68,
25 [17] Yamashita, N., and Fukushima, M., On Stationary Points of the Implicit Lagrangian for Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 84, pp. 653{663, [18] Chen, B., and Harker, P.T., Smooth Approximations to Nonlinear Complementarity Problems, SIAM Journal on Optimization, to appear. [19] De Luca, T., Facchinei, F., and Kanzow, C., A Semismooth Equation Approach to the Solution of Nonlinear Complementarity Problems, Mathematical Programming, to appear. [2] Facchinei, F., and Kanzow, C., On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian, Journal of Optimization Theory and Applications, to appear. [21] Jiang, H., Unconstrained Minimization Approaches to Nonlinear Complementarity Problems, Journal of Global Optimization, to appear. 24
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