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A New Derivative-Free Descent Method for the Nonlinear Complementarity Problem

1

Kenjiro Yamada , Nobuo Yamashita

2

2

and Masao Fukushima

July 22, 1998

Recently, much eort has been made in solving and analyzing the nonlinear complementarity problem (NCP) by means of a reformulation of the problem as an equivalent unconstrained optimization problem involving a merit function. In this paper, we propose a new merit function for the NCP and show several favorable properties of the proposed function. In particular, we give conditions under which the function provides a global error bound for the NCP and conditions under which its level sets are bounded. Moreover, we propose a new derivative-free descent algorithm for solving the NCP based on this function. We show that any accumulation point generated by the algorithm is a solution of the NCP under the monotonicity assumption on the problem. Also, we prove that the sequence generated by the algorithm converges linearly to the solution under the strong monotonicity assumption. The most interesting feature of the algorithm is that the search direction and the stepsize are adjusted simultaneously during the linesearch process, whereas a xed search direction is used in the linesearch process in earlier derivative-free algorithms proposed by Geiger and Kanzow, Jiang, Mangasarian and Solodov, and Yamashita and Fukushima. Making use of this particular feature, we can establish the linear convergence of the algorithm under more practical assumptions compared with the linearly convergent derivative-free algorithm recently proposed by Mangasarian and Solodov. Abstract.

Derivative-free method, nonlinear complementarity problem, merit function, global convergence, linear convergence.

Keywords.

1 2

IBM Japan, Ltd., 19-21, Nihonbashi Hakozaki-cho, Chuo-ku, Tokyo 103-8510 Japan. e-mail:

[email protected]

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto

606-8501, Japan. e-mail:

fnobuo,[email protected]

1.

Introduction

The nonlinear complementarity problem (NCP) [11; 20] is to nd a point x 2 Rn such that x  0; F (x)  0; xT F (x) = 0; (1) where F : Rn ! Rn is a given function. Throughout this paper, we assume that F is continuously dierentiable. Applications of the NCP can be found in many important elds such as mathematical programming, economics, engineering and mechanics [2; 11]. A variety of methods have been proposed for solving the NCP [9; 11; 20]. One of the powerful approaches for solving the NCP that have been studied intensively in recent years is to reformulate the NCP as an equivalent system of nonlinear equations [16; 21; 25] or as an equivalent unconstrained optimization problem [3; 5; 6; 10; 12; 14; 18; 22; 24]. The objective function of such an equivalent unconstrained optimization problem is called a merit function. In other words, a merit function for the NCP is a function whose global minima are coincident with the solutions of the NCP. To construct a merit function, the class of functions called NCP-functions have played an important role. A function  : R2 ! R is called an NCP-function if it satis es (a; b) = 0 () a  0; b  0; ab = 0: An NCP-function  is said to be nonnegative if (a; b)  0 for all (a; b)T 2 R2: It is clear that if  is a nonnegative NCP-function on R2, then the function 9 : Rn ! R de ned by n 9(x) = (xi; Fi(x)): i=1 is a merit function for the NCP. In this paper, we are particularly interested in the following merit functions constructed by some well-known NCP-functions. The natural residual 9NR : Rn ! Rn [11] is a merit function de ned by n (2) 9NR(x) = NR(xi; Fi (x)); i=1 where NR : R2 ! R is a nonnegative NCP-function given by NR (a; b) = minfa; bg2 : Note that the function NR is not dierentiable everywhere, hence neither is 9NR. So, the equivalent unconstrained optimization problem of minimizing the function 9NR cannot be solved by using general algorithms that require the dierentiability of the objective function. Mangasarian and Solodov [17] proposed the implicit Lagrangian 9MS : Rn ! R de ned by n 9MS (x) = MS (xi; Fi(x)); i=1

X

X

X

1

where MS : R2 ! R is a nonnegative NCP-function given by 1 (0b + a)2 0 a2 + (0a + b)2 0 b2 ; MS (a; b) = ab + + + 2 where  > 1 is a xed constant and (z)+ = maxf0; zg for z 2 R. The function MS is continuously dierentiable, and so is 9MS . Any stationary point of 9MS is a solution of the NCP whenever F is a P -function [4] (see also [24]). Another well-known merit function is the function 9FB : Rn ! R de ned by n 9FB (x) = FB (xi; Fi(x)); (3)

n

o

X

where FB : R2 ! R is given by

i=1

p

1 ( a2 + b2 0 a 0 b)2: (4) 2 The function FB is a nonnegative NCP-function introduced by Fischer [6] and called the FischerBurmeister function. This function has been extensively studied and shown to have a number of favorable properties. For example, F B is continuously dierentiable on R2, and hence 9FB is also continuously dierentiable. Moreover, any stationary point of 9FB is a solution of the NCP whenever F is a P0-function [5]. Note that this condition is weaker than the one required for the implicit Lagrangian to have the same property. For solving an equivalent unconstrained optimization problem based on a merit function, extensions of classical iterative methods such as Newton's method have been proposed. Recently, so-called derivative-free methods have attracted much attention, which do not require computation of derivatives of F and make use of the particular properties of a merit function [8; 10; 12; 18; 24; 27]. Derivative-free methods are particularly suitable for problems where the derivatives of F are not available or are extremely expensive to compute. For the variational inequality problem (VIP), which contains the NCP as a special case, Fukushima [8] proposed a derivative-free descent method based on the regularized gap function and showed its global convergence under the strong monotonicity assumption on the problem mapping. More recently, Yamashita, Taji and Fukushima [27] proposed a derivative-free descent method for solving the VIP based on the D-gap function and established global convergence of the method under similar assumptions. For the NCP, Yamashita and Fukushima [24] proposed a derivative-free descent method based on the implicit Lagrangian 9MS . They showed that their method is globally convergent under the strong monotonicity assumption on F . Jiang [12] proposed a derivative-free descent method based on the Fischer-Burmeister function. He showed that his method converges globally to a solution of the NCP if F is monotone. Geiger and Kanzow [10] proposed a similar method and proved its global convergence under somewhat stronger assumptions than [12]. Note however that convergence rate of the proposed methods has not been discussed for the above mentioned methods for the VIP and the NCP. Recently, Mangasarian and Solodov [18] slightly modi ed the derivative-free descent algorithm of [24] and showed that the sequence generated by the modi ed algorithm converges linearly to the solution of the NCP. However, for their algorithm to have a global convergence FB (a; b) =

2

property, the strong monotonicity assumption on F is needed. Moreover, in order to ensure the linear rate of convergence, some parameters used in the algorithm have to be speci ed by using usually unknown constants such as the modulus of strong monotonicity of F . In this paper, we propose the following merit function 9 : Rn ! R for the NCP : 9(x) =

X  (x ; F (x)); n

 i

i=1

i

(5)

where  : R2 ! R is a nonnegative NCP-function given by 1  (6)  (a; b) = (ab)2+ + ( a2 + b2 0 a 0 b)2 2 2 and   0 is a real parameter. The function 9 is a slight modi cation of the function proposed in [26] (see also [1]). When  = 0, the function  reduces to the Fischer-Burmeister function F B , so  is considered an extension of the Fischer-Burmeister function. We show that  inherits several favorable properties of the Fischer-Burmeister function. In addition, when  > 0, we can show that  enjoys an important property that the Fischer-Burmeister function does not have. In particular, we give conditions under which 9 provides a global error bound for the NCP and conditions under which it has bounded level sets. Moreover, we propose a new derivative-free descent algorithm for solving the NCP based on the function 9. We prove that this algorithm has a global convergence property if F is a monotone function, and that the sequence generated by the algorithm converges linearly to the solution if F is a strongly monotone function. In contrast with the algorithm proposed in [18], proper values of the parameters used in our algorithm do not rely on usually unknown constants such as the modulus of strong monotonicity of F . This paper is organized as follows. In Section 2, we review some de nitions and preliminary results that will be used in the subsequent analyses. In Section 3, we show some important properties of the proposed merit function. In Section 4, we present a derivative-free descent algorithm for solving the NCP and prove global convergence of the algorithm, while in Section 5, we prove its linear convergence. Section 6 makes some concluding remarks. Throughout the paper, we shall adopt the following notations: For a real-valued matrix A of any dimension, AT denotes its transpose. For a dierentiable vector-valued function F : Rn ! Rn , rF denotes the transposed n 2 n Jacobian matrix whose columns are the gradients of the components of F and the ith component function of F is denoted by Fi. The norm k 1 k denotes the Euclidean norm. For z 2 R, dze denotes the smallest integer no less than z . The level set of a function 9 : Rn ! R is denoted L(9; c) := fx 2 Rn j 9(x)  cg.

p

2.

Preliminaries

In this section, we recall some de nitions and preliminary results that will be used in the subsequent analyses. The following classes of functions F : Rn ! Rn play an important role in the NCP. De nition 2.1

Let

S

be a convex set in

Rn .

3

(a) F is monotone on S if (x 0 x0 )T (F (x) 0 F (x0))  0 for all x; x0 2 S:

(b) F is a P0-function on S if (x 0 x0i )(Fi (x) 0 Fi (x0 ))  0 for all x; x0 2 S; x 6= x0 : max in i

1

6 0i

x i =x

(c) F is a P -function on S if max (xi 0 x0i )(Fi (x) 0 Fi (x0 )) > 0 for all x; x0 2 S; x 6= x0:

1in

(d) F is strongly monotone with modulus  > 0 on S if (x 0 x0 )T (F (x) 0 F (x0 ))  kx 0 x0 k2 for all x; x0 2 S:

(e) F is a uniform P-function with modulus  > 0 on S if max (xi 0 x0i )(Fi (x) 0 Fi (x0 ))  kx 0 x0k2 for all x; x0 2 S:

1in

(f)

rF (x) is uniformly positive de nite with modulus  > 0 on S if dT rF (x)d  kdk2 for all x 2 S; d 2 Rn :

(g) F is Lipschitz continuous on S if there exists a constant L > 0 such that

kF (x) 0 F (x0)k  Lkx 0 x0k

for all x; x0 2 S:

2 When the properties mentioned in De nition 2.1 hold with S = Rn , we simply omit the words \on S". Note that F is a P0-function on S if F is monotone on S, and that F is a uniform P -function with modulus  > 0 on S if F is strongly monotone with modulus  > 0 on S. Also, when F is continuously dierentiable, F is strongly monotone with modulus  > 0 on S if and only if rF (x) is uniformly positive de nite with modulus  > 0 on S. We recall some concepts about the NCP which will be used when we discuss boundedness of the solution set of the NCP.

De nition 2.2 The NCP is said to be strictly feasible if there exists x^ > 0 such that F (^x) > 0. 2 The error bound is an important concept that indicates how close an arbitrary point is to the solution set of the NCP. Thus, an error bound may be used to provide stopping rules for an iterative method. The following lemma shows that the natural residual provides a global error bound for the NCP under suitable conditions, which will play an important role in obtaining the linear convergence result of the algorithm to be presented in Section 4. 4

([19]) Let 9NR be de ned by (2). Suppose that F is strongly monotone with modulus  > 0 and Lipschitz continuous with constant L > 0 on some convex set S  Rn . Let x3 be the unique solution of the NCP. Then the following inequality holds: Lemma 2.1

kx 0 x3k  L + 1 9NR(x) 12 for all

x 2 S:

2

In the next lemma, we summarize some important properties of the Fischer-Burmeister function F B de ned by (4). Lemma 2.2 (i)

([10; 13; 23]) The function F B de ned by (4) has the following properties:

F B (a; b) = 0 () a  0; b  0; ab = 0: F B (a; b)  0 for all (a; b)T

(ii) (iii) (iv) (v) (vi)

2 R2:

F B is continuously dierentiable for all (a; b)T

2 R2 .

@F B (a; b) @@bF B (a; b)  0 at any (a; b)T 2 R2: @a @F B (a; b) = 0 () @@bF B (a; b) = 0 () F B (a; b) = 0: @a

1  p 2 2 minfa; bg2   (a; b)  1 (p2 + 2)2 minfa; bg2 for all (a; b)T 2 R2. FB 2 2+2 2

2

Lemma 2.2 (i) and (v) mean that F B , @@aFB and @@bFB are all NCP-functions. Moreover, the above lemma indicates that the function 9F B de ned by (3) is nonnegative on Rn and that 9F B (x) = 0 if and only if x solves the NCP. Important results about the merit function 9F B are summarized in the following theorem. Theorem 2.1 (i)

([10; 23]) The function 9F B de ned by (3) has the following properties:

If the NCP has at least one solution, then the NCP is equivalent to the unconstrained optimization problem minx2Rn 9F B (x).

(ii) (iii)

If F is a P0 -function, any stationary point of 9F B is a solution of the NCP. Let 9NR be the natural residual de ned by (2). Then the following inequalities hold:

1  p 2 2 9 (x)  9 (x)  1 (p2 + 2)29 (x) for all x 2 Rn : (7) NR FB 2 2 + 2 NR 2 Moreover, if F is strongly monotone and Lipschitz continuous, then 9F B provides a global error bound for the NCP.

(iv)

If F is a uniform P -function, then the level set L(9F B ; c) is bounded for all c 2 R.

5

2

3.

A New Merit Function and Its Properties

In this section, we propose a new merit function and show that the function has several favorable properties. Let 9 : Rn ! R be de ned by (5) and (6), i.e., 9(x) =

n X i=1

 (xi ; Fi (x));

where  : R2 ! R is given by 1p   (a; b) = (ab)2+ + ( a2 + b2 0 a 0 b)2 2 2 and   0 is a real parameter. Since the function 0 reduces to the Fischer-Burmeister function F B , the function  inherits several favorable properties of the Fischer-Burmeister function. Moreover, when  > 0,  enjoys an important property the Fischer-Burmeister function F B does not have (see Lemma 3.1 (vi) below). Lemma 3.1 Let   0. The function  de ned by (6) has the following properties: (i)  (a; b) = 0 () a  0; b  0; ab = 0: (ii) (a; b)  0 for all (a; b)T 2 R2 : (iii)  is continuously dierentiable at any (a; b)T 2 R2 and @ (0; 0) @a @ (a; b) @a @ (a; b) @b

 = @ (0; 0) = 0; @b



p = b(ab)+ + p 2a 2 0 1 ( a2 + b2 0 a 0 b)  a b+ b  p 2 2

= a(ab)+ + p

a2 + b2

0 1 ( a + b 0 a 0 b)

(8) for (a; b) 6= (0; 0);

(9)

for (a; b) 6= (0; 0):

(10)

@ @ ( a; b)  (a; b)  0 for all (a; b)T 2 R2 : @a @b @ @ (v) ( a; b) = 0 ()  (a; b) = 0 ()  (a; b) = 0: @a @b If  > 0, then the function  enjoys the following additional property: (iv)

(vi)

If a ! 01 or b ! 01 or ab ! 1, then (a; b) ! 1.

Since (i){(iii) and (vi) can easily be shown, we only show (iv) and (v). We rst prove (iv). When (a; b) = (0; 0), @@a (a; b) @@b (a; b)  0 holds obviously. When (a; b) 6= (0; 0), it follows from (9) and (10) that Proof.

@ @ ( a; b)  (a; b) @a @b

p  = 2ab(ab)2+ + a(ab)+ p 2a 2 0 1 ( a2 + b2 0 a 0 b) a + b  b p + b(ab)+ p 2 2 0 1 ( a2 + b2 0 a 0 b) a +  a  b b p + p 2 2 0 1 p 2 2 0 1 ( a2 + b2 0 a 0 b)2: a +b a +b

6

(11)

Since

p 2a 2 0 1  0; p 2b 2 0 1  0; (12) a +b a +b it is sucient to show that the second and third terms of (11) are both nonnegative for all (a; b)T 6= (0; 0)T 2 R2. We rst show ab(ab)2+  0;

p p 2a 2 0 1 ( a2 + b2 0 a 0 b)  0: (13) a +b Consider p 2 three2 cases: a = 0, a < 0 and a > 0. When a = 0, (13) holds obviously. When a < 0, we we consider the case where a > 0. If b  0, have a + b 0 a 0 b > 0, and hence (13) holds. p 2 Now 2 we have (ab)+ = 0. If b > 0, then we have a + b 0 a 0 b < 0. It follows from (12) that (13) holds. Therefore (13) holds for all (a; b)T 6= (0; 0)T 2 R2. In a similar way, we can show that the inequality p  b b(ab)+ p 2 2 0 1 ( a2 + b2 0 a 0 b)  0 (14) a +b holds for all (a; b)T 6= (0; 0)T 2 R2. Consequently, it follows from (11){(14) that a(ab)+



@ @ ( a; b)  (a; b)  0: (15) @a @b We consider it follows from the de nition of  that p 2 (v).2 Suppose that (a; b) = 0.@Then  ab = 0 and a + b 0 a 0 b = 0. Thus we obtain @a (a; b) = 0 and @@b (a; b) = 0 from (9) and (10), respectively. Next we show that @@a (a; b) = 0 implies (a; b) = 0. Suppose that @@a (a; b) = 0. It

then follows from (9) that

 p p 2a 2 0 1 ( a2 + b2 0 a 0 b): (16) a +b Suppose that both sides of (16) do not vanish. Then we have b 6= 0. If b < 0, then the left hand side of (16) is not positive and the right hand side of (16) is positive. This is a contradiction. When b > 0, a > 0 must hold in order to have (ab)+ 6= 0. However, in this case, the left hand side of (16) is positive and the right hand side of (16) is negative. This is a contradiction, too. Therefore both sides of (16) must be equal to zero, and then it can easily be shown that (a; b) = 0 holds. In a similar way, we can show that @@b (a; b) = 0 implies (a; b) = 0. The proof is complete. 2 b(ab)+ = 0



As mentioned above, the Fischer-Burmeister function F B does not have a property like Lemma 3.1 (vi); more precisely, ab ! 1 does not necessarily imply F B (a; b) ! 1. Lemma 3.1 (vi) will turn out to be useful in showing the boundedness of level sets of 9. From the above lemma, we can easily deduce the following theorem. Let 9 be de ned by (5). Then, 9 (x)  0 for all x 2 Rn and 9 (x) = 0 if and only if x solves the NCP. Moreover, suppose that the NCP has at least one solution. Then, x is a global minimizer of 9 if and only if x solves the NCP. 2 Theorem 3.1

7

The above theorem says that the NCP can be reformulated as the unconstrained minimization problem: min 9 (x): (17) x2Rn  In practice, it is not easy to nd a global minimum of 9. Therefore, it is important to know under what conditions any stationary point of 9 is a global minimum. Let F be a P0 -function. Then, x3 2 Rn is a global minimum of the unconstrained optimization problem (17) if and only if x3 is a stationary point of 9 . Theorem 3.2

The theorem can be proved by using property (iv) of Lemma 3.1 and the same proof technique as that of Theorem 3.5 in [10]. 2 Proof.

Now we give some preliminary results that will be useful in obtaining the global and linear convergence results for the algorithm to be presented in the next section. We rst show that the functions 9 and 9NR are of the same order on any bounded set. Let 9 and 9NR be de ned by (5) and (2), respectively, and  bounded set S , the following inequality holds: Lemma 3.2

1  p 2 2 9 (x)  9 (x)    B2 + 1 (p2 + 2)2 9 (x) NR  2 2 + 2 NR 2 2

where B is a constant de ned by

B = sup



 0.

Then for any

for all x 2 S;

(18)



max fmaxfjxij; jFi (x)jgg < 1:

x2S 1in

By Theorem 2.1 (iii) and the de nitions of 9 and 9F B = 90, we have  2 2 1 9 (x) for all x 2 Rn : 9(x)  9F B (x)  2 p 2 + 2 NR Therefore the left-hand inequality of (18) holds. Next we consider the right-hand inequality of (18). Let us show that, for each i, the following inequality holds for all x 2 S : Proof.

(xiFi(x))+  B j minfxi; Fi(x)gj:

(19)

We rst consider the case where Fi(x)  xi. Then minfxi; Fi(x)g = xi. When Fi(x)  xi  0, it follows from the de nition of B that (xiFi(x))+ = xiFi(x) = Fi(x) j minfxi; Fi(x)gj  B j minfxi; Fi(x)gj: In the case where Fi (x)  0  xi, we have (xiFi(x))+ = 0 and hence (19) holds obviously. When 0  Fi(x)  xi, it follows from the de nition of B that (xiFi (x))+ = jxiFi(x)j  jxij2  B j minfxi ; Fi(x)gj: 8

In a similar way, we can show that (19) also holds when xi  Fi(x). Therefore (19) holds for all x 2 S . Thus it follows from Lemma 2.2 (vi) and (19) that for all i = 1; . .. ; n and x 2 S  (xi ; Fi (x))

= 2 (xiFi(x))2+ + F B (xi; Fi(x))    2 1p 2  2 B + 2 ( 2 + 2) minfxi; Fi(x)g2:

(20)

Consequently, from the de nitions of 9 and 9NR, we have the right-hand inequality of (18). The proof is complete. 2 Lemma 3.2 indicates that 9 is in the same order of the natural residual 9NR on a bounded set. Thus, from Lemma 2.1, 9 provides a global error bound for the NCP if F is strongly monotone and Lipschitz continuous. Moreover, we can show that the function 9 provides a global error bound without assuming Lipschitz continuity of F when  > 0. Note that 9F B does not have the same property under the strong monotonicity of F only. Let 9 be de ned by (5) and  > 0. Suppose that F is a uniform P-function with modulus . Then there exists a positive constant  such that Theorem 3.3

kx 0 x3k  9(x) 14

for all x 2 Rn ;

(21)

where x3 is the unique solution of the NCP. Proof.

Since F is a uniform P -function with modulus , we have kx 0 x3k2

where

 = = 

max (x 0 x3i )(Fi(x) 0 Fi(x3)) 1in i

max fxiFi(x) 0 x3i Fi(x) 0 Fi(x3)xi + x3i Fi (x3)g max fx F (x) 0 x3i Fi(x) 0 Fi(x3)xig 1in i i max  f(x F (x)) + (0Fi(x))+ + (0xi )+ g ; 1in i i i + 1in

(22)

i := maxf1; x3i ; Fi (x3 )g:

Now we show that there exists a positive constant  such that f(ab)+ + (0b)+ + (0a)+ g2  (a; b) for all (a; b)T 2 R2. First we establish the inequality (0a)2+ + (0b)2+ 

p

2

a2 + b2 0 a 0 b :

(23) (24)

Without loss of generality, we assume p 2a  b2. In the case where a  b  0, (24) holds obviously. In the case where a  0  b, we have a + b 0 a 0 b  0b  0, which implies that (0a)2+ + (0b)2+ = b2  9

p

2

a2 + b2 0 a 0 b :

p p In the case where 0  a  b, since a2 + b2 0 a 0 b  a2 + b2, we obtain (0a)2+ + (0b)2+ = a2 + b2 

2

p

a2 + b2 0 a 0 b :

We thus have shown that (24) holds for all (a; b)T 2 R2. Therefore, we have for all (a; b)T 2 R2 f(ab)+ + (0b)+ + (0a)+g2 = (ab)2+ + (0b)2+ + (0a)2+ + 2(ab)+ (0b)+ + 2(ab)+ (0a)+ + 2(0a)+ (0b)+  (ab)2+ + (0b)2+ + (0a)2+ +n((ab)2+ + (0b)2+ ) + ((ab)o2+ + (0a)2+ ) + ((0a)2+ + (0b)2+)  3 (ab)2+ + (0b)2+ + (0a)2+  p   3 (ab)2+ + a2 + b2 0 a 0 b 2  p     2 (ab)2+ + 12 a2 + b2 0 a 0 b 2 = (a; b); where

 6  := max ; 6 > 0;  and the third inequality follows from (24). Consequently, letting ^ := max i ; 1in and combining (22) and (23), we obtain 

kx 0 x3k2

1 2  1max  f  ( x ; F ( x ))g i  i i i n 1  ^ 21 1max (xi ; Fi (x)) 2 i n

 ^

1 2

(n X i=1

) 21

 (xi ; Fi (x))

= ^ 21 9(x; F (x)) 21 : The proof is complete.

2

Next we consider the boundedness of level sets of 9. The boundedness of level sets of a merit function is important since it ensures that the sequence generated by a descent method has at least one accumulation point. The following theorem gives conditions under which 9 has bounded level sets. Theorem 3.4

Suppose that either of the following conditions holds:

(a)

 > 0, F

(b)

  0 and F

is monotone and the NCP is strictly feasible. is strongly monotone.

10

Then the level set

L(9; c)

is bounded for al l

c 2 R.

Under condition (a), the boundedness of the level set L(9; c) is easily shown in a way similar to Theorem 4.1 in [15] by using Lemma 3.1 (vi). Under condition (b), the result can be established by slightly modifying the proof of Theorem 3.8 in [10]. 2 Proof.

Finally we state a preliminary result which is the key to establishing both global and linear convergence of the algorithm. We abbreviate the vectors ( @@a (x1; F1(x)); . .. ; @@a (xn ; Fn (x)))T and ( @@b (x1; F1(x)); . . . ; @@b (xn ; Fn(x)))T as @@a (x; F (x)) and @@b (x; F (x)), respectively.   0. Then the following inequality holds for all x 2 Rn : p @  ( x; F (x)) +  (x; F (x))k2  (2 0 2)4 9NR (x): k @ @a @b Suppose (a; b) 6= (0; 0). It then follows from (9) and (10) that

Lemma 3.3 Let

Proof.

2 @ (a; b) + @b (a; b) @a

 @



=

2  a+b p2 2 p 02 (a + b)(ab)+ + ( a + b 0 a 0 b) a2 + b2  a+b 2 p2 2 2 2 2 2



=  (a + b) (ab)+ + ( a + b 0 a 0 b) p 2 2 0 2 a +b   a+ p2 2 + 2(a + b)(ab)+( a + b 0 a 0 b) p 2 b 2 0 2 : (25) a +b Now we show that the inequality  a+b  p2 2 2(a + b)(ab)+( a + b 0 a 0 b) p 2 2 0 2  0 (26) a +b holds for all (a; b)T 6= (0; 0)T 2 R2. Note that paa2++bb2 0 2  0 holds for all (a;pb)T 6= (0; 0)T 2 R2. 2 2 When ab  0, we have (ab)+ = 0. When a > 0 and p b2> 0,2we have a+b > 0 and a + b 0a 0b < 0. When a < 0 and b < 0, we have a + b < 0 and a + b 0 a 0 b > 0. Therefore (26) holds for all (a; b)T 6= (0; 0)T 2 R2. Combining (25) and (26), we obtain for all (a; b)T 6= (0; 0)T 2 R2 2 2  a+b  @ p2 2 @  2 (a; b) + @b (a; b)  ( a + b 0 a 0 b) p 2 2 0 2 @a a +b  2 2  a+b 2 2 p p  minfa; bg 02 2 + 2 a2 + b2 2  p (2 0 2)2 minfa; bg2  p2 2p+ 2 = (2 0 2)4 minfa; bg2; (27) where the second inequality follows from Lemma 2.2 (vi) and the last inequality follows from the fact that p 2 0 2  2 0 pa2+ b 2 for all (a; b)T 6= (0; 0)T 2 R2: a +b Note that (27) also holds trivially for (a; b)T = (0; 0)T , and hence it holds for all (a; b)T 2 R2. Consequently, the lemma follows from the de nition of 9NR. 2 11

4.

A Descent Method and Global Convergence

In this section, we propose a derivative-free descent method based on the function . Furthermore, we prove global convergence of this method. We consider the following search direction: @ @ k ( x ; F (xk )) 0  (xk ; F (xk )); (28) @b @a where is a parameter such that 2 (0; 1). Although this search direction is not necessarily a descent direction of 9 at xk for any 2 (0; 1), we can choose > 0 suciently small so that dk ( ) is a descent direction of 9 at xk , provided that the monotonicity assumption on F is ful lled. We dk ( ) := 0

show this result in the following lemma.

Suppose that F is monotone and   0. If xk is not a solution of the NCP, then there exists (xk ) 2 (0; 1) such that, for all 2 [0; (xk )) the search direction dk ( ) de ned by (28) satis es the descent condition Lemma 4.1

r9(xk )T dk ( ) < 0:

(29)

Since F is continuously dierentiable, the function 9 is also continuously dierentiable by Lemma 3.1 (iii) and the gradient of 9 at xk is given by Proof.

 k  k r9(xk ) = @ (x ; F (xk )) + rF (xk ) @ (x ; F (xk )): @a @b

(30)

From (30) and the de nition of dk ( ), we have r9(xk )T dk ( ) @ @ @  k ( x ; F (xk ))T  (xk ; F (xk )) 0  (xk ; F (xk ))T rF (xk )  (xk ; F (xk )) = 0 @ @a @b @b @b 



 k  k  k + 0k @ (x ; F (xk ))k2 0 @ (x ; F (xk ))T rF (xk ) @ (x ; F (xk )) : @a @a @b Now let p(xk ) and q(xk ) be de ned by

p(xk ) := 0

and

(31)

@ k  k  k  k (x ; F (xk ))T @ (x ; F (xk )) 0 @ (x ; F (xk ))T rF (xk ) @ (x ; F (xk )) @a @b @b @b

q (xk ) := 0k

@ @ @ k ( x ; F (xk ))k2 0  (xk ; F (xk ))T rF (xk )  (xk ; F (xk )); @a @a @b

respectively. Then (31) is rewritten as r9(xk )T dk ( ) = p(xk ) + q(xk ): By the monotonicity of F , @@b (xk ; F (xk ))T rF (xk ) @@b (xk ; F (xk ))  0. If xk is not a solution of the NCP, we have @@a (xk ; F (xk ))T @@b (xk ; F (xk )) > 0 from Lemma 3.1 (iv) and (v). Therefore p(xk ) < 0. Let (xk ) be de ned by ( 0p(xk )=q(xk ) if q(xk ) 6= 0 and 0  0p(xk )=q(xk )  1, (xk ) := 1 otherwise. 12

For such (xk ), it is easily seen that, for all 2 [0; (xk )), the search direction dk ( ) satis es the descent condition r9(xk )T dk ( ) < 0. 2 The above lemma motivates us to propose the following descent algorithm. Algorithm 4.1 Step1.

Choose x0 2 Rn ;   0;  2 (0; 1); 2 (0; 1) and 2 (0; 1). Set k := 0.

Step2.

If 9 (xk )  , stop: xk is an approximate solution of the NCP.

Step3.

Compute xk+1 := xk + lk dk ( lk ), where

@ k  k (x ; F (xk )) 0 lk @ (x ; F (xk )) @b @a with lk being the smallest nonnegative integer l satisfying  k  k 9(xk + ldk ( l)) 0 9(xk )  0( l)2k @ (x ; F (xk )) + @ (x ; F (xk ))k2: @a @b dk ( lk ) := 0

Step4.

(32)

Set k := k + 1 and go to Step 2.

Note that the above algorithm has no need to compute the gradients of 9, and therefore there is also no need to compute the Jacobian of F . The most remarkable feature of Algorithm 4.1 is that not only the stepsize but also the search direction itself is adjusted during the backtracking search of Armijo-type, which may be regarded as a sort of curvilinear search. We are ready to state the global convergence result for Algorithm 4.1 under the monotonicity assumption on F . In the remainder of the paper, we assume that the parameter  used in Algorithm 4.1 is set to be zero and Algorithm 4.1 generates an in nite sequence fxk g. Suppose that F is monotone and   0. Then Algorithm 4.1 is well-de ned for any initial point x0 . Furthermore, if x3 is an accumulation point of the sequence fxk g generated by Algorithm 4.1, then x3 is a solution of the NCP. Theorem 4.1

We rst prove that Algorithm 4.1 is well-de ned. It suces to show that Step 3 is wellde ned. Assume to the contrary that there is no nonnegative integer l satisfying (32). It then follows that for any integer l  0 Proof.

 k  k 9(xk + ldk ( l)) 0 9(xk ) > 0( l)2k @ (x ; F (xk )) + @ (x ; F (xk ))k2: @a @b

Dividing the above inequality by l and passing to the limit l ! 1, we have k l k l k lim 9(x + d ( )) 0 9(x )  0: l!1

l

(33)

Since 9 is continuously dierentiable, 9 is locally Lipschitz continuous at xk , i.e., there exist L1 > 0 and  > 0 such that k9(xk ) 0 9(x0)k  L1kxk 0 x0k; for all x0 such that kxk 0 x0k < : 13

Therefore, we have k l k l k lim 9(x + d ( )) 0 9(x ) l!1

l ) 9  (xk + l dk (0)) 0 9 (xk ) 9(xk + l dk ( l )) 0 9 (xk + l dk (0)) + = llim !1

l

l 9(xk + ldk ( l)) 0 9(xk + ldk (0)) = r9(xk )T dk (0) + llim !1

l  r9(xk )T dk (0) + llim L kdk ( l ) 0 dk (0)k !1 1 = r9(xk )T dk (0):

(

(34)

It then follows from (33) and (34) that r9(xk )T dk (0)  0: This is a contradiction, because from Lemma 4.1, dk (0) must be a descent direction of 9 at xk if xk is not a solution of the NCP. Thus Algorithm 4.1 is well-de ned. Next we prove that any accumulation point x3 of fxk g is a solution of the NCP. Let fxk gk2K be a subsequence converging to x3. Then fxk gk2K is bounded and thus fdk ( lk )gk2K is bounded since  is continuously dierentiable. Without loss of generality, we assume dk ( lk ) ! d3 as k(2 K ) ! 1. Since 9 (xk ) decreases at each iteration, the right-hand side of (32) tends to 0. First we consider the case where flk gk2K is bounded. Then f lk gk2K does not approach 0, so we have  3  3 k @ (x ; F (x3)) + @ (x ; F (x3))k2 = 0: @a @b Therefore, it follows from Lemma 3.3 that 9NR(x3) = 0, i.e., x3 is a solution of the NCP. Next suppose that flk gk2K is unbounded, which implies f lk gk2K ! 0. When flk gk2K ! 1, it follows from Step 3 of Algorithm 4.1 that for all k 2 K  k  k (x ; F (xk )) + @ (x ; F (xk ))k2: 9(xk + lk 01dk ( lk 01)) 0 9(xk ) > 0( lk 01)2k @ @a @b

Dividing the above inequality by lk 01 and passing to the limit k ! 1, we have k lk 01 k lk 01 k lim 9(x + d ( )) 0 9(x )  0:

lk 01

k!1

(35)

Since 9 is continuously dierentiable, by the Mean-Value theorem there exists some ~k 2 [0; lk 01] such that 9(xk + lk 01dk ( lk 01)) 0 9(xk ) = lk 01r9(xk + ~k dk ( lk 01))T dk ( lk 01):

(36)

As k(2 K ) ! 1, we have f lk 01g ! 0, which in turn implies f~ k gk2K ! 0 and hence lim ~k dk ( lk 01) = 0:

k!1

14

(37)

It then follows from (36), (37) and the given assumptions that k lk 01 k lk 01 k lim 9(x + d ( )) 0 9(x ) = lim r9 (xk + ~k dk ( lk 01))T dk ( lk 01) k!1

lk 01

k!1



= r9(x3)T d3;

(38)

where ~k 2 [0; lk 01]. Combining (35) and (38), we obtain r9(x3)T d3  0: Since r9(x3)T d3  0 holds obviously, we have r9(x3)T d3 = 0. Moreover, since passing to the limit in (28) and (30) on the subsequence yields d3 = 0

and we obtain

@ 3 (x ; F (x3)) @b

@  3 ( x ; F (x3 )) + rF (x3)  (x3 ; F (x3 )); r9(x3) = @ @a @b

r9(x3)T d3 @ @ @  3 = 0 @ ( x ; F (x3 ))T  (x3 ; F (x3 )) 0  (x3 ; F (x3 ))T rF (x3)  (x3 ; F (x3 )) @a @b @b @b = 0: Consequently, by the monotonicity of F and Lemma 3.1 (iv), we have @ @ 3 ( x ; F (x3))T  (x3 ; F (x3 )) = 0: @a @b It then follows from Lemma 3.1 (i) and (v) that x3 is a solution of the NCP.

2

Combining the above theorem and Lemma 3.4, we obtain the following global convergence result. Theorem 4.2

Suppose that either of the following conditions (a) and (b) holds:

(a)

 > 0, F is monotone and the NCP is strictly feasible.

(b)

  0 and F is strongly monotone.

Then Algorithm 4.1 is well-de ned for any initial point x0 . Furthermore, the sequence fxk g generated by Algorithm 4.1 has at least one accumulation point and any accumulation point of fxk g is a solution of the NCP. 2

15

5.

Linear Convergence

In this section, we show that any sequence generated by Algorithm 4.1 converges linearly to the solution if F is strongly monotone. Before proving the linear convergence result, we give the following lemma. Lemma 5.1

Suppose that F is strongly monotone with modulus  >

0

and that 

x 2 Rn be any given vector. Then there exists an integer ^l  0 such that, for each set L(9 ; 9(x0 )) and for all l  ^l, the search direction dk ( l ) satis es 0

r9

(x)T dk ( l)  0

l

 0.

Let x in the level

2  @ @  k (x; F (x))k + k (x; F (x))k :

2 @a @b 0 Proof. It follows from Theorem 3.4 that the level set L(9 ; 9 (x )) is bounded. Since rF is continuous, there exists a constant  > 0 such that krF (x)k   for all x 2 L(9; 9(x0)):

(39)

From (28) and (30), we have r9(xk )T dk ( l) @ @ @  k = 0 @ ( x ; F (xk ))T  (xk ; F (xk )) 0  (xk ; F (xk ))T rF (xk )  (xk ; F (xk )) @a @b @b @b 0 l

 

 @  @ k @ k  k k 2 k T k k k (x ; F (x ))k + (x ; F (x )) rF (x ) (x ; F (x ))

@a @a  k  k (x ; F (xk ))k2 0 lk @ (x ; F (xk ))k2 0k @ @b @a @  k 0 l @ ( x ; F (xk ))T rF (xk )  (xk ; F (xk )) @a @b @ @ k 0k @b (x ; F (xk ))k2 0 lk @a (xk ; F (xk ))k2  k  k (x ; F (xk ))kk @ (x ; F (xk ))k; + l k @ @a @b

@b

(40)

where the rst inequality follows from Lemma 3.1 (iv) and the strong monotonicity of F and, the second inequality follows from the Cauchy-Schwartz inequality and (39). Now let ^l be de ned by   ^l := log 2 2 :  + 2 + 2 Then l satis es

l 

2  2 + 2 + 2

for all l  ^l:

(41)

Since (41) implies  0 2  0, the right-hand side of (40) can be rewritten as l

@ @ @  k 0k @ ( x ; F (xk ))k2 0 l k  (xk ; F (xk ))k2 + l  k  (xk ; F (xk ))kk  (xk ; F (xk ))k @b @a @a @b

=

2 l l  (xk ; F (xk ))k + k @ (xk ; F (xk ))k 0  0 0 2 k @ @a @b 2

16

!

 (xk ; F (xk ))k2 k @ @b

l

@ @  k 0 2 k @ ( x ; F (xk ))k2 + l ( + 1)k  (xk ; F (xk ))kk  (xk ; F (xk ))k @a @a @b

2 l @  k ( x ; F (xk ))k + k  (xk ; F (xk ))k = 0 2 k @ @a @b 0 s

12

s

l l @  k 0 @ 2 k @ ( x ; F (xk ))k 0  0 k  (xk ; F (xk ))kA @a 2 @b   q  k  k + l( + 1) 0 l(2 0 l) k @ (x ; F (xk ))kk @ (x ; F (xk ))k @a @b

for all l  ^l. Since (41) implies q

(42)

q

 2l( 2 + 2 + 2) 0 2l q = 2l( + 1)2 = l( + 1); the last term of (42) is nonpositive for all l  ^l. Consequently, it follows from (40) and (42) that for all l  ^l r9

l (2 0 l )

(xk )T dk ( l)  0

2 @ k @ k k k k @a (x ; F (x ))k + k @b (x ; F (x ))k :

l

2

Now the linear convergence result for Algorithm 4.1 is presented. Theorem 5.1 Suppose that F is strongly monotone and   0. Suppose also that F

2

is Lipschitz

and be chosen to satisfy < . Then the sequence generated by Algorithm 4.1 converges R-linearly to the solution of the NCP and the sequence f9  (xk )g converges Q-linearly to zero. continuous on any bounded set. Let the parameters

fxk g

We rst show that the sequence f9(xk )g converges Q-linearly to zero. By Theorem 3.4, the level set L(9; 9(x0)) is bounded. Then since F , rF and r9 are all Lipschitz continuous on L(9; 9(x0)), there exists a constant L2 > 0 such that kr9(x) 0 r9(x )k  L2kx 0 x k for all x; x 2 L(9; 9(x0)): (43) Since the sequence f9(xk )g is nonincreasing, fxk g  L(9; 9(x0)). It then follows from (43) and the Cauchy-Schwartz inequality that for each t 2 [0; 1] Proof.

0

9

(xk

+ tdk ( l)) 0 9



(xk )

0

0

Zt

r9(xk + sdk ( l))T dk ( l)ds = 0 k T k l = tr9  (x ) d ( ) Z t T + r9(xk + sdk ( l)) 0 r9(xk ) dk ( l)ds 0

 tr9 

(xk )T dk ( l) + L

Zt

2

= t r9

(xk )T dk ( l) +

17

0

L2 t

2

skdk ( l )k2 ds

kdk ( l)k2



:

(44)

By Lemma 3.1 (iv) and l 2 (0; 1), we have kdk ( l)k2 @  k ( x ; F (xk )) 0 l  (xk ; F (xk ))k2 = k 0 @ @b @a

 k  k  k  k (x ; F (xk ))k2 + 2lk @ (x ; F (xk ))k2 + 2 l @ (x ; F (xk ))T @ (x ; F (xk )) = k @ @b @a @a @b

@ @ @  k  k @ ( x ; F (xk ))k2 + k  (xk ; F (xk ))k2 + 2  (xk ; F (xk ))T  (xk ; F (xk )) @b @a @a @b

@  k = k @ ( x ; F (xk )) +  (xk ; F (xk ))k2 : @a @b

(45)

Let ^l be as speci ed in the proof of Lemma 5.1. Then, from (44), (45) and Lemma 5.1, we have for all l  ^l 9(xk + ldk ( l)) 0 9(xk ) ! 2 L l  @ k l @ k 2 k k k l 2 l  0 2 k @a (x ; F (x ))k + k @b (x ; F (x ))k + 2 kd ( )k l @ L l  k  l 0 2 k @ ( x ; F (xk )) +  (xk ; F (xk ))k2 + 2 kdk ( l )k2 @a @b 2 l @  k  0 2 ( l 0 L2 l)k @ ( x ; F (xk )) +  (xk ; F (xk ))k2 ; @a @b

!

(46) where the second inequality follows from the Cauchy-Schwartz inequality and the last inequality follows from (45). By (46), condition (32) is satis ed whenever l

0 2 ( l 0 L2 l)  0( l)2: Since this inequality can be rewritten as  l  2 +1 L ; 2 condition (32) is satis ed for all l  l, where l is de ned by    l := max l^; log (2 + L2 ) :

(47) (48)

Notice that l does not depend on k. Thus, we have lk  lfor all k since lk is the smallest nonnegative integer l satisfying (32). It then follows from (32) and Lemma 3.3 that  k  k 9(xk+1 ) 0 9(xk )  0( lk )2k @ (x ; F (xk )) + @ (x ; F (xk ))k2 @a @b

@  k ( x ; F (xk )) +  (xk ; F (xk ))k2  0( l)2k @ @b p @a 4 2l k  0(2 0 2)  9NR(x ): Since the level set L(9; 9(x0)) is bounded, it follows from Lemma 3.2 that   p 1 k 2 2 9(x )  2 B + 2 ( 2 + 2) 9NR(xk ) for all xk 2 L(9; 9(x0));

18

(49) (50)

where B is given by B = sup



o

n

max maxfjxki j; jFi(xk )jg 1in

Therefore from (49) and (50) we have 9

(xk+1) 0 9



(xk )  0

Consequently, we obtain

x 2 L(9; 9 (x

0



)) :

p 2(2 0p 2)4  2l9 (xk ):  B 2 + ( 2 + 2)2

(51)

p 4 2l 2(2 0 1 0 2 p2)  2 9(xk )  9(xk+1)  0: B + ( 2 + 2) p p On the other hand, since ( 2 + 2)2 > 2(2 0 2)4;  2l 2 (0; 1) and B2  0, we have p 4 2l 2(2 0 0 < 1 0 2 p2)  2 < 1: B + ( 2 + 2) Therefore the sequence f9(xk )g converges Q-linearly to zero. Next we prove that the sequence fxk g converges R-linearly to the unique solution x3 of the NCP. From Lemma 2.1 and Lemma 3.2, there exists a positive constant  such that kx 0 x3k  9(x) 12 for all x 2 L(9; 9(x0)): Since the sequence f9(xk )g converges Q-linearly to zero, the sequence fxk g converges R-linearly to the solution of the NCP. 2 !

The conditions in the above theorem are weaker than those in [18]. For the linear convergence, we only need the condition < , which does not involve unknown parameters. Also, note that some parameters used in Algorithm 4.1 do not rely on usually unknown constants such as the modulus of strong monotonicity of F . 6.

Concluding Remarks

In this paper, we have proposed a merit function for the NCP and shown its properties. We have also proposed a new derivative-free descent algorithm based on this function for solving the NCP and proven global and linear convergence theorems under appropriate assumptions. The most interesting part of Algorithm 4.1 is Step 3, where the search direction and the stepsize are adjusted simultaneously, whereas search directions in the algorithms proposed in [10; 12; 18; 24] are xed during a linesearch process. This particular feature enables us to establish the linear convergence of the algorithm under more practical assumptions compared with the algorithm of [18]. Derivative-free descent algorithms have also been proposed for solving the VIP [8; 27]. However, the convergence rate results for those algorithms have been scarce. Also, those algorithms generally use parameters that have to be speci ed by using unknown constants. It is an interesting future topic to extend the results of this paper to the more general class of VIPs. 19

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21