Lagrangian Globalization Methods for Nonlinear ... - Semantic Scholar

Lagrangian Globalization Methods for Nonlinear Complementarity Problems 1 Xiaojun Chen2 Department of Mathematics and Computer Science Shimane University Matsue 690-8504, Japan E-mail: [email protected] Liqun Qi School of Mathematics The University of New South Wales Sydney, New South Wales 2052, Australia E-mail: [email protected] Yufei Yang 3 Department of Applied Mathematics Hunan University Changsha, Hunan 410082, China October 29, 1999

Abstract This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation Φ(x) = 0 arising from nonlinear complementarity problems NCP and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving NCP, but to find x ˆ such that kΦ(ˆ x)k < kΦ(¯ x)k when the NCP has a solution and x ¯ is a stationary point but not a solution.

Key Words: Nonlinear complementarity problem, global convergence, Lagrangian. AMS Subject Classification. 90C30, 65K10

1

This work is supported by the Australian Research Council. The work of this author is partly supported by the Scientific Research Grant C11640119 from the Japanese Ministry of Education. 3 Present address: School of Mathematics, The University of New South Wales, Sydney, New South Wales 2052, Australia, E-mail: [email protected] 2

1

Introduction

We consider the following nonlinear complementarity problem NCP(F): find a vector x ∈ Rn such that x ≥ 0, F (x) ≥ 0, xT F (x) = 0 (1)

where F : Rn → Rn be continuously differentiable. The NCP(F) can be equivalently reformulated as a system of nonlinear equations: Φ(x) = 0,

(2)

where Φ : Rn → Rn is locally Lipschitz but not differentiable. During the last few years, many methods have been developed for the solution of NCP based on (2), e.g., see [1, 3, 4, 6, 7, 12, 13] and references therein. In particular, NCP functions have attracted much more attention and are widely used in the context of nonlinear complementarity problems. A common global strategy in these algorithms is the line search technique that strictly reduces the value of the least square merit function 1 Ψ(x) = Φ(x)T Φ(x) (3) 2 at every iteration. At the present a strongest global convergence theorem is considered to have the following result. Each accumulation point of the iterative sequence is a stationary point of Ψ. However, a stationary point of Ψ(x) is only a local minimizer of Ψ(x), which is not necessary a solution of (2). To ensure every stationary point of Ψ is a solution of (2), additional conditions are assumed. For example, F is assumed to be at least a P0 function, i.e., max (xi − yi )(Fi (x) − Fi (y)) ≥ 0,

i:xi 6=yi

for all x, y ∈ Rn , x 6= y.

In general, it is difficult to ensure each accumulation point of the iterative sequence generated by these methods is a solution of the NCP. Recently, Nazareth [8], Nazareth and Qi [9] proposed a Lagrangian globalization (LG) method for the solution of smooth equations g(x) = 0. The main strategy of the method is first to employ a standard algorithm, for example, Newton method (or one of its variants), to solve the smooth equations. If it stacks at a point, say x¯, which is not a solution of g(x) = 0, a (more expensive) LG-type algorithm is employed to find a new point, say xˆ, with kg(ˆ x)k < kg(¯ x)k. Then the Newton algorithm can be reinitiated at xˆ. In this paper, we extend the LG method to nonsmooth equations arising from NCP(F). The LG method associates an objective function f (x) with the system (2) to give an equality constrained nonlinear programming problem of the form: min f (x) s.t. Φ(x) = 0,

2

(4)

where f : Rn → R is assumed to be continuously differentiable. For example, f (x) = αeT x where e is the vector of all ones, whereas α 6= 0 is a constant. See [9] for specific choices about f . The Lagrangian and augmented Lagrangian of (4) are respectively L(x, λ) := f (x) + λT Φ(x)

(5)

and

1 (6) Pc (z) := Pc (x, λ) := L(x, λ) + ckΦ(x)k2 , 2 where λ ∈ Rn is the Lagrange multiply vector, c is a nonnegative real parameter and z = (x, λ) ∈ R2n . We propose a descent algorithm for solving the unconstrained nonsmooth optimization problem min2n Pc (z). (7) z∈R

We prove that, under mild conditions, a negative generalized gradient direction of the object function Pc is a descent direction of Pc at any noncritical point of Pc . In this paper, we obtain the following result: For any x¯ with Φ(¯ x) 6= 0, we can find a xˆ such that kΦ(ˆ x)k < kΦ(¯ x)k if the NCP(F) has a solution. The organization of this paper is as follows. In the next section, we will shortly review some basic properties of semismooth functions and present some preliminary results which are closely related to problem (7). By analyzing the properties of a NCP function, we prove the descent property of the generalized gradients of Pc in Section 3. Section 4 contains a detailed description of a descent algorithm along with the corresponding well-defineness and convergence theory. Moreover, we give an example which shows how to find a new starting point xˆ such that kΦ(ˆ x)k < kΦ(¯ x)k when the NCP(F) has a solution but Φ(¯ x) 6= 0. Notation: For a given function G : Rn → Rm , we denote by Gi its i-th component function. If G is continuously differentiable, ∇G(x) is the transposed Jacobian of G at x ∈ Rn . If G is directional differentiable, we denote the directional derivative of G at x ∈ Rn in the direction d ∈ Rn by G′ (x; d). The symbol k · k indicates the Euclidean vector norm or its associated matrix norm. The set {0, 1, 2, · · ·} of all nonnegative integers is denoted by N, and the set of all nonnegative real numbers is denoted by R+ .

2

Preliminaries

In this section, we first recall several definitions and results which will be used in our subsequent analysis, and then prove some preliminary results which are closely related to problem (7). Suppose G : Rn → Rm is a locally Lipschitzian function, then G is differentiable almost everywhere on Rn . Denote the set of points at which G is differentiable by DG . 3

Let ∂G(x) be the generalized Jacobian defined by Clarke in [2]. Then ∇G(xk )T }. ∂G(x) = co{ lim k

(8)

x →x xk ∈DG

Lemma 2.1 ( see [2] ) Let G : Rn → R and p : Rn → Rm be Lipschitzian near x. Moreover, g : Rm → R is assumed to be Lipschitzian near p(x). Then (i) 0 ∈ ∂G(x) if G attains a (local) minimum at x. (ii) ∂G(·) is upper semicontinuous in the sense that {xk → xˆ, y k ∈ ∂G(xk ) and y k → yˆ} ⇒ {ˆ y ∈ ∂G(ˆ x)}. (iii) The composite function G = g◦p is Lipschitzian near x and ∂G(x) ⊆ co{∂g(p(x))∂p(x)}. We now introduce the definition and properties of a semismooth operator, see Qi and Sun [11] for details. If G is locally Lipschitzian at x and for any h ∈ Rn , lim

V ∈∂G(x+th′ ) h′ →h, t↓0

{V h′ }

(9)

exists, then we say that G is semismooth at x. We note that semismooth functions are directionally differentiable and that the directional derivative G′ (x; h) of G at x ∈ Rn in the direction h ∈ Rn is equal to the limit in (9). Furthermore, the following assertion holds. Lemma 2.2 Suppose G : Rn → Rm is a locally Lipschitzian function. Then G is semismooth at x if and only if for any V ∈ ∂G(x + h), h → 0, V h − G′ (x; h) = o(khk). Last property motivates the following definition. If G is semismooth at x and for any V ∈ ∂G(x + h), h → 0, V h − G′ (x; h) = O(khk2), then we call G is strongly semismooth at x. The following results can be found in Fischer [4]. Lemma 2.3 Suppose that the function p : Rn → Rm is semismooth at x and that the function g : Rm → R is semismooth at p(x). Then (i) The composite function G = g ◦ p is directionally differentiable at x and G′ (x; d) = g ′(p(x); p′ (x; d)),

d ∈ Rn .

(ii) Furthermore, if p and g are strongly semismooth at x and p(x), respectively. Then the composite function G = g ◦ p is also strongly semismooth. 4

It is not difficult to deduce the generalized derivative of (6) as follows: ∂x Pc (x, λ) = ∂x L(x, λ) + c∂Φ(x)T Φ(x) = ∇f (x) + ∂Φ(x)T λ + c∂Φ(x)T Φ(x), ∂λ Pc (x, λ) = Φ(x).

(10)

We call z = (x, λ) ∈ R2n is a critical point of Pc if 0 ∈ ∂x Pc (x, λ), 0 = ∂λ Pc (x, λ) = Φ(x).

(11)

Proposition 2.1 (i) If x∗ is not a solution of NCP(F), then for any λ ∈ Rn , (x∗ , λ) is not a critical point of Pc . (ii) If (x∗ , λ∗ ) is a critical point of Pc , then x∗ is a solution of NCP(F). Proof. The conclusion directly follows from the fact that ∂λ Pc (¯ x, λ) = Φ(¯ x), for any n λ∈R . 2 Proposition 2.2 If x∗ is a solution of NCP(F) and ∇f (x∗ ) belongs to the range space of an element H ∗ ∈ ∂Φ(x∗ )T . Then there exists λ∗ ∈ Rn such that (x∗ , λ∗ ) is a critical point of Pc for any c ∈ R+ . Proof. Since x∗ is a solution of NCP(F), we have ∂λ Pc (x∗ , λ) = Φ(x∗ ) = 0 for any λ ∈ Rn . Let λ∗ be the vector such that H ∗ λ∗ = −∇f (x∗ ). Then ∇f (x∗ ) + H ∗ λ∗ + cH ∗ Φ(x∗ ) = 0, for any c ∈ R+ .

Hence (x∗ , λ∗ ) is a critical point of Pc for any c ∈ R+ .

3

2

Descent property of generalized gradients

The Fischer-Burmeister function φ(a, b) :=

√

a2 + b2 − a − b

has been used for solving the NCP(F), e.g., see [3, 4, 6, 13]. To define a descent direction for Pc , we use its absolute function √ ψ(a, b) := |( a2 + b2 − √a − b| (12) a + b − a2 + b2 , if a > 0, b > 0, √ = 2 2 a + b − a − b, otherwise. 5

It is obvious that ψ is a NCP function, i.e., ψ(a, b) = 0 ⇐⇒ a ≥ 0, b ≥ 0 and ab = 0. In the rest part of this paper, we consider the nonsmooth equation (2) with 



ψ(x1 , F1 (x))   .. .  Φ(x) :=  .  ψ(xn , Fn (x))

(13)

In this section, we investgate the semismoothness of ψ, Φ and Pc , and give estimates of generalized Jacobians of Φ and generalized gradients of Pc . In particular, we prove that a negative generalized gradient of Pc at z = (x, λ) is a descent direction of Pc if z is not a critical point of Pc and the related λ is nonpositive. Some properties of ψ are stated in the following proposition. Proposition 3.1 (i) ψ is locally Lipschitzian, strongly semismooth in R2 and ψ 2 is continuously differentiable in R2 . (ii) If ψ(a, b) 6= 0, then ψ is continuously differentiable at (a, b) ∈ R2 , and (1 − √ 2a 2 , 1 − √ 2b 2 )T , if a > 0, b > 0, a +b a +b ∇ψ(a, b) =  √ b  (√ a − 1, − 1)T , if a < 0 or b < 0. a2 + b2 a2 + b2   

(14)

If ψ(a, b) = 0, then the generalized Jacobian of ψ at (a, b) ∈ R2 is   

{(ρ, 0) | ρ ∈ [−1, 1]}, if a = 0, b > 0, ∂ψ(a, b) =  {(0, ρ) | ρ ∈ [−1, 1]}, if a > 0, b = 0,  Ω, if a = 0, b = 0,

(15)

where Ω = co{Ω1 ∪ Ω2 },

Ω1 = {(1 − ξ, 1 − η) | ξ ≥ 0, η ≥ 0, ξ 2 + η 2 = 1} Ω2 = {(ξ − 1, η − 1) | ξ ≤ 0 or η ≤ 0, ξ 2 + η 2 = 1} (iii) The directional derivative of ψ at (a, b) ∈ R2 in the direction v = (v1 , v2 )T is

ψ ′ ((a, b); v) =

            

∂ψ v + ∂ψ v , ∂a 1 ∂b 2 |v1 |, |vq2 |, | v12 + v22 − (v1 + v2 )|,

if if if if

ψ(a, b) 6= 0, a = 0, b > 0, a > 0, b = 0, a = 0, b = 0.

(16)

Proof. (i) Note that | · | and φ(a, b) are both locally Lipschitzian, strongly semismooth. By Lemma 2.3 (ii), ψ is locally Lipschitzian, strongly semismooth. Moreover, we can 6

directly deduce that ψ 2 is continuously differentiable from the fact that ψ 2 = φ2 and φ2 is continuously differentiable. (ii) (14) and (15) are deduced from the definitions of ψ and the generalized Jacobian. (iii) If ψ(a, b) 6= 0, the conclusion directly follows from the fact that ψ is differentiable at (a, b). If a = 0 and b > 0, then ψ ′ ((a, b); v) = lim t↓0

ψ(a + tv1 , b + tv2 ) − ψ(a, b) t q

| (tv1 )2 + (b + tv2 )2 − (tv1 + (b + tv2 ))| t t↓0 2|v1 (b + tv2 )| = lim q t↓0 | (tv )2 + (b + tv )2 + (tv + (b + tv ))| 1 2 1 2 = |v1 |. = lim

Similarly, we can show the case : a > 0 and b = 0. Finally, if a = b = 0, then q

| (tv1 )2 + (tv2 )2 − (tv1 + tv2 )| ψ ′ ((a, b); v) = lim t t↓0 q = | v12 + v22 − (v1 + v2 )|.

2

The introduction of the following index sets will be very convenient to estimate generalized Jacobians of Φ and generalized gradients of Pc . Let I1 (x) I01 (x) I0 (x) I(x)

= {i | xi > 0, Fi (x) > 0}, = {i | xi = 0, Fi (x) > 0}, = {i | xi = 0, Fi (x) = 0}, = I0 (x) ∪ I01 (x) ∪ I10 (x),

I2 (x) = {i | xi < 0 or Fi (x) < 0}, I10 (x) = {i | xi > 0, Fi (x) = 0}, ¯ I(x)

= {1, 2, · · · , n}\I(x) = I1 (x) ∪ I2 (x).

By Proposition 3.1, Lemma 2.1 and Lemma 2.3, we can deduce the following propositions. Proposition 3.2 (i) Φ is semismooth in Rn if F is continuously differentiable in Rn , and Φ is strongly semismooth in Rn if F ′ is Lipschitzian in Rn . (ii) Ψ is continuously differentiable in Rn . (iii) Pc is semismooth in Rn if F is continuously differentiable in Rn , and Pc is strongly semismooth in Rn if F ′ is Lipschitzian in Rn . Proposition 3.3 Each element Hi ∈ ∂Φi (x)T and V ∈ ∂x Pc (x, λ) can be written as follows: Hi = ai (x)ei + bi (x)∇F (x), 7

(17)

V = ∇f (x) + H(λ + cΦ(x))

(18)

where H = (H1 , H2 , · · · , Hn ) and

ai (x) =

 xi  , 1− q   2 2   (x ) + F (x) i i    xi   − 1,  q           

(xi )2 + Fi (x)2 ρi , 0, ξi ,

 Fi (x)   , 1− q   2 + F (x)2   (x ) i i     Fi (x)   q − 1,

bi (x) =            

(xi )2 + Fi (x)2

0, ρi , ηi ,

with ρi ∈ [−1, 1] and (ξi , ηi ) ∈ Ω.

if i ∈ I1 (x), if i ∈ I2 (x), if i ∈ I01 (x), if i ∈ I10 (x), if i ∈ I0 (x),

(19)

if i ∈ I1 (x), if i ∈ I2 (x),

(20)

if i ∈ I01 (x), if i ∈ I10 (x), if i ∈ I0 (x)

Remark 3.1 Noting the definition (8) of the generalized Jacobian and the expressions ¯ of Hi = ai (x)ei + bi (x)∇Fi (x) for i ∈ I(x). If i ∈ I01 (x), by considering xi → 0+ and xi → 0− , respectively, we deduce that (ai (x), bi (x)) = (ρi , 0) with every ρi ∈ [−1, 1]. If i ∈ I10 (x), (ai (x), bi (x)) = (0, ρi ) with ρi = 1 or ρi = −1 or every ρi ∈ [−1, 1]. Moreover, if i ∈ I0 (x), from the boundness of ai (x) and bi (x) in I2 (x), it is easy to deduce that there exists at least an element (ξi , ηi ) ∈ Ω2 such that (ai (x), bi (x)) = (ξi , ηi ). Proposition 3.4 The directional derivative of Φi (x) = ψ(xi , Fi (x)) at x in the direction d is given by  T ¯  if i ∈ I(x),  Hi d,    |di |, if i ∈ I01 (x), Φ′i (x; d) = |∇F (x)T d|, (21) if i ∈ I10 (x),   q i    | (d )2 + (∇F (x)T d)2 − (d + ∇F (x)T d)|, if i ∈ I (x). i i i i 0

Proof. The conclusion directly follows from Lemma 2.3(i) and Proposition 3.1(iii).

2

Note that for any i ∈ I(x), Φi (x) = 0. It is not difficult ! to deduce that the directional d derivative of Pc (z) at z ∈ R2n in the direction q = with d, h ∈ Rn can be written h as Pc′(z; q) = Φ(x)T h + ∇f (x)T d + (λ + cΦ(x))T Φ′ (x; d) P P λi Φ′i (x; d). (λi + cΦi (x))Φ′i (x; d) + = Φ(x)T h + ∇f (x)T d + ¯ i∈I(x)

i∈I(x)

8

(22)

From (17)–(20), we can deduce V T d = ∇f (x)T d + (λ + cΦ(x))T H T d P P λi HiT d (λi + cΦi (x))HiT d + = ∇f (x)T d + ¯ i∈I(x)

T

= ∇f (x) d +

P

¯ i∈I(x)

(λi +

i∈I(x) P λi cΦi (x))Φ′i (x; d) + i∈I(x)

(23)

HiT d.

Thus, by (22) and (23), we have Pc′ (z; q) − V T d − Φ(x)T h P λi (Φ′i (x; d) − HiT d). =

(24)

i∈I(x)

Moreover, from (17)–(21), we can obtain        

Φ′i (x; d) − HiT d =       

0, |di | − ρi di, |∇F (x)T d| − ρi ∇Fi (x)T d, q i | (di )2 + (∇Fi (x)T d)2 − (di + ∇Fi (x)T d)| −ξi di − ηi ∇Fi (x)T d,

¯ if i ∈ I(x), if i ∈ I01 (x), if i ∈ I10 (x),

(25)

if i ∈ I0 (x)

with ρi ∈ [−1, 1] and (ξi, ηi ) ∈ Ω. Denote Ω3 = {(ξ, η) | (ξ + 1)2 + (η + 1)2 ≤ 1} ∪ {(ξ, η) | − 1 ≤ ξ ≤ 0, −1 ≤ η ≤ 0}. It is obvious that Ω2 ⊂ Ω3 ⊂ Ω. The following result plays a crucial role in proving the descent property of generalized gradients of Pc . Proposition 3.5 For any (s, t) ∈ Ω3 , √ | a2 + b2 − a − b| − as − bt ≥ 0.

(26)

Proof. Let u = 1 + s and w = 1 + t. Assume first that a ≥ 0 and b ≥ 0. Note that for any (s, t) ∈ Ω3 , s ≤ 0 and t ≤ 0. Thus √ √ | a2 + b2 − a − b| − as − bt = a + b − √a2 + b2 − as − bt ≥ a + b − a2 + b2 ≥ 0. Assume then that a < 0 and b < 0. If u < 0 and w < 0, then

a2 + b2 − (au + bw)2 = (1 − u2 )a2 + (1 − w 2 )b2 − 2uwab ≥ w 2 a2 + u2 b2 − 2uwab ≥ 0. 9

(27)

That is,

√

a2 + b2 − au − bw ≥ 0.

(28)

If u ≥ 0 or w ≥ 0, it is obvious that (28) holds. Therefore, for any (s, t) ∈ Ω3 , √ √ | a2 + b2 − a − b| − as − bt = √a2 + b2 − a − b − as − bt = √a2 + b2 − a(1 + s) − b(1 + t) = a2 + b2 − au − bw ≥ 0. Assume now that a < 0 and b ≥ 0. If u < 0 and w > 0, then (27) holds and hence (28) holds. If u ≥ 0 or w ≤ 0, it is obvious that (28) holds. Hence, for any (s, t) ∈ Ω3 , √ √ | a2 + b2 − a − b| − as − bt = √a2 + b2 − a − b − as − bt = √a2 + b2 − a(1 + s) − b(1 + t) = a2 + b2 − au − bw ≥ 0. Finally, consider the case b < 0 and a ≥ 0. Similar to the above proof, we can also deduce √ √ | a2 + b2 − a − b| − as − bt = a2 + b2 − a − b − as − bt ≥ 0. This proves the assertion.

2

In Proposition 3.3, we give the expressions of the generalized Jacobians of Φ(x) at x ∈ Rn and the generalized gradients of Pc (z) at z ∈ R2n . In particully, the analysis in Remark 3.1 will be very helpful to the choice of the parameters from the computational point of view. In fact, we obtain the following result. Theorem 3.1 Assume that!z = (x, λ) is not a critical point of Pc and that λ is non−V positive. Let q = where V ∈ ∂x Pc (z) with ρi ∈ [−1, 1] and (ξi , ηi ) ∈ Ω3 in −Φ(x) (17)–(20). Then Pc′ (z; q) < 0. That is, q is a descent direction of Pc at z. Proof. From Proposition 3.5 and (25), we deduce Φ′i (x; −V ) − HiT (−V ) ≥ 0, ∀i ∈ I(x).

(29)

Hence, by (24), we have Pc′ (z; q) ≤ −kV k2 − kΦ(x)k2 = −kqk2 < 0. This shows that q is a descent direction of Pc (z) at z = (x, λ).

10

(30)

2

4

Algorithm and its convergence

We first give an algorithmic framework for the solution of the nonsmooth equation Φ(x) = 0. Algorithm 4.1 Let r ∈ (0, 1) and x0 ∈ Rn . Step 1. Use a globally convergent algorithm to solve the nonsmooth equation Φ(x) = 0 with x0 as the starting point. Let x¯ be the solution candidate that is returned. If Φ(¯ x) = 0, then stop. If x¯ is only a local minimum of Ψ, then go to next step. Step 2(LG phase). Use a globally convergent minimization algorithm applied to the augmented Lagrangian Pc (x, λ) defined by (6), starting from x¯ until a point xˆ is found such that kΦ(ˆ x)k ≤ rkΦ(¯ x)k. Set x0 to xˆ and return to Step 1.

We now present a LG algorithm which can be used in LG phase of the above algorithm. Algorithm 4.2 (LG algorithm) Step 0. Choose σ, β, r ∈ (0, 1) and initial vector z 0 = (x0 , λ0 ) ∈ R2n with λ0 ≤ 0. Set k := 0. Step 1. If kΦ(xk )k ≤ rkΦ(¯ x)k, then let xˆ := xk , stop. Step 2. Choose V k ∈ ∂x Pc (z k ) with ρi ∈ [−1, 1] and (ξi , ηi ) ∈ Ω3 in (17)–(20). Denote −V k −Φ(xk )

k

q =

!

.

Step 3. Determine tk = β mk , where mk is the smallest nonnegative integer m such that (31) holds for t = β m : Pc (z k + β m q k ) − Pc (z k ) ≤ −σβ m kq k k2 .

(31)

Step 4. Set z k+1 := z k + tk q k , k := k + 1. Go to Step 1. Proposition 4.1 Algorithm 4.2 is well-defined. That is, there exists a finite nonnegative integer m such that (31) holds if kΦ(xk )k > rkΦ(¯ x)k.

11

Proof. From the construction of Algorithm 4.2 and (30), we deduce Pc′ (z k ; q k ) ≤ −kq k k2 < 0. If the proposition does not hold, then ∀m ∈ N, Pc (z k + β m q k ) − Pc (z k ) > −σβ m kq k k2 . Hence

Pc (z k + β m q k ) − Pc (z k ) ≥ −σkq k k2 . m→∞ βm lim

That is Pc′ (z k ; q k ) ≥ −σkq k k2 ≥ σPc′ (z k ; q k ).

This contradicts Pc′(z k ; q k ) < 0 and σ ∈ (0, 1).

2

The following is the global convergence theorem of Algorithm 4.2 whose proof is motivated by Theorem 2.5 in [10]. Theorem 4.1 Assume that at any accumulation point z ∗ = (x∗ , λ∗ ) of {z k } produced by Algorithm 4.2 , the strict complementarity condition holds, i.e., I0 (x∗ ) = ∅. Then 0 ∈ ∂Pc (x∗ , λ∗ ). That is, x∗ is a solution of NCP(F) (1). Proof. Let subsequence {z k }k∈K of {z k } converge to z ∗ . From (17)–(20) and the continuity of ∇f, ∇F and Φ, we deduce that {q k }k∈K ! is bounded. Without loss of ∗ −V . Then, by Lemma 2.1 (ii), we generality, we assume that lim q k = q ∗ = −Φ(x∗ ) k∈K have V ∗ ∈ ∂x Pc (z ∗ ). If 0 ∈ / ∂Pc (z ∗ ), then q ∗ 6= 0 and q ∗ is a descent direction of Pc at z ∗ from I0 (x∗ ) = ∅ and Theorem 3.1. Choose a constant σ1 ∈ (σ, 1). From the proof of Proposition 4.1, there exists a constant t∗ > 0 such that Pc (z ∗ + t∗ q ∗ ) − Pc (z ∗ ) ≤ −t∗ σ1 kq ∗ k2 .

Since {z k }k∈K → z ∗ , {q k }k∈K → q ∗ , Pc (·) is continuous and σ1 > σ, there exists a positive integer k0 such that for all k ∈ K and k ≥ k0 , Pc (z k + t∗ q k ) − Pc (z k ) ≤ −t∗ σkq k k2 . This shows tk ≥ t∗ for all k ∈ K and k ≥ k0 . Therefore, Pc (z k+1 ) − Pc (z k ) ≤ −tk σkq k k2 ≤ −t∗ σkq k k2 .

(32)

Note that {Pc (z k )} is monotonically decreasing, thus lim Pc (z k ) = Pc (z ∗ ). But, by k→∞

(32), we have 0 ≤ −t∗ σkq ∗ k2 . This contradicts t∗ σkq ∗ k2 > 0. This contradiction shows 0 ∈ ∂Pc (x∗ , λ∗ ). By Proposition 2.1(ii), x∗ is a solution of NCP(F). 12

2 Example 4.1 Consider the NCP(F), where

F (x) =

                    

Then

Φ(x) =

                

1( x2 2 2 x2 + x + 1 − x − x − 1),

if x ≤ 1,

1( x2 2 2 2 −x + 5x − 1 + x − 5x + 1),

if 1 < x < 4,

x2 − 25 x − 29 6,

if x ≥ 4.

x2 + 1,

if x ≤ 1,

|x2 − 4x + 1|,

if 1 < x < 4,

x − 23 x − 29 6 − 2

r

2 x2 + (x2 − 52 x − 29 6) ,

if x ≥ 4.

1 Ψ(x) = kΦ(x)k2 . 2 √ Note that Ψ(x) has two local minimums: x = 0 and x = 2 + 3, in which the latter is a solution of Φ(x) = 0, but the former is not. Suppose that at Step 1 of Algorithm 4.1, we get the solution candidate x¯ = 0 of Φ(x) = 0, then we employ Algorithm 4.2 to produce a new point xˆ with kΦ(ˆ x)k ≤ rkΦ(¯ x)k and r ∈ (0, 1). In the implementation of the algorithm, we choose f (x) = αx, initial vector (x0 , λ0 ) = (0, 0) and constants α = −3, c = 2, σ = 0.8, β = 0.5, r = 0.9. If i ∈ I10 (x), we choose ρi = 1. Then Φ(¯ x) = 1. By implementing the algorithm in Matlab, at the 4-th iteration, we obtain x4 = 3.48602108,

kΦ(x4 )k = 0.791741350 ≤ rkΦ(¯ x)k.

We note that in the implementation of the algorithm, we can avoid the mutiply vector λ → ∞ by adjust parameters α and c so that the search direction on x is suitable.

References [1] X. Chen, L. Qi, and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67(1998) 519–540. 13

[2] F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983). [3] T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Prog., 75(1996) 407-439. [4] A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Prog., 76(1997) 513-532, [5] P.T. Harker and J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Prog., 48(1990) 161-220. [6] C. Kanzow and H. Pieper, Jacobian Smoothing methods for nonlinear complementarity problems, SIAM J. Optim., 9(1999) 342-373. [7] J.J. Mor´e, Global methods for nonlinear complementarity problems, Math. Oper. Res., 21(1996) 589-614. [8] J.L. Nazareth, Lagrangian globalization: Solving nonlinear equations via constrained optimization, J. Renegar, M. Shub and S. Smale (Eds. ), The Mathematics of Numerical Analysis, the American Society, Providence, Rhode Island, USA, Lectures in Applied Mathematics, 32(1996) 533-542. [9] J.L. Nazareth and L. Qi, Globalization of Newton’s methods for solving nonlinear equations, J. Numer. Algebra Appl. 3 (1996) 239-249. [10] E. Polak, D.Q. Mayne and Y. Wardi, On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems, SIAM J. Control & Optim., 21(1983) 179-204. [11] L. Qi and J. Sun, A nonsmooth version of Newton’s method, Math. Prog., 58(1993) 353-367. [12] N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Math. Prog., 76(1997) 469-491. [13] Y.F. Yang and D.H. Li, A globally and superlinearly convergent method for monotone complementarity problems with nonsmooth functions, Chinese Advanced Math. 6(1998) 555-557.

14