Properties of a Family of Generalized NCP-Functions and a Derivative Free Algorithm for Complementarity Problems

Properties of a Family of Generalized NCP-Functions and a Derivative Free Algorithm for Complementarity

Problems

Sheng-Long Hu

Department of Mathematics School of Science, Tianjin University

Tianjin 300072, P.R. China

Zheng-Hai Huang

Department of Mathematics School of Science, Tianjin University

Tianjin 300072, P.R. China Email: huangzhenghai@tju.edu.cn

Jein-Shan Chen

Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677 Email: jschen@math.ntnu.edu.tw

September 7, 2008; Revised: October 24, 2008

∗Corresponding Author. The author’s work is partially supported by the National Natural Science Foundation of China (Grant No. 10571134 and No. 10871144) and the Natural Science Foundation of Tianjin (Grant No. 07JCYBJC05200).

†Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

Abstract

In this paper, we propose a new family of NCP-functions and the corresponding merit functions, which are the generalization of some popular NCP-functions and the related merit functions. We show that the new NCP-functions and the corresponding merit functions possess a system of favorite properties. Specially, we show that the new NCP-functions are strongly semismooth, Lipschitz continuous, and continuously dif- ferentiable; and that the corresponding merit functions have SC¹ property (i.e., they are continuously differentiable and their gradients are semismooth) and LC¹ property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions. Based on the new NCP-functions and the corresponding merit functions, we investigate a derivative free algorithm for the nonlinear comple- mentarity problem and discuss its global convergence. Some preliminary numerical results are reported.

Key words: Complementarity problem, NCP-function, merit function, derivative free algorithm.

AMS subject classifications (2000): 90C33, 90C56, 65K10.

1 Introduction

In the last decades, people have put a lot of their energy and attention on the com- plementarity problem due to its various applications in operation research, economics, and engineering (see, for examples, [10, 13, 23]). The nonlinear complementarity problem (NCP) is to find a point x ∈ <ⁿ such that

x ≥ 0, F (x) ≥ 0, x^TF (x) = 0, (1.1)

where F : <ⁿ → <ⁿ is a continuously differentiable mapping with F := (F₁, F₂, . . . , F_n)^T. Many solution methods have been developed to solve NCP (1.1), for examples, [3, 5, 13, 14, 15, 16, 17, 19, 23, 27, 28]. For more details, please refers to the excellent monograph [9]. One of the most popular methods is to reformulate the NCP (1.1) as a unconstrained optimization problem and then to solve the reformulated problem by the unconstrained optimization technique. This kind of methods is called the merit function method, where the merit function is generally constructed by some NCP-function.

Definition 1.1 A function φ : <² → < is called an NCP-function [2, 18, 25, 26], if it satisfies

φ(a, b) = 0 ⇐⇒ a ≥ 0, b ≥ 0, ab = 0.

Furthermore, if φ(a, b) ≥ 0 for all (a, b) ∈ <² then the NCP-function φ is called a nonnegative NCP-function. In addition, if a function Ψ : <ⁿ → < is nonnegative and Ψ(x) = 0 if and only if x solves the NCP, then Ψ is called a merit function for the NCP.

If φ is an NCP-function, then it is easy to see that the function Ψ : <ⁿ → < defined by Ψ(x) := P_n

i=1φ²(xi, Fi(x)) is a merit function for the NCP. Thus, finding a solution of the NCP is equivalent to finding a global minimum of the unconstrained minimization min_x∈<ⁿΨ(x) with optimal value 0.

Many NCP-functions have been proposed in the literature. Among them, the FB function is one of the most popular NCP-functions, which is defined by

φ(a, b) :=√

a²+ b² − a − b, ∀(a, b) ∈ <².

One of the main generalization of FB function was given by Kanzow and Kleinmichel [18]:

φ_θ(a, b) :=p

(a − b)²+ θab − a − b, θ ∈ (0, 4), ∀(a, b) ∈ <². (1.2) Another main generalization was given by Chen and Pan [5]:

φ_p(a, b) := p^p

|a|^p+ |b|^p− a − b, p ∈ (1, ∞), ∀(a, b) ∈ <². (1.3)

It has been proved in [3, 4, 5, 6, 18, 22] that the functions φθ given in (1.2) and φp given in (1.3) possess a system of favorite properties, such as, strong semismoothness, Lipschitz continuity, and continuous differentiability. It has also been proved that the corresponding merit functions of φ_θand φ_p have SC¹property (i.e., they are continuously differentiable and their gradients are semismooth) and LC¹ property (i.e., they are continuously differentiable and their gradients are Lipschitz continuous) under suitable assumptions.

Motivated by Kanzow and Kleinmichel [18] and Chen and Pan [5], we introduce in this paper the following functions:

φ_θp(a, b) := p^p

θ(|a|^p+ |b|^p) + (1 − θ)|a − b|^p− a − b, p > 1, θ ∈ (0, 1], (a, b) ∈ <². (1.4) and

Ψ_θp(x) := 1 2

Xn i=1

φ²_θp(x_i, F_i(x)). (1.5)

Is the function φ_θp an NCP-function? If it is, do the functions given by (1.4) and (1.5) have the same properties as those known functions mentioned above? Furthermore, how is the numerical behavior of the merit function methods based on the functions defined by (1.4) and (1.5)?

In this paper, we will answer the questions mentioned above partly. Firstly, we show that the function φ_θpdefined by (1.4) is an NCP-function; and discuss some favorite properties of the NCP-function (1.4) and its nonnegative NCP-function, including strong semismoothness, Lipschitz continuity, and continuous differentiability. Since the function φ_θp defined by (1.4) is an NCP-function, it follows that the function Ψ_θp defined by (1.5) is a merit function associated to the NCP-function φθp. We also show that the merit function Ψθp has SC¹ property and LC¹ property. Secondly, we investigate a derivative free method based on the functions defined by (1.4) and (1.5) and show its global convergence. (Note: usually the nonsmooth Newton method is faster than the derivative free method for solving NCPs.

However, the derivative free algorithm may overcome the case where strong conditions are sometimes needed to guarantee that the Jacobian of the merit function is nonsingular or very expensive to compute.) Thirdly, we report the preliminary numerical results for test problems from MCPLIB. The preliminary numerical results show, on the average, that the algorithm works better when θ = 1 (according to the FB-type function), θ = 0.9 and θ = 0.25, and when p = 1.1 or p = 2 or p = 20 generally.

The rest of this paper are organized as follows. Various properties of the new NCP- function (1.4) and the nonnegative NCP-function associated to (1.4) are established in the next section. In Section 3, some properties of the merit function defined by (1.5) are an- alyzed. In Section 4, we investigate a derivative free algorithm for the NCP and show its global convergence. Some preliminary numerical results are reported in Section 5 and final conclusions are given in the last section.

Throughout this paper, unless stated otherwise, all vectors are column vectors, the sub- script T denotes transpose, <ⁿ denotes the space of n-dimensional real column vectors, and

<ⁿ₊ (respectively, <ⁿ₊₊) denotes the nonnegative (respectively, positive) orthant in <ⁿ. For any vectors u, v ∈ <ⁿ, we write (u^T, v^T)^T as (u, v) for simplicity. For x ∈ <ⁿ, we use x ≥ 0 (respectively, x > 0) to mean x ∈ <ⁿ₊ (respectively, x ∈ <ⁿ₊₊). We use “:=” to mean ”be defined as”. We denote by kuk the 2-norm of u and kuk_p the p-norm with p > 1. We use

∇F to denote the gradient of F (while ^{∂F (x)}_∂xi denotes to the i-th component of the gradient of F ) and ∇²F to denote the second order derivative of F . We use α = o(β) (respectively, α = O(β)) to mean ^α_β tends to zero (respectively, bounded uniformly) as β → 0.

2 Properties of the New NCP-Function

In this section, we show that the function φθp defined by (1.4) is an NCP-function, and discuss its properties which are similar to those obtained in [3, 5] for the function φp defined by (1.3). We also study a nonnegative NCP-function associated with φ_θp, and discuss its properties. In addition, we discuss the semismooth-related properties due to its importance in semismooth and smooth analysis [8, 10, 15, 16, 20, 24].

For convenience, we define η_θp(a, b) := p^p

θ(|a|^p+ |b|^p) + (1 − θ)|a − b|^p, p > 1, θ ∈ (0, 1], (a, b) ∈ <². (2.1) The proofs of the following propositions are trivial, we omit their proofs here.

Proposition 2.1 The function φ_θp defined by (1.4) is an NCP-function.

Proposition 2.2 The function ηθp defined by (2.1) is a norm on <² for all p > 1, θ ∈ (0, 1].

Now, we briefly introduce the concept of semismoothness, which was originally introduced by Mifflin [20] for functionals and was extended to vector valued functions by Qi and Sun [24]. A locally Lipschitz function F : <ⁿ → <^m, which has the generalized Jacobian ∂F (x) in the sense of Clarke [8], is said to be semismooth (or strongly semismooth) at x ∈ <ⁿ, if F is directionally differentiable at x and

F (x + h) − F (x) − V h = o(khk) (or = O(khk²) holds for any V ∈ ∂F (x + h).

Proposition 2.3 Let φθp be defined by (1.4), then for all θ ∈ (0, 1] and p > 1,

(i) φ_θpis sub-additive, i.e., φ_θp((a, b)+(c, d)) ≤ φ_θp(a, b)+φ_θp(c, d) for all (a, b), (c, d) ∈ <²;

(ii) φθp is positive homogenous, i.e., φθp(α(a, b)) = αφθp(a, b) for all (a, b) ∈ <² and α > 0;

(iii) φ_θp is a convex function on <², i.e., φ_θp(α(a, b) + (1 − α)(c, d)) ≤ αφ_θp(a, b) + (1 − α)φ_θp(c, d) for all (a, b), (c, d) ∈ <² and α ∈ [0, 1];

(iv) φ_θp is Lipschitz continuous on <²;

(v) φ_θp is continuously differentiable on <²\{(0, 0)};

(vi) φθp is strongly semismooth on <².

Proof. By using φ_θp((a, b)) = η_θp(a, b) − (a + b) and Proposition 2.2, we can obtain that the results (i), (ii), and (iii) hold.

Consider the result (iv). Since η_θp is a norm on <² from Proposition 2.2 and any two norms in finite dimensional space are equivalent, it follows that there exists a positive con- stant κ such that

ηθp(a, b) ≤ κk(a, b)k, ∀(a, b) ∈ <²,

where k · k represents the Euclidean norm on <². Hence, for all (a, b), (c, d) ∈ <²,

|φ_θp(a, b) − φ_θp(c, d)| = |η_θp(a, b) − (a + b) − η_θp(c, d) + (c + d)|

≤ |η_θp(a, b) − η_θp(c, d)| + |a − c| + |b − d|

≤ η_θp(a − c, b − d) +√

2k(a − c, b − d)k

≤ κk(a − c, b − d)k +√

2k(a − c, b − d)k

= (κ +√

2)k(a − c, b − d)k.

Hence, φ_θp is Lipschitz continuous with Lipschitz constant κ +√

2, i.e., the result (iv) holds.

Consider the result (v). If (a, b) 6= (0, 0), then ηθp(a, b) 6= 0 by Proposition 2.2. By a direct calculation, we get

∂φθp(a, b)

∂a = θsgn(a)|a|^p−1+ (1 − θ)sgn(a − b)|a − b|^p−1

η_θp(a, b)^p−1 − 1; (2.2)

∂φθp(a, b)

∂b = θsgn(b)|b|^p−1− (1 − θ)sgn(a − b)|a − b|^p−1

η_θp(a, b)^p−1 − 1, (2.3)

where sgn(·) is the symbol function. It is easy to see from (2.2) and (2.3) that the result (v) holds.

Consider the result (vi). Since φθp is a convex function by the result (iii), we get that it is a semismooth function. Noticing that φ_θp is continuously differentiable except (0, 0), it is

sufficient to prove that it is strongly semismooth at (0, 0). For any (h, k) ∈ <²\{(0, 0)}, φθp

is differentiable at (h, k), and hence, ∇φ_θp(h, k) =

³∂φθp(h,k)

∂a ,^∂φθp_∂b^(h,k)

´_T . So,

φ_θp((0, 0) + (h, k)) − φ_θp(0, 0) −

µ∂φ_θp(h, k)

∂a ,∂φ_θp(h, k)

∂b

¶ µ h k

¶

= p^p

θ(|h|^p+ |k|^p) + (1 − θ)|h − k|^p− (h + k)

−(sgn(h)|h|^p−1+ sgn(h − k)|h − k|^p−1 η_θp(h, k)^p−1 − 1)h

−(sgn(k)|k|^p−1− sgn(h − k)|h − k|^p−1 η_θp(h, k)^p−1 − 1)k

= p^p

θ(|h|^p+ |k|^p) + (1 − θ)|h − k|^p

−sgn(h)|h|^p−1h + sgn(k)|k|^p−1k + sgn(h − k)|h − k|^p−1(h − k) η_θp(h, k)^p−1

= p^p

θ(|h|^p+ |k|^p) + (1 − θ)|h − k|^p− |h|^p+ |k|^p + |h − k|^p η_θp(h, k)^p−1

= η_θp(h, k) − |h|^p + |k|^p+ |h − k|^p ηθp(h, k)^p−1

= η_θp(h, k)^p − (|h|^p + |k|^p+ |h − k|^p) ηθp(h, k)^p−1

= 0

= O(k(h, k)k²).

Thus, we obtain that φθp is strongly semismooth.

We complete the proof. 2

Proposition 2.4 Let φθp be defined by (1.4) and {(a^k, b^k)} ⊆ <². Then, |φθp(a^k, b^k)| → ∞ if one of the following conditions is satisfied.

(i). a^k→ −∞; (ii). b^k→ −∞; (iii). a^k → ∞ and b^k → ∞.

Proof. (i) Suppose that a^k → −∞. If {b^k} is bounded from above, then the result holds trivially. When b^k → ∞, we have −a^k > 0 and b^k > 0 for all k sufficiently large, and hence,

pp

θ(|a^k|^p+ |b^k|^p) + (1 − θ)|a^k− b^k|^p− b^k≥ p^p

θ|b^k|^p+ (1 − θ)|b^k|^p− b^k= 0.

This, together with −a^k → ∞ and the definition of φθp, implies that the result holds.

(ii) For the case of b^k → −∞, a similar analysis yields the result of the proposition.

(iii) Suppose that a^k → ∞ and b^k → ∞. Since p > 1 and θ ∈ (0, 1], we have (1 − θ)|a^k− b^k|^p ≤ (1 − θ)(|a^k|^p+ |b^k|^p) for all sufficiently large k. Thus, for all sufficiently large k,

pp

θ(|a^k|^p+ |b^k|^p) + (1 − θ)|a^k− b^k|^p ≤ p^p

|a^k|^p+ |b^k|^p,

and hence,

(a^k+ b^k) − p^p

θ(|a^k|^p+ |b^k|^p) + (1 − θ)|a^k− b^k|^p ≥ (a^k+ b^k) −p^p

|a^k|^p+ |b^k|^p. By [5, Lemma 3.1] we know that (a^k + b^k) − p^p

|a^k|^p + |b^k|^p → ∞ as k → ∞ when the condition (iii) is satisfied. Thus, we obtain that

|φθp(a^k, b^k)| = (a^k+ b^k) − p^p

θ(|a^k|^p+ |b^k|^p) + (1 − θ)|a^k− b^k|^p → ∞

as k → ∞, which completes the proof. 2

Now, we define a nonnegative function, associated with the function φ_θp, as follows.

ψ_θp(a, b) := 1

2φ²_θp(a, b), p > 1, θ ∈ (0, 1], (a, b) ∈ <². (2.4) Proposition 2.5 Let ψ_θp be defined by (2.4), then for all θ ∈ (0, 1] and p > 1,

(i) ψ_θp is an NCP-function;

(ii) ψ_θp(a, b) ≥ 0 for all (a, b) ∈ <²;

(iii) ψ_θp is continuously differentiable on <²; (vi) ψ_θp is strongly semismooth on <²;

(v) ^∂ψθp_∂a^(a,b) · ^∂ψθp_∂b^(a,b) ≥ 0 for all (a, b) ∈ <², where the equality holds if and only if φ_θp(a, b) = 0;

(vi) ^∂ψθp_∂a^(a,b) = 0 ⇐⇒ ^∂ψθp_∂b^(a,b) = 0 ⇐⇒ φθp(a, b) = 0.

Proof. By the definition of ψ_θp, it is easy to see that the results (i) and (ii) hold.

Consider the result (iii). By using Proposition 2.3 and the definition of ψ_θp, it is sufficient to prove that ψ_θp is differentiable at (0, 0) and the gradient is continuous at (0, 0). In fact, for all (a, b) ∈ <²\{(0, 0)}, we have,

|φ_θp(a, b)| =

¯¯

¯p^p

θ(|a|^p + |b|^p) + (1 − θ)|a − b|^p− a − b

¯¯

¯

≤

¯¯

¯p^p

θ|a|^p+p^p

θ|b|^p+p^p

(1 − θ)|a − b|^p

¯¯

¯ + |a| + |b|

≤ |a| + |b| + |a − b| + |a| + |b|

≤ 3(|a| + |b|),

where the second inequality follows from p > 1 and the third inequality follows from θ ∈ (0, 1]. Hence,

ψ_θp(a, b) − ψ_θp(0, 0) = 1

2φ²_θp(a, b) ≤ 1

2(3(|a| + |b|))² ≤ O(|a|²+ |b|²).

Thus, similar to that of [7, Proposition 1], we can get that ψθpis differentiable at (0, 0) with

∇ψθp(0, 0) = (0, 0)^T. Now, we prove that for all (a, b) ∈ <²\{(0, 0)},

¯¯

¯¯θsgn(a)|a|^p−1+ (1 − θ)sgn(a − b)|a − b|^p−1 η_θp(a, b)^p−1

¯¯

¯¯ ≤ 1, (2.5)

¯¯

¯¯θsgn(b)|b|^p−1− (1 − θ)sgn(a − b)|a − b|^p−1 η_θp(a, b)^p−1

¯¯

¯¯ ≤ 1. (2.6)

In fact,

¯¯

¯¯θsgn(a)|a|^p−1+ (1 − θ)sgn(a − b)|a − b|^p−1 η_θp(a, b)^p−1

¯¯

¯¯

≤ θ|a|^p−1+ (1 − θ)|a − b|^p−1 η_θp(a, b)^p−1

= θ^1p|θ^1pa|^p−1+ (1 − θ)^1p|(1 − θ)^1p(a − b)|^p−1 η_θp(a, b)^p−1

≤ ((θ^1p)^p+ ((1 − θ)^1p)^p)^1p((|θ^1pa|^p−1)^p−1p + (|(1 − θ)^1p(a − b)|^p−1)^p−1p )^p−1p ηθp(a, b)^p−1

= (θ + (1 − θ))(x^p+ z^p)^p−1p ηθp(a, b)^p−1

= (x^p+ z^p)^p−1p (x^p+ y^p+ z^p)^p−1p

= ( x^p+ z^p

x^p+ y^p+ z^p)^p−1p

≤ 1,

where x := |θ^1pa|^p, y := |θ^1pb|^p, z := |(1 − θ)^1p(a − b)|^p; the first inequality follows from the triangle inequality; the second inequality follows from the well-known H¨older inequality; the second equality follows from the definitions of x and z; the third equality follows from the definitions of ηθp(a, b), x, y and z; and the third inequality follows from the fact that x, y and z are all nonnegative. So, (2.5) holds. Similar analysis will derive that (2.6) holds.

Thus, it follows from (2.5) and (2.6) that both ^∂φθp_∂a^(a,b) and ^∂φθp_∂b^(a,b) are uniformly bounded. Since φ_θp(a, b) → 0 as (a, b) → (0, 0), we get the desired result.

Consider the result (iv). Since the composition of strongly semismooth function is also strongly semismooth (see [11, Theorem 19]), by Proposition 2.3(vi) and the definition of ψθp

we obtain that the desired result holds.

Consider the result (v). It is obvious that ^∂ψθp_∂a^(a,b) = 0 when (a, b) = (0, 0). Now, suppose that (a, b) 6= (0, 0). Since

∂ψ_θp(a, b)

∂a ·∂ψ_θp(a, b)

∂b = ∂φ_θp(a, b)

∂a · ∂φ_θp(a, b)

∂b · φ_θp(a, b)², (2.7)

by (2.2), (2.3), (2.5), and (2.6), we obtain that ^∂φθp_∂a^(a,b) ≤ 0 and ^∂φθp_∂b^(a,b) ≤ 0 for all (a, b) ∈ <², that is, the first result of (v) holds. In addition, from (2.7) it is obvious that the sufficient condition of the second result of (v) holds. Now, we suppose that ^∂ψθp_∂a^(a,b) · ^∂ψθp_∂b^(a,b) = 0.

Then, it is sufficient to prove that φ_θp(a, b) = 0 when ^∂φθp_∂a^(a,b) · ^∂φθp_∂b^(a,b) = 0. Suppose that

∂φθp(a,b)

∂a = 0 without loss of generality. From the proof of (iii) in this proposition, it is easy to see that it must be y = 0, and hence, b = 0. After a simple symbol discussion for (2.2), we may get a ≥ 0. Hence φθp(a, b) = 0 by Proposition 2.1. So, the result (v) holds.

Consider the result (vi). Since

∂ψ_θp(a, b)

∂a = ∂φ_θp(a, b)

∂a φ_θp(a, b), ∂ψ_θp(a, b)

∂b = ∂φ_θp(a, b)

∂b φ_θp(a, b), the result (vi) is immediately satisfied from the above analysis.

We complete the proof. 2

Lemma 2.1 [21, Theorem 3.3.5] If f : D ⊆ <ⁿ→ <^m has a second derivative at each point of a convex set D₀ ⊆ D, then k∇f (y) − ∇f (x)k ≤ sup_0≤t≤1k∇²f (x + t(y − x))k · ky − xk.

Theorem 2.1 The gradient function of the function ψ_θp defined by (2.4) with p ≥ 2, θ ∈ (0, 1] is Lipschitz continuous, that is, there exists a positive constant L such that

k∇ψθp(a, b) − ∇ψθp(c, d)k ≤ Lk(a, b) − (c, d)k (2.8) holds for all (a, b), (c, d) ∈ <².

Proof. It follows from the definition of ψθp and the proof of Proposition 2.5(iii) that

∇ψθp(a, b) = ∇φθp(a, b)φθp(a, b) when (a, b) 6= (0, 0), and ∇ψθp(0, 0) = (0, 0)^T. From Propo- sition 2.5(iii) we know that ψ_θp is continuous differentiable. The proof is divided into the following three cases.

Case 1. If (a, b) = (c, d) = (0, 0), it follows from Proposition 2.5 that ∇ψ_θp(0, 0) = (0, 0), and hence, (2.8) holds for all positive number L.

Case 2. Consider the case that one of (a, b) and (c, d) is (0, 0), but not all. We assume that (a, b) 6= (0, 0) and (c, d) = (0, 0) without loss of generality. Then,

k∇ψ_θp(a, b) − ∇ψ_θp(c, d)k = k∇ψ_θp(a, b) − (0, 0)k

= k∇φθp(a, b)φθp(a, b) − (0, 0)k

= k∇φ_θp(a, b)kφ_θp(a, b)

= k∇φ_θp(a, b)k|φ_θp(a, b) − φ_θp(0, 0)|

≤ Lk(a, b) − (0, 0)k,

where the inequality follows from the fact that {k∇φθp(a, b)k} is uniformly bounded on <² (which can be obtained from the proof of Proposition 2.5(iii)) and φθpis Lipschitz continuous on <² given in Proposition 2.3 (iv). Hence, (2.8) holds for some positive constant L.

Case 3. If both (a, b) and (c, d) are not (0, 0), we will use Lemma 2.1 to prove (2.8) holds for this case. For simplicity, we denote

ˆh1 := θsgn(a)|a|^p−1+ (1 − θ)sgn(a − b)|a − b|^p−1 η^p−1_θp (a, b) ; ˆh₂ := θsgn(b)|b|^p−1− (1 − θ)sgn(a − b)|a − b|^p−1

η_θp^p−1(a, b) ; ˆa₁ := (θ|a|^p−2+ (1 − θ)|a − b|^p−2)η^p_θp(a, b);

ˆa₂ := −ˆh²₁η_θp^2p−2(a, b);

ˆb₁ := −(1 − θ)|a − b|^p−2η_θp^p (a, b);

ˆb2 := −ˆh₁ˆh2η^2p−2_θp (a, b);

ˆc1 := (θ|b|^p−2+ (1 − θ)|a − b|^p−2)η_θp^p (a, b);

ˆc₂ := −ˆh²₂η_θp^2p−2(a, b).

When (a, b) 6= (0, 0), by a direct calculation, we have

∂²ψ_θp(a, b)

∂a² = (ˆh₁− 1)²+ (p − 1) ˆa₁+ ˆa₂

η_θp^2p−1(a, b)(η_θp(a, b) − (a + b));

∂²ψ_θp(a, b)

∂a∂b = (ˆh₁− 1)(ˆh₂− 1) + (p − 1) ˆb₁+ ˆb₂

η^2p−1_θp (a, b)(η_θp(a, b) − (a + b));

∂²ψ_θp(a, b)

∂b² = (ˆh₂− 1)²+ (p − 1) ˆc₁+ ˆc₂

η_θp^2p−1(a, b)(η_θp(a, b) − (a + b));

∂²ψθp(a, b)

∂b∂a = ∂²ψθp(a, b)

∂a∂b ,

where the last equality follows from the fact that ^∂2ψ_∂a∂b^θp(a,b) and ^∂2ψ_∂b∂a^θp(a,b) are continuous when (a, b) 6= (0, 0). Since p ≥ 2 and η_θp(·, ·) is a norm on <² by Proposition 2.2, it is easy to verify that

|a + b| ≤ |a| + |b| ≤ p^p

|a|^p + |b|^p+p^p

|a|^p+ |b|^p = 2k(a, b)k_p ≤ 2κ^∗η_θp(a, b), where κ^∗ > 0 is a constant depending on θ and p.

ˆa₁

η_θp^2p−2(a, b) = θ|a|^p−2+ (1 − θ)|a − b|^p−2 η^p−2_θp (a, b)

= θ|a|^p−2

η_θp^p−2(a, b) + (1 − θ)|a − b|^p−2 η_θp^p−2(a, b)

≤ θ^2p + (1 − θ)^2p

≤ 2.

Similarly, we have

|ˆb₁|

η^2p−2_θp (a, b) ≤ 1; ˆc₁

η_θp^2p−2(a, b) ≤ 2.

These, together with the results |ˆh₁| ≤ 1 and |ˆh₂| ≤ 1 given in Proposition 2.5, yield

|ˆa₂|

η_θp^2p−2(a, b) ≤ 1; |ˆb₂|

η_θp^2p−2(a, b) ≤ 1; |ˆc₂|

η_θp^2p−2(a, b) ≤ 1.

Thus,

¯¯

¯¯∂²ψ_θp(a, b)

∂a²

¯¯

¯¯ =

¯¯

¯¯

¯(ˆh₁− 1)² + (p − 1) ˆa₁+ ˆa₂

η^2p−1_θp (a, b)(η_θp(a, b) − (a + b))

¯¯

¯¯

¯

≤ |(ˆh₁− 1)²| + (p − 1) ï¯

¯¯

¯

ˆa1+ ˆa2

η_θp^2p−1(a, b)η_θp(a, b)

¯¯

¯¯

¯+

¯¯

¯¯

¯

ˆa1+ ˆa2

η^2p−1_θp (a, b)(a + b)

¯¯

¯¯

¯

!

≤ 4 + (1 + 2κ^∗)(p − 1)

à ˆa₁

η_θp^2p−2(a, b) + |ˆa₂| η^2p−2_θp (a, b)

!

≤ 4 + 3(1 + 2κ^∗)(p − 1);

¯¯

¯¯∂²ψ_θp(a, b)

∂a∂b

¯¯

¯¯ =

¯¯

¯¯

¯(ˆh₁− 1)(ˆh₂ − 1) + (p − 1) ˆa₁+ ˆa₂

η_θp^2p−1(a, b)(η_θp(a, b) − (a + b))

¯¯

¯¯

¯

≤ |(ˆh₁− 1)(ˆh₂ − 1)|

+(p − 1) ï¯

¯¯

¯

ˆb1 + ˆb2

η_θp^2p−1(a, b)ηθp(a, b)

¯¯

¯¯

¯+

¯¯

¯¯

¯

ˆb1+ ˆb2

η_θp^2p−1(a, b)(a + b)

¯¯

¯¯

¯

!

≤ 4 + (1 + 2κ^∗)(p − 1)

à |ˆb₁|

η_θp^2p−2(a, b) + |ˆb₂| η^2p−2_θp (a, b)

!

≤ 4 + 2(1 + 2κ^∗)(p − 1);

¯¯

¯¯∂²ψ_θp(a, b)

∂b²

¯¯

¯¯ =

¯¯

¯¯

¯(ˆh₂− 1)² + (p − 1) ˆc₁+ ˆc₂

η^2p−1_θp (a, b)(η_θp(a, b) − (a + b))

¯¯

¯¯

¯

≤ |(ˆh₂− 1)²| + (p − 1) ï¯

¯¯

¯

ˆc₁+ ˆc₂

η_θp^2p−1(a, b)η_θp(a, b)

¯¯

¯¯

¯+

¯¯

¯¯

¯

ˆc₁+ ˆc₂

η^2p−1_θp (a, b)(a + b)

¯¯

¯¯

¯

!

≤ 4 + (1 + 2κ^∗)(p − 1) Ã

ˆc1

η_θp^2p−2(a, b) + |ˆc2| η^2p−2_θp (a, b)

!

≤ 4 + 3(1 + 2κ^∗)(p − 1).

Hence, there exists a positive constant L such that (2.8) holds by Lemma 2.1.

Combining Cases 1–3, we complete the proof. 2

Remark 2.1 It should be noted that ∇ψθp is not Lipschitz continuous for all θ ∈ (0, 1] when p ∈ (1, 2). In fact, if we fixed θ = 1. For (a, b) 6= (0, 0) and (c, d) 6= (0, 0), we have

k∇ψ_1p(a, b) − ∇ψ_1p(c, d)k

= k∇φ1p(a, b)φ1p(a, b) − ∇φ1p(c, d)φ1p(c, d)k

≥

¯¯

¯¯sgn(a)|a|^p−1 k(a, b)k^p−1p

φ_1p(a, b) − sgn(c)|c|^p−1 k(c, d)k^p−1p

φ_1p(c, d) + φ_1p(c, d) − φ_1p(a, b)

¯¯

¯¯

≥

¯¯

¯¯sgn(a)|a|^p−1 k(a, b)k^p−1p

φ_1p(a, b) − sgn(c)|c|^p−1 k(c, d)k^p−1p

φ_1p(c, d)

¯¯

¯¯ − |φ^1p(c, d) − φ_1p(a, b)|

≥

¯¯

¯¯sgn(a)|a|^p−1 k(a, b)k^p−1p

φ_1p(a, b) − sgn(c)|c|^p−1 k(c, d)k^p−1p

φ_1p(c, d)

¯¯

¯¯ − (κ +

√2)k(c, d) − (a, b)k,

where κ +√

2 is given in Proposition 2.3(iv). If we let (a, b) = (1, −n), (c, d) = (−1, −n) with n ∈ (1, ∞), we have

¯¯

¯¯sgn(a)|a|^p−1 k(a, b)k^p−1p

φ1p(a, b) − sgn(c)|c|^p−1 k(c, d)k^p−1p

φ1p(c, d)

¯¯

¯¯

=

√p

1 + n^p+ (n − 1) (1 + n^p)^(p−1)/p +

√p

1 + n^p+ (n + 1) (1 + n^p)^(p−1)/p

= 2

√p

1 + n^p+ n (1 + n^p)^(p−1)/p

≥ 4n

(1 + n^p)^(p−1)/p

= 4n^2−pn^p−1 (1 + n^p)^(p−1)/p

= 4n^2−p

(1 + (1/n)^p)^(p−1)/p

≥ n^2−p,

where the first and the second inequalities follow from 2 > p > 1 and n > 1. Since k(a, b) − (c, d)k = 2 and n ∈ (1, ∞), form the above inequalities it is easy to verify that ∇ψ1p is not Lipschitz continuous.

3 Properties of Merit Function

In this section, we consider the merit function for the NCP defined by (1.5), and then discuss its several important properties. These properties provide the theoretical basis for the algorithm we discussed in the next section. In addition, we also discuss the semismooth- related properties of the merit function.

Define

Φ_θp(x) :=





φ_θp(x₁, F₁(x)) . . . φ_θp(x_n, F_n(x))



 . (3.1)

Then, the merit function defined by (1.5) can be written as

Ψ_θp(x) = 1

2kΦ_θp(x)k² = Xn

i=1

ψ_θp(x_i, F_i(x)). (3.2)

Proposition 3.1 (i) The function ψ_θp defined by (2.4) with p ≥ 2, θ ∈ (0, 1] is an SC¹ function. Hence, if every F_i is an SC¹ function, then the function Ψ_θp defined by (3.2) with p ≥ 2, θ ∈ (0, 1] is also an SC¹ function.

(ii) If every F_i is an LC¹ function, then the function Φ_θp defined by (3.1) with p > 1, θ ∈ (0, 1] is strongly semismooth.

(iii) The function ψ_θp defined by (2.4) with p ≥ 2, θ ∈ (0, 1] is an LC¹ function. Hence, if every F_i is an LC¹function, then the function Ψ_θpdefined by (3.2) with p ≥ 2, θ ∈ (0, 1]

is also an LC¹ function.

Proof. (i) By Proposition 2.5, it is sufficient to prove that ∇ψ_θp is semismooth. It is obvious from the proof of Theorem 2.1 that ∇ψ_θp(a, b) is continuously differentiable when (a, b) 6= (0, 0), so we only need to show the semismoothness of ∇ψ_θp(a, b) at (0, 0). For any (h₁, h₂) ∈ <²\{(0, 0)}, we know that ∇ψ_θp is differentiable at (h₁, h₂), and hence, we only need to show that

∇ψ_θp(h₁, h₂) − ∇ψ_θp(0, 0) − ∇²ψ_θp(h₁, h₂) · (h₁, h₂)^T = o(k(h₁, h₂)k). (3.3) In fact, let ˆa₁, ˆa₂, ˆb₁, ˆb₂, ˆc₁, ˆc₂ be similarly defined as those in Theorem 2.1 with (a, b) being replaced by (h₁, h₂). Denote

ˆh3 := (p − 1) ˆa₁+ ˆa₂

η_θp^2p−1(h₁, h₂)φθp(h1, h2);

ˆh4 := (p − 1) ˆb1+ ˆb₂

η_θp^2p−1(h1, h2)φ_θp(h₁, h₂);

ˆh5 := (p − 1) ˆc₁+ ˆc₂

η_θp^2p−1(h₁, h₂)φθp(h1, h2), and

m₁ : = (θ|h₁|^p−2+ (1 − θ)|h₁− h₂|^p−2)η^p_θp(h₁, h₂)h₁− ˆh²₁η_θp^2p−2(h₁, h₂)h₁;

m2 : = (1 − θ)|h1− h2|^p−2η_θp^p (h1, h2)h2+ ˆh1ˆh2η_θp^2p−2(h1, h2)h2; m₃ : = (θ|h₁|^p−2+ (1 − θ)|h₁− h₂|^p−2)η^p_θp(h₁, h₂)h₁

−(1 − θ)|h₁− h₂|^p−2η_θp^p (h₁, h₂)h₂; m₄ : = ˆh₁ˆh₂η_θp^2p−2(h₁, h₂)h₂+ ˆh²₁η_θp^2p−2(h₁, h₂)h₁;

m₅ : = (θsgn(h₁)|h₁|^p−1+ (1 − θ)sgn(h₁− h₂)|h₁− h₂|^p−1)η_θp^p (h₁, h₂);

m6 : = ˆh1ˆh2η_θp^2p−2(h1, h2)h2+ ˆh²₁η_θp^2p−2(h1, h2)h1. Then,

µ H₁ H2

¶ :=

à ˆh₁− 1 ˆh₂− 1

!

· φ_θp(h₁, h₂) − µ 0

0

¶

− Ã

(ˆh1− 1)²+ ˆh3 (ˆh1− 1)(ˆh2− 1) + ˆh4

(ˆh₁− 1)(ˆh₂− 1) + ˆh₄ (ˆh₂− 1)²+ ˆh₅

!

· µ h1

h₂

¶ .

and hence,

H₁ = (ˆh₁− 1)φ_θp(h₁, h₂) − ((ˆh₁− 1)²+ ˆh₃)h₁− ((ˆh₁− 1)(ˆh₂− 1) + ˆh₄)h₂

= (ˆh₁− 1)φ_θp(h₁, h₂) − ˆh₃h₁− ˆh₄h₂− (ˆh₁− 1)((ˆh₁− 1)h₁+ (ˆh₂− 1)h₂)

= (ˆh1− 1)φθp(h1, h2) − ˆh3h1− ˆh4h2− (ˆh1− 1)φθp(h1, h2)

= −(p − 1) Ã

ˆa₁+ ˆa₂

η_θp^2p−1(h₁, h₂)h₁+ ˆb₁+ ˆb₂ η_θp^2p−1(h₁, h₂)h₂

!

φ_θp(h₁, h₂)

= −(p − 1)φ_θp(h₁, h₂) Ã

m1 − m2

η^2p−1_θp (h₁, h₂)

!

= −(p − 1)φθp(h1, h2)

à m₃ − m₄ η^2p−1_θp (h₁, h₂)

!

= −(p − 1)φ_θp(h₁, h₂)

à m₅ − m₆ η^2p−1_θp (h1, h2)

!

= −(p − 1)φ_θp(h₁, h₂) Ã

ˆh₁− ˆh₁ˆh₁h₁+ ˆh₂h₂ η_θp(h₁, h₂)

!

= −(p − 1)φ_θp(h₁, h₂)(ˆh₁− ˆh₁)

= 0,

where the third equality follows from ˆh₁h₁+ ˆh₂h₂ = η_θp given in the proof of Proposition 2.3 and the definition of φ_θp, the fourth equality follows from the definitions of ˆh₃, ˆh₄, the fifth equality follows from the definitions of ˆa₁, ˆa₂, ˆb₁, ˆb₂, and the eighth equality follows from ˆh1h1+ ˆh2h2 = ηθp given in the proof of Proposition 2.3.

Similar analysis yields H2 = 0. Thus, ∇ψθp is semismooth. Furthermore, ψθp is an SC¹ function.

(ii) Since the LC¹ function is strongly semismooth and the composition of strongly semismooth function is also strongly semismooth, it follows from Proposition 2.3(vi) that the desired results holds.

(iii) By using the above results, it is easy that the result (iii) holds.

We complete the proof. 2

Remark 3.1 The results of Proposition 3.1(i)(iii) do not hold when p ∈ (1, 2) for all θ ∈ (0, 1] since ∇ψ_θp is not locally Lipschitz continuous in general. For example, let (a, b) = (¹_n, −1) and (c, d) = (−¹_n, −1), similar to Remark 2.1, we can obtain that ∇ψ_θp is not Lipschitz continuous in any neighborhood of (0, −1).

Definition 3.1 Let F : <ⁿ→ <ⁿ.

• F is said to be monotone if (x − y)^T(F (x) − F (y)) ≥ 0 for all x, y ∈ <ⁿ.

• F is said to be strongly monotone with modulus µ > 0 if (x − y)^T(F (x) − F (y)) ≥ µkx − yk² for all x, y ∈ <ⁿ.

• F is said to be a P₀-function if max_{1≤j≤n,xi6=yi}(x_i − y_i)(F_i(x) − F_i(y)) ≥ 0 for all x, y ∈ <ⁿ and x 6= y.

• F is said to be a uniform P -function with modulus µ > 0 if max1≤j≤n(xi− yi)(Fi(x) − F_i(y)) ≥ µkx − yk² for all x, y ∈ <ⁿ.

Proposition 3.2 Let Ψθp : <ⁿ → < be defined by (3.2) with p > 1, θ ∈ (0, 1]. Then Ψθp(x) ≥ 0 for all x ∈ <ⁿ and Ψθp(x) = 0 if and only if x solves the NCP (1.1). Moreover, suppose that the solution set of the NCP (1.1) is nonempty, then x is a global minimizer of Ψ_θp if and only if x solves the NCP (1.1).

Proof. The result follows from Proposition 2.5 immediately. 2

Proposition 3.3 Let Ψθp: <ⁿ→ < be defined by (3.2) with p > 1, θ ∈ (0, 1]. Suppose that F is either a monotone function or a P0-function, then every stationary point of Ψθp is a global minima of min_x∈<ⁿΨ_θp(x); and therefore solves the NCP (1.1).

Proof. By using Proposition 2.5 and [5, Lemma 2.1], the proof of the proposition is similar to the one given in [5, Proposition 3.4]. We omit it here. 2