## 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 SC*^{1} property (i.e., they
*are continuously differentiable and their gradients are semismooth) and LC*^{1} 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*^{T}*F (x) = 0,* (1.1)

*where F : <*^{n}*→ <*^{n}*is a continuously differentiable mapping with F := (F*_{1}*, F*_{2}*, . . . , F** _{n}*)

*. 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.*

^{T}*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=1**φ*^{2}*(x**i**, F**i**(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∈<}^{n}*Ψ(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) :=√*

*a*^{2}*+ b*^{2} *− 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 φ**θ* *given in (1.2) and φ**p* 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*_{θ}*and φ*_{p}*have SC*^{1}property (i.e., they are continuously differentiable and
*their gradients are semismooth) and LC*^{1} 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

X*n*
*i=1*

*φ*^{2}_{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** _{θp}*defined 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*

*defined by (1.4) is an NCP-function, it follows that the function Ψ*

_{θp}*defined by (1.5) is a merit function*

_{θp}*associated to the NCP-function φ*

*θp*. We also show that the merit function Ψ

*θp*

*has SC*

^{1}

*property and LC*

^{1}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)}_{∂x}_{i}*denotes to the i-th component of the gradient*
*of F ) and ∇*^{2}*F 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 φ**θp* defined by (1.4) is an NCP-function, and
*discuss its properties which are similar to those obtained in [3, 5] for the function φ**p* 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 η**θp* *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(khk*^{2})
*holds for any V ∈ ∂F (x + h).*

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

*(i) φ*_{θp}*is 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) φ**θp* *is positive homogenous, i.e., φ**θp**(α(a, b)) = αφ**θp**(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) φ**θp* *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*

*η**θp**(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 η**θp**(a, b) 6= 0 by Proposition 2.2. By a*
direct calculation, we get

*∂φ**θp**(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)

*∂φ**θp**(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 φ**θp* 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)}, φ**θp*

*is differentiable at (h, k), and hence, ∇φ*_{θp}*(h, k) =*

³*∂φ**θp**(h,k)*

*∂a* *,*^{∂φ}^{θp}_{∂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−1}*h + sgn(k)|k|*^{p−1}*k + 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}*η**θp**(h, k)*^{p−1}

= *η*_{θp}*(h, k)*^{p}*− (|h|*^{p}*+ |k|*^{p}*+ |h − k|** ^{p}*)

*η*

*θp*

*(h, k)*

^{p−1}= 0

*= O(k(h, k)k*^{2}*).*

*Thus, we obtain that φ**θp* is strongly semismooth.

We complete the proof. *2*

*Proposition 2.4 Let φ**θp* *be defined by (1.4) and {(a*^{k}*, b*^{k}*)} ⊆ <*^{2}*. Then, |φ**θp**(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,*

p*p*

*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 φ**θp*, 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,*

p*p*

*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

*|φ**θp**(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

2*φ*^{2}_{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)* ^{∂ψ}^{θp}_{∂a}^{(a,b)}*·* ^{∂ψ}^{θp}_{∂b}^{(a,b)}*≥ 0 for all (a, b) ∈ <*^{2}*, where the equality holds if and only if*
*φ*_{θp}*(a, b) = 0;*

*(vi)* ^{∂ψ}^{θp}_{∂a}^{(a,b)}*= 0 ⇐⇒* ^{∂ψ}^{θp}_{∂b}^{(a,b)}*= 0 ⇐⇒ φ**θp**(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

2*φ*^{2}_{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 ψ**θp**is differentiable at (0, 0) with*

*∇ψ**θp**(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*^{1}^{p}*|θ*^{1}^{p}*a|*^{p−1}*+ (1 − θ)*^{1}^{p}*|(1 − θ)*^{1}^{p}*(a − b)|*^{p−1}*η*_{θp}*(a, b)*^{p−1}

*≤* *((θ*^{1}* ^{p}*)

^{p}*+ ((1 − θ)*

^{1}

*)*

^{p}*)*

^{p}^{1}

^{p}*((|θ*

^{1}

^{p}*a|*

*)*

^{p−1}

^{p−1}

^{p}*+ (|(1 − θ)*

^{1}

^{p}*(a − b)|*

*)*

^{p−1}

^{p−1}*)*

^{p}

^{p−1}

^{p}*η*

*θp*

*(a, b)*

^{p−1}= *(θ + (1 − θ))(x*^{p}*+ z** ^{p}*)

^{p−1}

^{p}*η*

*θp*

*(a, b)*

^{p−1}= *(x*^{p}*+ z** ^{p}*)

^{p−1}

^{p}*(x*

^{p}*+ y*

^{p}*+ z*

*)*

^{p}

^{p−1}

^{p}= ( *x*^{p}*+ z*^{p}

*x*^{p}*+ y*^{p}*+ z** ^{p}*)

^{p−1}

^{p}*≤ 1,*

*where x := |θ*^{1}^{p}*a|*^{p}*, y := |θ*^{1}^{p}*b|*^{p}*, z := |(1 − θ)*^{1}^{p}*(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 η*

*θp*

*(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 ^{∂φ}^{θp}_{∂a}* ^{(a,b)}* and

^{∂φ}

^{θp}

_{∂b}*are uniformly*

^{(a,b)}*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 ψ**θp*

we obtain that the desired result holds.

Consider the result (v). It is obvious that ^{∂ψ}^{θp}_{∂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 ^{∂φ}^{θp}_{∂a}^{(a,b)}*≤ 0 and* ^{∂φ}^{θp}_{∂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 ^{∂ψ}^{θp}_{∂a}^{(a,b)}*·* ^{∂ψ}^{θp}_{∂b}* ^{(a,b)}* = 0.

*Then, it is sufficient to prove that φ*_{θp}*(a, b) = 0 when* ^{∂φ}^{θp}_{∂a}^{(a,b)}*·* ^{∂φ}^{θp}_{∂b}* ^{(a,b)}* = 0. Suppose that

*∂φ**θp**(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 φ**θp**(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 D*_{0} *⊆ D, then k∇f (y) − ∇f (x)k ≤ sup*_{0≤t≤1}*k∇*^{2}*f (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∇ψ**θp**(a, b) − ∇ψ**θp**(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 ψ**θp* and the proof of Proposition 2.5(iii) that

*∇ψ**θp**(a, b) = ∇φ**θp**(a, b)φ**θp**(a, b) when (a, b) 6= (0, 0), and ∇ψ**θp**(0, 0) = (0, 0)** ^{T}*. From Propo-

*sition 2.5(iii) we know that ψ*

*is continuous differentiable. The proof is divided into the following three cases.*

_{θp}*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∇φ**θp**(a, b)φ**θp**(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∇φ**θp**(a, b)k} is uniformly bounded on <*^{2}
*(which can be obtained from the proof of Proposition 2.5(iii)) and φ**θp*is 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

*ˆh*1 := *θsgn(a)|a|*^{p−1}*+ (1 − θ)sgn(a − b)|a − b|*^{p−1}*η*^{p−1}_{θp}*(a, b)* ;
*ˆh*_{2} := *θsgn(b)|b|*^{p−1}*− (1 − θ)sgn(a − b)|a − b|*^{p−1}

*η*_{θp}^{p−1}*(a, b)* ;
*ˆa*_{1} *:= (θ|a|*^{p−2}*+ (1 − θ)|a − b|*^{p−2}*)η*^{p}_{θp}*(a, b);*

*ˆa*_{2} *:= −ˆh*^{2}_{1}*η*_{θp}^{2p−2}*(a, b);*

*ˆb*_{1} *:= −(1 − θ)|a − b|*^{p−2}*η*_{θp}^{p}*(a, b);*

*ˆb*2 *:= −ˆh*_{1}*ˆh*2*η*^{2p−2}_{θp}*(a, b);*

*ˆc*1 *:= (θ|b|*^{p−2}*+ (1 − θ)|a − b|*^{p−2}*)η*_{θp}^{p}*(a, b);*

*ˆc*_{2} *:= −ˆh*^{2}_{2}*η*_{θp}^{2p−2}*(a, b).*

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

*∂*^{2}*ψ*_{θp}*(a, b)*

*∂a*^{2} *= (ˆh*_{1}*− 1)*^{2}*+ (p − 1)* *ˆa*_{1}*+ ˆa*_{2}

*η*_{θp}^{2p−1}*(a, b)(η*_{θp}*(a, b) − (a + b));*

*∂*^{2}*ψ*_{θp}*(a, b)*

*∂a∂b* *= (ˆh*_{1}*− 1)(ˆh*_{2}*− 1) + (p − 1)* *ˆb*_{1}*+ ˆb*_{2}

*η*^{2p−1}_{θp}*(a, b)(η*_{θp}*(a, b) − (a + b));*

*∂*^{2}*ψ*_{θp}*(a, b)*

*∂b*^{2} *= (ˆh*_{2}*− 1)*^{2}*+ (p − 1)* *ˆc*_{1}*+ ˆc*_{2}

*η*_{θp}^{2p−1}*(a, b)(η*_{θp}*(a, b) − (a + b));*

*∂*^{2}*ψ**θp**(a, b)*

*∂b∂a* = *∂*^{2}*ψ**θp**(a, b)*

*∂a∂b* *,*

where the last equality follows from the fact that ^{∂}^{2}^{ψ}_{∂a∂b}^{θp}* ^{(a,b)}* and

^{∂}^{2}

^{ψ}

_{∂b∂a}

^{θp}*are continuous when*

^{(a,b)}*(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.*

*ˆa*_{1}

*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*^{2}^{p}*+ (1 − θ)*^{2}^{p}

*≤ 2.*

Similarly, we have

*|ˆb*_{1}*|*

*η*^{2p−2}_{θp}*(a, b)* *≤ 1;* *ˆc*_{1}

*η*_{θp}^{2p−2}*(a, b)* *≤ 2.*

*These, together with the results |ˆh*_{1}*| ≤ 1 and |ˆh*_{2}*| ≤ 1 given in Proposition 2.5, yield*

*|ˆa*_{2}*|*

*η*_{θp}^{2p−2}*(a, b)* *≤ 1;* *|ˆb*_{2}*|*

*η*_{θp}^{2p−2}*(a, b)* *≤ 1;* *|ˆc*_{2}*|*

*η*_{θp}^{2p−2}*(a, b)* *≤ 1.*

Thus,

¯¯

¯¯*∂*^{2}*ψ*_{θp}*(a, b)*

*∂a*^{2}

¯¯

¯¯ =

¯¯

¯¯

¯*(ˆh*_{1}*− 1)*^{2} *+ (p − 1)* *ˆa*_{1}*+ ˆa*_{2}

*η*^{2p−1}_{θp}*(a, b)(η*_{θp}*(a, b) − (a + b))*

¯¯

¯¯

¯

*≤ |(ˆh*_{1}*− 1)*^{2}*| + (p − 1)*
Ã¯¯

¯¯

¯

*ˆa*1*+ ˆa*2

*η*_{θp}^{2p−1}*(a, b)η*_{θp}*(a, b)*

¯¯

¯¯

¯+

¯¯

¯¯

¯

*ˆa*1*+ ˆa*2

*η*^{2p−1}_{θp}*(a, b)(a + b)*

¯¯

¯¯

¯

!

*≤ 4 + (1 + 2κ*^{∗}*)(p − 1)*

Ã *ˆa*_{1}

*η*_{θp}^{2p−2}*(a, b)* + *|ˆa*_{2}*|*
*η*^{2p−2}_{θp}*(a, b)*

!

*≤ 4 + 3(1 + 2κ*^{∗}*)(p − 1);*

¯¯

¯¯*∂*^{2}*ψ*_{θp}*(a, b)*

*∂a∂b*

¯¯

¯¯ =

¯¯

¯¯

¯*(ˆh*_{1}*− 1)(ˆh*_{2} *− 1) + (p − 1)* *ˆa*_{1}*+ ˆa*_{2}

*η*_{θp}^{2p−1}*(a, b)(η*_{θp}*(a, b) − (a + b))*

¯¯

¯¯

¯

*≤ |(ˆh*_{1}*− 1)(ˆh*_{2} *− 1)|*

*+(p − 1)*
Ã¯¯

¯¯

¯

*ˆb*1 *+ ˆb*2

*η*_{θp}^{2p−1}*(a, b)η**θp**(a, b)*

¯¯

¯¯

¯+

¯¯

¯¯

¯

*ˆb*1*+ ˆb*2

*η*_{θp}^{2p−1}*(a, b)(a + b)*

¯¯

¯¯

¯

!

*≤ 4 + (1 + 2κ*^{∗}*)(p − 1)*

Ã *|ˆb*_{1}*|*

*η*_{θp}^{2p−2}*(a, b)* + *|ˆb*_{2}*|*
*η*^{2p−2}_{θp}*(a, b)*

!

*≤ 4 + 2(1 + 2κ*^{∗}*)(p − 1);*

¯¯

¯¯*∂*^{2}*ψ*_{θp}*(a, b)*

*∂b*^{2}

¯¯

¯¯ =

¯¯

¯¯

¯*(ˆh*_{2}*− 1)*^{2} *+ (p − 1)* *ˆc*_{1}*+ ˆc*_{2}

*η*^{2p−1}_{θp}*(a, b)(η*_{θp}*(a, b) − (a + b))*

¯¯

¯¯

¯

*≤ |(ˆh*_{2}*− 1)*^{2}*| + (p − 1)*
Ã¯¯

¯¯

¯

*ˆc*_{1}*+ ˆc*_{2}

*η*_{θp}^{2p−1}*(a, b)η*_{θp}*(a, b)*

¯¯

¯¯

¯+

¯¯

¯¯

¯

*ˆc*_{1}*+ ˆc*_{2}

*η*^{2p−1}_{θp}*(a, b)(a + b)*

¯¯

¯¯

¯

!

*≤ 4 + (1 + 2κ*^{∗}*)(p − 1)*
Ã

*ˆc*1

*η*_{θp}^{2p−2}*(a, b)* + *|ˆc*2*|*
*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 ∇ψ**θp* *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∇φ**1p**(a, b)φ**1p**(a, b) − ∇φ**1p**(c, d)φ**1p**(c, d)k*

*≥*

¯¯

¯¯*sgn(a)|a|*^{p−1}*k(a, b)k*^{p−1}*p*

*φ*_{1p}*(a, b) −* *sgn(c)|c|*^{p−1}*k(c, d)k*^{p−1}*p*

*φ*_{1p}*(c, d) + φ*_{1p}*(c, d) − φ*_{1p}*(a, b)*

¯¯

¯¯

*≥*

¯¯

¯¯*sgn(a)|a|*^{p−1}*k(a, b)k*^{p−1}*p*

*φ*_{1p}*(a, b) −* *sgn(c)|c|*^{p−1}*k(c, d)k*^{p−1}*p*

*φ*_{1p}*(c, d)*

¯¯

¯*¯ − |φ*^{1p}*(c, d) − φ*_{1p}*(a, b)|*

*≥*

¯¯

¯¯*sgn(a)|a|*^{p−1}*k(a, b)k*^{p−1}*p*

*φ*_{1p}*(a, b) −* *sgn(c)|c|*^{p−1}*k(c, d)k*^{p−1}*p*

*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−1}*p*

*φ**1p**(a, b) −* *sgn(c)|c|*^{p−1}*k(c, d)k*^{p−1}*p*

*φ**1p**(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−p}*n*^{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 ∇ψ**1p* *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}*(x*_{1}*, F*_{1}*(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

2*kΦ*_{θp}*(x)k*^{2} =
X*n*

*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 SC*^{1}
*function. Hence, if every F*_{i}*is an SC*^{1} *function, then the function Ψ*_{θp}*defined by (3.2)*
*with p ≥ 2, θ ∈ (0, 1] is also an SC*^{1} *function.*

*(ii) If every F*_{i}*is an LC*^{1} *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 LC*^{1} *function. Hence, if*
*every F*_{i}*is an LC*^{1}*function, then the function Ψ*_{θp}*defined by (3.2) with p ≥ 2, θ ∈ (0, 1]*

*is also an LC*^{1} *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*

*(h*

_{1}

*, h*

_{2}

*) ∈ <*

^{2}

*\{(0, 0)}, we know that ∇ψ*

_{θp}*is differentiable at (h*

_{1}

*, h*

_{2}), and hence, we only need to show that

*∇ψ*_{θp}*(h*_{1}*, h*_{2}*) − ∇ψ*_{θp}*(0, 0) − ∇*^{2}*ψ*_{θp}*(h*_{1}*, h*_{2}*) · (h*_{1}*, h*_{2})^{T}*= o(k(h*_{1}*, h*_{2}*)k).* (3.3)
*In fact, let ˆa*_{1}*, ˆa*_{2}*, ˆb*_{1}*, ˆb*_{2}*, ˆc*_{1}*, ˆc*_{2} *be similarly defined as those in Theorem 2.1 with (a, b) being*
*replaced by (h*_{1}*, h*_{2}). Denote

*ˆh*3 *:= (p − 1)* *ˆa*_{1}*+ ˆa*_{2}

*η*_{θp}^{2p−1}*(h*_{1}*, h*_{2})*φ**θp**(h*1*, h*2);

*ˆh*4 *:= (p − 1)* *ˆb*1*+ ˆb*_{2}

*η*_{θp}^{2p−1}*(h*1*, h*2)*φ*_{θp}*(h*_{1}*, h*_{2});

*ˆh*5 *:= (p − 1)* *ˆc*_{1}*+ ˆc*_{2}

*η*_{θp}^{2p−1}*(h*_{1}*, h*_{2})*φ**θp**(h*1*, h*2*),*
and

*m*_{1} *: = (θ|h*_{1}*|*^{p−2}*+ (1 − θ)|h*_{1}*− h*_{2}*|*^{p−2}*)η*^{p}_{θp}*(h*_{1}*, h*_{2}*)h*_{1}*− ˆh*^{2}_{1}*η*_{θp}^{2p−2}*(h*_{1}*, h*_{2}*)h*_{1};

*m*2 *: = (1 − θ)|h*1*− h*2*|*^{p−2}*η*_{θp}^{p}*(h*1*, h*2*)h*2*+ ˆh*1*ˆh*2*η*_{θp}^{2p−2}*(h*1*, h*2*)h*2;
*m*_{3} *: = (θ|h*_{1}*|*^{p−2}*+ (1 − θ)|h*_{1}*− h*_{2}*|*^{p−2}*)η*^{p}_{θp}*(h*_{1}*, h*_{2}*)h*_{1}

*−(1 − θ)|h*_{1}*− h*_{2}*|*^{p−2}*η*_{θp}^{p}*(h*_{1}*, h*_{2}*)h*_{2};
*m*_{4} *: = ˆh*_{1}*ˆh*_{2}*η*_{θp}^{2p−2}*(h*_{1}*, h*_{2}*)h*_{2}*+ ˆh*^{2}_{1}*η*_{θp}^{2p−2}*(h*_{1}*, h*_{2}*)h*_{1};

*m*_{5} *: = (θsgn(h*_{1}*)|h*_{1}*|*^{p−1}*+ (1 − θ)sgn(h*_{1}*− h*_{2}*)|h*_{1}*− h*_{2}*|*^{p−1}*)η*_{θp}^{p}*(h*_{1}*, h*_{2});

*m*6 *: = ˆh*1*ˆh*2*η*_{θp}^{2p−2}*(h*1*, h*2*)h*2*+ ˆh*^{2}_{1}*η*_{θp}^{2p−2}*(h*1*, h*2*)h*1*.*
Then,

µ *H*_{1}
*H*2

¶ :=

Ã *ˆh*_{1}*− 1*
*ˆh*_{2}*− 1*

!

*· φ*_{θp}*(h*_{1}*, h*_{2}*) −*
µ 0

0

¶

*−*
Ã

*(ˆh*1*− 1)*^{2}*+ ˆh*3 *(ˆh*1*− 1)(ˆh*2*− 1) + ˆh*4

*(ˆh*_{1}*− 1)(ˆh*_{2}*− 1) + ˆh*_{4} *(ˆh*_{2}*− 1)*^{2}*+ ˆh*_{5}

!

*·*
µ *h*1

*h*_{2}

¶
*.*

and hence,

*H*_{1} *= (ˆh*_{1}*− 1)φ*_{θp}*(h*_{1}*, h*_{2}*) − ((ˆh*_{1}*− 1)*^{2}*+ ˆh*_{3}*)h*_{1}*− ((ˆh*_{1}*− 1)(ˆh*_{2}*− 1) + ˆh*_{4}*)h*_{2}

*= (ˆh*_{1}*− 1)φ*_{θp}*(h*_{1}*, h*_{2}*) − ˆh*_{3}*h*_{1}*− ˆh*_{4}*h*_{2}*− (ˆh*_{1}*− 1)((ˆh*_{1}*− 1)h*_{1}*+ (ˆh*_{2}*− 1)h*_{2})

*= (ˆh*1*− 1)φ**θp**(h*1*, h*2*) − ˆh*3*h*1*− ˆh*4*h*2*− (ˆh*1*− 1)φ**θp**(h*1*, h*2)

*= −(p − 1)*
Ã

*ˆa*_{1}*+ ˆa*_{2}

*η*_{θp}^{2p−1}*(h*_{1}*, h*_{2})*h*_{1}+ *ˆb*_{1}*+ ˆb*_{2}
*η*_{θp}^{2p−1}*(h*_{1}*, h*_{2})*h*_{2}

!

*φ*_{θp}*(h*_{1}*, h*_{2})

*= −(p − 1)φ*_{θp}*(h*_{1}*, h*_{2})
Ã

*m*1 *− m*2

*η*^{2p−1}_{θp}*(h*_{1}*, h*_{2})

!

*= −(p − 1)φ**θp**(h*1*, h*2)

Ã *m*_{3} *− m*_{4}
*η*^{2p−1}_{θp}*(h*_{1}*, h*_{2})

!

*= −(p − 1)φ*_{θp}*(h*_{1}*, h*_{2})

Ã *m*_{5} *− m*_{6}
*η*^{2p−1}_{θp}*(h*1*, h*2)

!

*= −(p − 1)φ*_{θp}*(h*_{1}*, h*_{2})
Ã

*ˆh*_{1}*− ˆh*_{1}*ˆh*_{1}*h*_{1}*+ ˆh*_{2}*h*_{2}
*η*_{θp}*(h*_{1}*, h*_{2})

!

*= −(p − 1)φ*_{θp}*(h*_{1}*, h*_{2}*)(ˆh*_{1}*− ˆh*_{1})

*= 0,*

*where the third equality follows from ˆh*_{1}*h*_{1}*+ ˆh*_{2}*h*_{2} *= η** _{θp}* given in the proof of Proposition

*2.3 and the definition of φ*

_{θp}*, the fourth equality follows from the definitions of ˆh*

_{3}

*, ˆh*

_{4}, the

*fifth equality follows from the definitions of ˆa*

_{1}

*, ˆa*

_{2}

*, ˆb*

_{1}

*, ˆb*

_{2}, and the eighth equality follows from

*ˆh*1

*h*1

*+ ˆh*2

*h*2

*= η*

*θp*given in the proof of Proposition 2.3.

*Similar analysis yields H*2 *= 0. Thus, ∇ψ**θp* *is semismooth. Furthermore, ψ**θp* *is an SC*^{1}
function.

*(ii) Since the LC*^{1} 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 − yk*^{2} *for all x, y ∈ <*^{n}*.*

*• F is said to be a P*_{0}*-function if max*_{1≤j≤n,x}_{i}_{6=y}_{i}*(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 max**1≤j≤n**(x**i**− y**i**)(F**i**(x) −*
*F*_{i}*(y)) ≥ µkx − yk*^{2} *for all x, y ∈ <*^{n}*.*

*Proposition 3.2 Let Ψ**θp* *: <*^{n}*→ < be defined by (3.2) with p > 1, θ ∈ (0, 1]. Then*
Ψ*θp**(x) ≥ 0 for all x ∈ <*^{n}*and Ψ**θp**(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 Ψ**θp**: <*^{n}*→ < be defined by (3.2) with p > 1, θ ∈ (0, 1]. Suppose that*
*F is either a monotone function or a P*0*-function, then every stationary point of Ψ**θp* *is a*
*global minima of min*_{x∈<}* ^{n}*Ψ

_{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*