Journal of Computational and Applied Mathematics, vol. 232, pp. 455-471, 2009

**A R-linearly convergent derivative-free algorithm for the NCPs** **based on the generalized Fischer-Burmeister merit function**

**A R-linearly convergent derivative-free algorithm for the NCPs**

Jein-Shan Chen ^{1}
Department of Mathematics
National Taiwan Normal University

Taipei, Taiwan 11677 E-mail: jschen@math.ntnu.edu.tw

Hung-Ta Gao

Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677 E-mail: kleinmankao@gmail.com

Shaohua Pan

School of Mathematical Sciences South China University of Technology

Guangzhou 510640, China E-mail: shhpan@scut.edu.cn

July 10, 2008

**Abstract. In the paper [4], the authors proposed a derivative-free descent algorithm for**
the nonlinear complementarity problems (NCPs) by the generalized Fischer-Burmeister
*merit function: ψ*_{p}*(a, b) =* ^{1}_{2}[*∥(a, b)∥**p* *− (a + b)]*^{2}, and observed that the choice of the
*parameter p has a great inﬂuence on the numerical performance of the algorithm. In this*
paper, we analyze the phenomenon theoretically for a derivative-free descent algorithm
*which is based on a penalized form of ψ** _{p}* and uses a diﬀerent direction from [4]. More
speciﬁcally, we show that the algorithm proposed is globally convergent and has a locally

*R-linear convergence rate, and furthermore, its convergence rate will become worse when*

*the parameter p decreases. Numerical results are also reported for the test problems from*MCPLIB, which further verify the obtained theoretical results.

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce. The author’s work is partially supported by National Science Council of Taiwan.

**Key Words. Nonlinear complementarity problem, NCP-function, merit function, global**
error bound, convergence rate.

**1** **Introduction**

*The nonlinear complementarity problem (NCP) is to ﬁnd a point x∈ IR** ^{n}* such that

*x≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩ = 0,* (1)

where *⟨·, ·⟩ is the Euclidean inner product and F = (F*1*, . . . , F** _{n}*)

*is a map from IR*

^{T}*to IR*

^{n}

^{n}*. We assume that F is continuously diﬀerentiable throughout this paper. The NCP*has attracted much attention because of its wide applications in the ﬁelds of economics, engineering, and operations research [5, 12], to name a few.

Many methods have been proposed to solve the NCP; see [9, 12, 18] and the references therein. One of the most powerful and popular methods is to reformulate the NCP as a system of nonlinear equations [16, 19, 24], or an unconstrained minimization problem [6, 7, 8, 10, 13, 14, 20, 23]. The objective function that can constitute an equivalent unconstrained minimization problem is called a merit function, whose global minima are coincident with the solutions of the original NCP. To construct a merit function, a class of functions, called NCP-functions and deﬁned below, plays a signiﬁcant role.

**Definition 1.1 A function ϕ : IR**^{2} *→ IR is called an NCP-function if it satisﬁes*

*ϕ(a, b) = 0* *⇐⇒ a ≥ 0, b ≥ 0, ab = 0.* (2)

The Fischer-Burmeister (FB) function is a well-known NCP-function deﬁned as
*ϕ*_{FB}*(a, b) =√*

*a*^{2}*+ b*^{2}*− (a + b),* (3)

by which the NCP can be reformulated as a system of nonsmooth equations:

Φ_{FB}*(x) =*

*ϕ*_{FB}*(x*_{1} *, F*_{1}*(x))*

*·*

*·*

*·*

*ϕ*_{FB}*(x*_{n}*, F*_{n}*(x))*

*= 0.* (4)

Thus, the function Ψ_{FB} : IR^{n}*→ IR*+ deﬁned as below is a merit function for the NCP:

Ψ_{FB}*(x) :=* 1

2*∥Φ*FB*(x)∥*^{2} =

∑*n*
*i=1*

*ψ*_{FB}*(x*_{i}*, F*_{i}*(x)),* (5)

*where ψ*_{FB} : IR^{2} *→ IR*+ *is the square of ϕ*_{FB}, i.e.,
*ψ*_{FB}*(a, b) =* 1

2

*√*

*a*^{2}*+ b*^{2}*− (a + b)*^{}^{2}*.* (6)
Consequently, the NCP is equivalent to an unconstrained minimization problem:

*x*min*∈IR** ^{n}*Ψ

_{FB}

*(x).*(7)

Recently, an extension of FB function was considered in [2, 3, 4] by the authors. More
*speciﬁcally, we deﬁne the generalized FB function ϕ** _{p}* : IR

^{2}

*→ IR by*

*ϕ*_{p}*(a, b) :=∥(a, b)∥**p**− (a + b),* (8)
*where p > 1 is an arbitrary ﬁxed real number and* *∥(a, b)∥**p* *denotes the p-norm of (a, b),*
i.e., *∥(a, b)∥**p* = ^{p}

√*|a|** ^{p}*+

*|b|*

^{p}*. In other words, in the function ϕ*

*, we replace the 2-norm*

_{p}*of (a, b) in the FB function by a more general p-norm. The function ϕ*

*p*is still an NCP-

*function, which naturally induces another NCP-function ψ*

*: IR*

_{p}^{2}

*→ IR*+ given by

*ψ*_{p}*(a, b) :=* 1

2*|ϕ**p**(a, b)|*^{2}*.* (9)

*For any given p > 1, the function ψ**p*is shown to possess all favorable properties of the FB
*function ψ*_{FB}*; see [2, 3, 4]. For example, ψ** _{p}* is also continuously diﬀerentiable everywhere
on IR

^{2}

*. Like ϕ*

_{FB}, the operator Φ

*: IR*

_{p}

^{n}*→ IR*

*deﬁned as*

^{n}Φ_{p}*(x) =*

*ϕ*_{p}*(x*_{1} *, F*_{1}*(x))*

*·*

*··*
*ϕ*_{p}*(x*_{n}*, F*_{n}*(x))*

(10)

yields a family of merit functions Ψ* _{p}* : IR

^{n}*→ IR*+ for the NCP Ψ

_{p}*(x) :=*1

2*∥Φ**p**(x)∥*^{2} =

∑*n*
*i=1*

*ψ*_{p}*(x*_{i}*, F*_{i}*(x)).* (11)

In this paper, we study the following merit function Ψ* _{α,p}*: IR

^{n}*→ IR for the NCP:*

Ψ_{α,p}*(x) :=*

∑*n*
*i=1*

*ψ*_{α,p}*(x*_{i}*, F*_{i}*(x)),* (12)

*where ψ**α,p* : IR^{2} *→ IR*+ is an NCP-function deﬁned by
*ψ*_{α,p}*(a, b) :=* *α*

2(max*{0, ab})*^{2}*+ ψ*_{p}*(a, b) =* *α*

2*(ab)*^{2}_{+}+ 1

2(*∥(a, b)∥**p**− (a + b))*^{2} (13)

*with α* *≥ 0 being a real parameter. When α = 0, the function ψ**α,p* *reduces to ψ** _{p}*. Hence,

*ψ*

_{α,p}*is an extension of ψ*

_{p}*. Besides, ψ*

_{α,p}*also extends the function ψ*

*studied in [25] by*

_{α}*Yamada, Yamashita, and Fukushima which corresponds to p = 2. Indeed, ψ*

*was ever*

_{α,p}*studied in [3] by one of the authors (see ψ*4 therein), but there was no investigation on

*property of error bound. In this paper, we present more favorable properties of ψ*

*, and particularly, the conditions under which Ψ*

_{α,p}*provides a global error bound for the NCP.*

_{α,p}*With these results, we propose a derivative-free descent algorithm based on ϕ** _{α,p}* and

*establish its global convergence and local R-linear convergence rate. Moreover, we also*

*analyze the inﬂuence of p on the convergence rate of the proposed algorithm theoretically*and obtain the conclusion that the convergence rate of the algorithm will become worse

*when the value of p decreases. Thus, this paper can be viewed as a follow-up of [3] and [4].*

This paper is organized as follows. In Section 2, we review some deﬁnitions and prelim-
inary results to be used in the subsequent analysis. In Section 3, we show some important
properties of the proposed merit function. In Section 4, we propose a derivative-free al-
gorithm associated with Ψ_{α,p}*, prove its global convergence and the R-linear convergence*
*rate, and analyze the inﬂuence of p on the convergence rate. Some numerical experiments*
are reported in Section 5, and we make concluding remarks in Section 6.

Throughout this paper, IR^{n}*denotes the space of n-dimensional real column vectors*
and ^{T}*denotes transpose. For every diﬀerentiable function f : IR*^{n}*→ IR, ∇f(x) denotes*
*the gradient of f at x. For every diﬀerentiable mapping F = (F*_{1}*, . . . , F** _{n}*)

*: IR*

^{T}

^{n}*→ IR*

*,*

^{n}*∇F (x) = (∇F*1*(x) . . .* *∇F**n**(x)) denotes the transpose Jacobian of F at x. We denote*
by*∥x∥**p* *the p-norm of x and by∥x∥ the Euclidean norm of x. The level set of a function*
Ψ : IR^{n}*→ IR is denoted by L(Ψ, c) := {x ∈ IR*^{n}*| Ψ(x) ≤ c}. In addition, we also use the*
natural residual merit function Ψ_{NR} : IR^{n}*→ IR*+ deﬁned by

Ψ_{NR}*(x) :=* 1
2

∑*n*
*i=1*

*ϕ*^{2}

NR*(x*_{i}*, F*_{i}*(x)),* (14)

*where ϕ*_{NR} : IR^{2} *→ IR denotes the minimum NCP-function min{a, b}. Unless otherwise*
*stated, in the sequel, we always suppose that p is a ﬁxed real number in (1,∞).*

**2** **Preliminaries**

*This section mainly recalls some concepts about the mapping F that will be used later.*

**Definition 2.1 Let F = (F**_{1}*, . . . , F** _{n}*)

^{T}*with F*

*: IR*

_{i}

^{n}*→ IR for i = 1, . . . , n. We say that*

**(a) F is monotone if***⟨x − y, F (x) − F (y)⟩ ≥ 0 for all x, y ∈ IR*

^{n}*.*

**(b) F is strongly monotone with modulus µ > 0 if***⟨x − y, F (x) − F (y)⟩ ≥ µ∥x − y∥*^{2} *for*
*all x, y* *∈ IR*^{n}*.*

**(c) F is a P**_{0}*-function if max*

1*≤i≤n*
*xi̸=yi*

*(x*_{i}*− y**i**)(F*_{i}*(x)− F**i**(y))≥ 0 for all x, y ∈ IR*^{n}*and x̸= y.*

**(d) F is a uniform P -function with modulus µ > 0 if max**

1*≤i≤n**(x**i* *− y**i**)(F**i**(x)− F**i**(y))* *≥*
*µ∥x − y∥*^{2} *for all x, y∈ IR*^{n}*.*

**(e)** *∇F (x) is uniformly positive deﬁnite with modulus µ > 0 if d*^{T}*∇F (x)d ≥ µ∥d∥*^{2} *for*
*all x∈ IR*^{n}*and d∈ IR*^{n}*.*

**(f ) F is Lipschitz continuous if there exists a constant L > 0 such that**∥F (x) − F (y)∥ ≤*L∥x − y∥ for all x, y ∈ IR*^{n}*.*

*From Deﬁnition 2.1, it is easy to see that F is a uniform P -function with modulus*
*µ > 0 if F is strongly monotone with modulus µ > 0, and F is a P*_{0}*-function if F is*
*monotone. In addition, when F is continuously diﬀerentiable, the following results hold:*

*1. F is monotone if and only if* *∇F (x) is positive semideﬁnite for all x ∈ IR** ^{n}*.

*2. F is strongly monotone if and only if*

*∇F (x) is uniformly positive deﬁnite.*

**3** **Properties of the Merit Function**

*In this section, we study some favorable properties of the merit function ψ** _{α,p}*which will be
used in the subsequent analysis, and then present some mild conditions under which the
merit function Ψ

*α,p*has bounded level sets and provides a global error bound, respectively.

*The following lemma states that ψ*_{α,p}*enjoys many favorable properties as ψ** _{p}* holds.

*Furthermore, when α > 0, it has an important property that ψ** _{p}* does not have (see
Lemma 3.1(f)). Although most results of the lemma were investigated in [3, Prop. 3.3]

*where only p being integer was considered, we here provide more detailed arguments for*
*general case where p is any real number greater than one.*

**Lemma 3.1 The function ψ**_{α,p}*deﬁned by (13) has the following favorable properties:*

**(a) ψ**_{α,p}*is an NCP-function and ψ*_{α,p}*≥ 0 for all (a, b) ∈ IR*^{2}*.*

**(b) ψ**_{α,p}*is continuously diﬀerentiable everywhere, and moreover, if (a, b)* *̸= (0, 0),*

*∇**a**ψ*_{α,p}*(a, b) = αb(ab)*_{+}+

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b),*

*∇**b**ψ*_{α,p}*(a, b)* *= αa(ab)*_{+}+

(*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b);*

(15)

*and otherwise* *∇**a**ψ*_{α,p}*(0, 0) =∇**b**ψ*_{α,p}*(0, 0) = 0.*

**(c) For p**≥ 2, the gradient of ψ*α,p* *is Lipschitz continuous on any nonempty bounded set*
*S, i.e., there exists L > 0 such that for any (a, b), (c, d)∈ S,*

*∥∇ψ**α,p**(a, b)− ∇ψ**α,p**(c, d)∥ ≤ L∥(a, b) − (c, d)∥.*

**(d)** *∇**a**ψ*_{α,p}*(a, b)* *· ∇**b**ψ*_{α,p}*(a, b)* *≥ 0 for any (a, b) ∈ IR*^{2}*, and furthermore, the equality*
*holds if and only if ψ*_{α,p}*(a, b)=0.*

**(e)** *∇**a**ψ**α,p**(a, b) = 0⇐⇒ ∇**b**ψ**α,p**(a, b) = 0⇐⇒ ψ**α,p**(a, b) = 0.*

**(f ) Suppose that α > 0. If a**→ −∞ or b → −∞ or ab → ∞, then ψ*α,p**(a, b)→ ∞.*

**Proof. Parts (a), (b) and (f) directly follow from the deﬁnition of ψ*** _{α,p}* and Proposition
3.2 (a)–(c) and Lemma 3.1 of [4]. It remains to show parts (c)–(e).

*(c) Notice that the functions a(ab)*_{+} *and b(ab)*_{+}*for any a, b∈ IR are Lipschitz continuous*
*on any nonempty bounded set S, whereas ϕ*_{p}*(a, b) is Lipschitz continuous on IR*^{2} by [4,
Proposition 3.1 (e)]. Therefore, by the expression of *∇ψ**α,p**(a, b) and the boundedness of*

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

and

(*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*,*

it is not hard to verify that the gradient *∇ψ**α,p**(a, b) is Lipschitz continuous on S for*
*p≥ 2.*

*(d) If (a, b) = (0, 0), part (d) clearly holds. Now suppose that (a, b)̸= (0, 0). Then,*

*∇**a**ψ*_{α,p}*(a, b)· ∇**b**ψ*_{α,p}*(a, b) =*

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

) (*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*^{2}_{p}*(a, b)*
*+α*^{2}*ab(ab)*_{+}^{2}*+ αa(ab)*_{+}

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b)*
*+αb(ab)*_{+}

(*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b).* (16)

Since

*ab(ab)*_{+}^{2} *≥ 0,* *sgn(a)· |a|*^{p−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1 ≤ 0, and* *sgn(b)· |b|*^{p−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1 ≤ 0,* (17)
it suﬃces to show that the last two terms of (16) are nonnegative. We next claim that

*αa(ab)*_{+}

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b)≥ 0, ∀ (a, b) ̸= (0, 0).* (18)

*If a≤ 0, then ϕ**p**(a, b)≥ 0, which together with the second inequality in (17) implies that*
*(18) holds. If a > 0 and b > 0, then ϕ*_{p}*(a, b) < 0, which implies (18) by a similar reason.*

*If a > 0 and b≤ 0, then (ab)*+ = 0, and hence (18) holds. Similarly, we have that
*αb(ab)*_{+}

(*sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b)≥ 0, ∀ (a, b) ̸= (0, 0).*

Consequently, *∇**a**ψ*_{α,p}*(a, b)· ∇**b**ψ*_{α,p}*(a, b)* *≥ 0. From (16), ∇**a**ψ*_{α,p}*(a, b)· ∇**b**ψ*_{α,p}*(a, b)=0 if*
and only if *{a = 0 or (a ≥ 0 and b = 0) or ϕ**p**(a, b)=0} and {b = 0 or (b ≥ 0 and a=0) or*
*ϕ*_{p}*(a, b) = 0} and {ab=0}. Thus, ∇**a**ψ*_{α}*(a, b)·∇**b**ψ*_{α,p}*(a, b) = 0 if and only if ψ*_{α,p}*(a, b) = 0.*

*(e) If ψ*_{α,p}*(a, b) = 0, then ab = 0 and ϕ*_{p}*(a, b) = 0 by part (a), which in turn implies*
that *∇**a**ψ**α,p**(a, b) = 0 and* *∇**b**ψ**α,p**(a, b) = 0. Next, we claim that∇**a**ψ**α,p**(a, b) = 0 implies*
*ψ*_{α,p}*(a, b) = 0. Suppose that* *∇**a**ψ*_{α,p}*(a, b) = 0. Then,*

*αb(ab)*_{+}=*−*

(*sgn(a)· |a|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 1*

)

*ϕ*_{p}*(a, b).* (19)

*We can verify that the equality (19) implies b = 0, a≥ 0 or b > 0, a = 0. Under the two*
*cases, we both have ψ*_{α,p}*(a, b) = 0. Similarly,* *∇**b**ψ*_{α,p}*(a, b) = 0 also implies ψ*_{α,p}*(a, b) = 0.*

*2*

*Notice that ab* *→ ∞ does not necessarily imply ψ**p**(a, b)* *→ ∞ which means ψ**p* does
*not share Lemma 3.1(f). In fact, for α = 0, the lemma needs to be modiﬁed as “if*
*(a* *→ ∞) or (b → ∞) or (a → ∞ and b → ∞), then ψ**α,p**(a, b)* *→ ∞”. As we will see*
later, Lemma 3.1(f) is useful for proving that the level sets of Ψ* _{α,p}* are bounded. Besides,
by Lemma 3.1(a), we immediately have the following theorem.

**Theorem 3.1 Let Ψ**_{α,p}*be deﬁned as in (12). Then Ψ*_{α,p}*(x)* *≥ 0 for all x ∈ IR*^{n}*and*
Ψ*α,p**(x) = 0 if and only if x solves the NCP. Moreover, if the NCP has at least one*
*solution, then x is a global minimizer of Ψ*_{α,p}*if and only if x solves the NCP.*

Theorem 3.1 indicates that the NCP can be recast as the unconstrained minimization:

*x*min*∈IR** ^{n}*Ψ

_{α,p}*(x).*(20)

In general, it is hard to ﬁnd a global minimum of Ψ* _{α,p}*. Therefore, it is important to know
under what conditions a stationary point of Ψ

*is a global minimum. Using Lemma 3.1(d) and the same proof techniques as in [10, Theorem 3.5], we can estabish that each stationary point of Ψ*

_{α,p}

_{α,p}*is a global minimum only if F is a P*

_{0}-function.

**Theorem 3.2 Let F be a P**_{0}*-function. Then x*^{∗}*∈ IR*^{n}*is a global minimum of the*
*unconstrained optimization problem (20) if and only if x*^{∗}*is a stationary point of Ψ*_{α,p}*.*

From the following theorem, we see that the unconstrained minimization problem
*(20) has a stationary point under rather weak conditions of the mapping F . Since similar*
results and analogous analysis can be found in [3, Proposition 4.1], [10, Theorem 3.8] and
[15, Theorem 4.1], we here omit the proof.

**Theorem 3.3 The function Ψ**_{α,p}*has bounded level sets* *L(Ψ**α,p**, c) for all c∈ IR, if F is*
*monotone and the NCP is strictly feasible (i.e., there exists ˆx > 0 such that F (ˆx) > 0)*
*when α > 0, or F is a uniform P -function when α≥ 0.*

In what follows, we will show that the merit functions Ψ* _{p}*, Ψ

_{NR}and Ψ

*have the same order on every bounded set. For this purpose, we need the following crucial technical*

_{α,p}*lemma, which generalizes the important property of ϕ*

_{FB}proved by Tseng in [22].

**Lemma 3.2 Let ϕ*** _{p}* : IR

^{2}

*→ IR be deﬁned as in (8). Then for any p > 1 we have*

(2*− 2*^{1}* ^{p}*)

*| min{a, b}| ≤ |ϕ*

*p*

*(a, b)| ≤ (2 + 2*

^{1}

*)*

^{p}*| min{a, b}|.*(21)

**Proof. Without loss of generality, suppose a***≥ b. We will prove the desired results by*

*considering the following two cases: (1) a + b≤ 0 and (2) a + b > 0.*

*Case(1): a + b≤ 0. In this case, we have*

*|ϕ**p**(a, b)| ≥ ∥(a, b)∥**p* *≥ |b| = | min{a, b}| ≥ (2 − 2*^{p}^{1})*| min{a, b}|.* (22)
*On the other hand, since a≥ b and a + b ≤ 0, we have |b| ≥ |a|. Then*

*|ϕ**p**(a, b)| ≤ ∥(a, b)∥**p**− 2b = (2 + 2*^{p}^{1})*|b| = (2 + 2*^{p}^{1})*| min{a, b}|.* (23)
*Case(2): a + b > 0. If ab=0, then (21) clearly holds. Thus, we discuss by two subcases:*

*(i) ab < 0. In this subcase, we have a > 0, b < 0, and* *|a| > |b|. Consequently,*

*ϕ*_{p}*(a, b)≤ |a| + |b| − (a + b) = −2b = 2| min{a, b}| ≤ (2 + 2*^{1}* ^{p}*)

*| min{a, b}|,*(24) and

*ϕ*_{p}*(a, b)≥ ∥(a, b)∥**∞**− (a + b) = −b = | min{a, b}| ≥ (2 − 2*^{1}* ^{p}*)

*| min{a, b}|.*(25)

*(ii) ab > 0. Now we have a≥ b > 0. Since for any p > 1 there holds that*

0*≥ ϕ**p**(a, b)* *≥ ∥(a, b)∥*_{∞}*− (a + b) = a − (a + b) = −b = − min{a, b},*
we immediately obtain that

*|ϕ**p**(a, b)| ≤ | min{a, b}| ≤ (2 + 2*^{1}* ^{p}*)

*| min{a, b}|.*(26)

*On the other hand, since ϕ*_{p}*(a, b)≤ 0, it follows that*

*|ϕ**p**(a, b)| = a + b − ∥(a, b)∥**p* *= b*

[(*a*
*b* + 1

)

*−*^{((}*a*
*b*

)_{p}

+ 1

)*1/p*]

*.*

*Let f (t) = t + 1− (t** ^{p}* + 1)

^{1/p}*for t≥ 1. Then*

*f*^{′}*(t) = 1−*^{(} *t*^{p}*t** ^{p}* + 1

)^{p}^{−1}_{p}

*.*

*Notice that f*^{′}*(t) > 0 for t≥ 1, and f(1) = 2 − 2*^{1}* ^{p}*, and hence we obtain that

*|ϕ**p**(a, b)| ≥ (2 − 2*^{1}^{p}*)b = (2− 2*^{1}* ^{p}*)

*| min{a, b}| for any p > 1.*(27) All the aforementioned inequalities (22)-(27) imply that (21) holds.

*2*

**Proposition 3.1 Let Ψ***p**, Ψ*_{NR} *and Ψ**α,p* *be deﬁned as in (11), (14) and (12), respectively.*

*Let S be an arbitrary bounded set. Then, for any p > 1, we have*

(

2*− 2*^{1}^{p}^{)}^{2}Ψ_{NR}*(x)≤ Ψ**p**(x)≤*^{(}2 + 2^{p}^{1}^{)}^{2}Ψ_{NR}*(x)* *for all x∈ IR** ^{n}* (28)

*and*

(

2*− 2*^{1}^{p}^{)}^{2}Ψ_{NR}*(x)≤ Ψ**α,p**(x)≤*^{(}*αB*^{2}+ (2 + 2^{1}* ^{p}*)

^{2}

^{)}Ψ

_{NR}

*(x)*

*for all x∈ S,*(29)

*where B is a constant deﬁned by B = max*

1≤i≤n

{

sup

*x**∈S**{max {|x**i**|, |F**i**(x)|}}*

}

*<∞.*

**Proof. The inequality in (28) is direct by Lemma 3.2 and the deﬁnitions of Ψ***p* and Ψ_{NR}.
In addition, from Lemma 3.2 and the deﬁnition of Ψ* _{α,p}*, it follows that

Ψ_{α,p}*(x)≥*^{(}2*− 2*^{p}^{1}^{)}^{2}Ψ_{NR}*(x)* *for all x∈ IR*^{n}*.*

*We next prove the inequality on the right hand side of (29). We claim that, for each i,*
*(x*_{i}*F*_{i}*(x))*_{+}*≤ B| min{x**i**, F*_{i}*(x)}| for all x ∈ S.* (30)
*Without loss of generality, suppose F*_{i}*(x)≥ x**i**. If F*_{i}*(x)≥ x**i* *≥ 0, it follows that*

*(x*_{i}*F*_{i}*(x))*_{+}*= x*_{i}*F*_{i}*(x) = F*_{i}*(x)| min{x**i**, F*_{i}*(x)}| ≤ B| min{x**i**, F*_{i}*(x)}|.*

*If F*_{i}*(x)≥ 0 ≥ x**i**, then (x*_{i}*F*_{i}*(x))*_{+}= 0. If 0 *≥ F**i**(x)≥ x**i*, it follows that
*(x*_{i}*F*_{i}*(x))*_{+} =*|x**i**F*_{i}*(x)| ≤ |x**i**|*^{2} *≤ B| min{x**i**, F*_{i}*(x)}|.*

*Thus, (30) holds for all x∈ S. By Lemma 3.2 and (30), for all i = 1, . . . , n and x ∈ S,*
*ψ*_{α,p}*(x*_{i}*, F*_{i}*(x))* *≤*^{{}*αB*^{2}+ (2 + 2^{1}* ^{p}*)

^{2}

^{}}min

*{x*

*i*

*, F*

_{i}*(x)}*

^{2}

*holds for any p > 1. The proof is then complete by the deﬁnition of Ψ** _{α,p}* and Ψ

_{NR}.

*2*

From Proposition 3.1, we immediately obtain the following result.

**Corollary 3.1 Let Ψ***p* *and Ψ**α,p* *be deﬁned by (12) and (11), respectively, and S be any*
*bounded set. Then, for any p > 1 and all x∈ S, we have the following inequalities:*

(2*− 2*^{1}* ^{p}*)

^{2}

(

*αB*^{2}+ (2 + 2^{1}* ^{p}*)

^{2}

^{)}

Ψ_{α,p}*(x)≤ Ψ**p**(x)≤* (2 + 2^{1}* ^{p}*)

^{2}(2

*− 2*

^{p}^{1})

^{2}

Ψ_{α,p}*(x)*

*where B is the constant deﬁned as in Proposition 3.1.*

Since Ψ_{p}*, Ψ*_{NR} and Ψ* _{α,p}* have the same order on a bounded set, one will provide a
global error bound for the NCP as long as the other one does. As below, we show that
Ψ

_{α,p}*provides a global error bound without the Lipschitz continuity of F when α > 0.*

**Theorem 3.4 Let Ψ**_{α,p}*be deﬁned as in (12). Suppose that F is a uniform P -function*
*with modulus µ > 0. If α > 0, then there exists a constant κ*_{1} *> 0 such that*

*∥x − x*^{∗}*∥ ≤ κ*1Ψ*α,p**(x)*^{1}^{4} *for all x∈ IR** ^{n}*;

*if α = 0 and S is any bounded set, there exists a constant κ*_{2} *> 0 such that*

*∥x − x*^{∗}*∥ ≤ κ*2

(

max

{

Ψ*α,p**(x),*

√

Ψ*α,p**(x)*

})^{1}

2 *for all x∈ S;*

*where x*^{∗}*= (x*^{∗}_{1}*,· · · , x*^{∗}_{n}*) is the unique solution for the NCP.*

**Proof. Since F is a uniform P -function, the NCP has the unique solution, and moreover,***µ∥x − x*^{∗}*∥*^{2} *≤ max*

1*≤i≤n**(x− x*^{∗}*)(F*_{i}*(x)− F**i**(x** ^{∗}*))

= max

1*≤i≤n**{x**i**F*_{i}*(x)− x*^{∗}*i**F*_{i}*(x)− x**i**F*_{i}*(x*^{∗}*) + x*^{∗}_{i}*F*_{i}*(x** ^{∗}*)

*}*

= max

1*≤i≤n**{x**i**F**i**(x)− x*^{∗}*i**F**i**(x)− x**i**F**i**(x** ^{∗}*)

*}*

*≤ max*

1*≤i≤n**τ*_{i}*{(x**i**F*_{i}*(x))*_{+}+ (*−F**i**(x))*_{+}+ (*−x**i*)_{+}*},* (31)
*where τ** _{i}* := max

*{1, x*

^{∗}*i*

*, F*

_{i}*(x*

*)*

^{∗}*}. We next prove that for all (a, b) ∈ IR*

^{2},

(*−a)*+2

+ (*−b)*+2 *≤ [∥(a, b)∥**p**− (a + b)]*^{2}*.* (32)

*Without loss of generality, suppose a* *≥ b. If a ≥ b ≥ 0, then (32) holds obviously. If*
*a≥ 0 ≥ b, then ∥(a, b)∥**p**− (a + b) ≥ −b ≥ 0, which in turn implies that*

(*−a)*+2

+ (*−b)*+2

*= b*^{2} *≤ [∥(a, b)∥**p**− (a + b)]*^{2}*.*
If 0*≥ a ≥ b, then (−a)*+2

+ (*−b)*+2

*= a*^{2}*+ b*^{2} *≤ [∥(a, b)∥**p**− (a + b)]*^{2}. Hence, (32) follows.

*Suppose that α > 0. Using the inequality (32), we then obtain that*

*[(ab)*_{+}+ (*−a)*++ (*−b)*+]^{2} *= (ab)*^{2}_{+}+ (*−b)*^{2}++ (*−a)*^{2}+*+ 2(ab)*_{+}(*−a)*+

+2(*−a)*+(*−b)*+*+ 2(ab)*_{+}(*−b)*+

*≤ (ab)*^{2}_{+}+ (*−b)*^{2}_{+}+ (*−a)*^{2}_{+}*+ (ab)*^{2}_{+}+ (*−a)*^{2}_{+}
+(*−a)*^{2}++ (*−b)*^{2}+*+ (ab)*^{2}_{+}+ (*−b)*^{2}+

*≤ 3*^{[}*(ab)*^{2}_{+}+ (*∥(a, b)∥**p* *− (a + b))*^{2}^{]}

*≤ τ*^{[}*α*

2*(ab)*^{2}_{+}+ 1

2(*∥(a, b)∥**p**− (a + b))*^{2}^{]}

*= τ ψ*_{α,p}*(a, b)* *for all (a, b)∈ IR*^{2}*,* (33)
*where τ := max*

{6
*α, 6*

}

*> 0. Combining (33) with (31) and letting ˆτ = max*

1*≤i≤n**τ** _{i}*, we get

*µ∥x − x*

^{∗}*∥*

^{2}

*≤ max*

1*≤i≤n**τ*_{i}*{τψ**α,p**(x*_{i}*, F*_{i}*(x))}*^{1/2}

*≤ ˆττ** ^{1/2}* max

1*≤i≤n**ψ*_{α,p}*(x*_{i}*, F (x))*^{1/2}

*≤ ˆττ*^{1/2}

{ _{n}

∑

*i=1*

*{ψ**α,p**(x*_{i}*, F*_{i}*(x))*

}_{1/2}

= ˆ*τ τ** ^{1/2}*Ψ

_{α,p}*(x, F (x))*

^{1/2}*.*

*From this, the ﬁrst desired result follows immediately by setting κ*_{1} :=^{[}ˆ*τ τ*^{1/2}*/µ*^{]}* ^{1/2}*.

*Suppose that α = 0. From the proof of Proposition 3.1, the inequality (30) holds.*

*Combining with equations (31)–(32), it then follows that for all x∈ S,*
*µ∥x − x*^{∗}*∥*^{2} *≤ max*

1*≤i≤n**τ*_{i}^{[}*B| min{x**i**, F*_{i}*(x)}| + (ψ**p**(x*_{i}*, F*_{i}*(x)))*^{1/2}^{]}

*≤ ˆτ max*

1*≤i≤n*

[*√*

2 ˆ*Bψ*_{p}*(x*_{i}*, F*_{i}*(x)) + (ψ*_{p}*(x*_{i}*, F*_{i}*(x)))*^{1/2}^{]}

*≤* *√*

2ˆ*τ ˆB*

(

Ψ_{p}*(x) +*

√

Ψ_{p}*(x)*

)

*≤ 4ˆτ ˆB max*

{

Ψ_{p}*(x),*

√

Ψ_{p}*(x)*

}

= 4ˆ*τ ˆB max*

{

Ψ_{α,p}*(x),*

√

Ψ_{α,p}*(x)*

}

where ˆ*B = B/(2− 2*^{1}^{p}*) and the second inequality is from Lemma 3.2. Letting κ*_{2} :=

2^{[}*τ ˆ*ˆ*B/µ*^{]}* ^{1/2}*, we obtain the desired result from the above inequality.

*2*

The following lemma is needed for the proof of Proposition 3.2, which plays a crucial role in showing the convergence rate of the algorithm described in the next section.

**Lemma 3.3 For all (a, b)**̸= (0, 0) and p > 1, we have the following inequality:

(*sgn(a)· |a|*^{p}^{−1}*+ sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)_{2}

*≥*^{(}2*− 2*^{1}^{p}^{)}^{2}*.*

**Proof. If a = 0 or b = 0, the inequality holds obviously. Then we complete the proof***by considering three cases: (i) a > 0 and b > 0, (ii) a < 0 and b < 0, and (iii) ab < 0.*

*Case (i): Without loss of generality, we suppose a≥ b > 0. Then*

*|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}* ^{−1}* =

(^{a}* _{b}*)

_{p}

_{−1}+ 1

((^{a}* _{b}*)

_{p}+ 1^{)}^{1}^{−}

1
*p*

*.* (34)

*Let f (t) :=* *t*^{p}* ^{−1}*+ 1

*(t*

*+ 1)*

^{p}^{1}

^{−}^{1}

^{p}*for any t > 0. By computation, we have that*

*f*^{′}*(t) =* *t*^{p}^{−2}*(p− 1)(1 − t)*

*(t** ^{p}*+ 1)

^{2}

*∀t > 0.*

*Since f*^{′}*(t) < 0 for t≥ 1 and f(1) = 2*^{1}^{p}*, it follows that f (t)≤ 2*^{1}^{p}*for t* *≥ 1. Therefore,*

*|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}^{−1}*≤ 2*^{1}^{p}*for p > 1,*
which in turn implies that 2*−* *|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}^{−1}*≥ 2 − 2*^{1}^{p}*for p > 1. Squaring both sides*
then leads to the desired inequality.

Case (ii): By similar arguments as in case (i), we obtain
2*− 2*^{1}^{p}*≤ 2 −|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}^{−1}*≤ 2 +* *|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}^{−1}*for p > 1,*
from which the result follows immediately.

Case (iii): Again, we suppose *|a| ≥ |b| and therefore have*
2^{1}^{p}*≥* *|a|*^{p}* ^{−1}*+

*|b|*

^{p}

^{−1}*∥(a, b)∥*^{p}^{p}^{−1}*≥* *|a|*^{p}^{−1}*− |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*for p > 1.*

Thus 2*− 2*^{1}^{p}*≤ 2 −|a|*^{p}^{−1}*− |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*for p > 1 and the desired result is also satisﬁed.* *2*

**Proposition 3.2 Let ψ**_{α,p}*be given as in (13). Then, for all x∈ IR*^{n}*and p > 1,*

*∥∇**a**ψ*_{α,p}*(x, F (x)) +∇**b**ψ*_{α,p}*(x, F (x))∥*^{2} *≥ 2*^{(}2*− 2*^{p}^{1}^{)}^{2}Ψ_{p}*(x),*
*and particularly, for all x belonging to any bounded set S and p > 1,*

*∥∇**a**ψ*_{α,p}*(x, F (x)) +∇**b**ψ*_{α,p}*(x, F (x))∥*^{2} *≥* 2(2*− 2*^{1}* ^{p}*)

^{4}

(

*αB*^{2}+ (2 + 2^{1}* ^{p}*)

^{2}

)Ψ_{α,p}*(x)*
*where B is deﬁned as in Proposition 3.1 and*

*∇**a**ψ*_{α,p}*(x, F (x)) :=*

(

*∇**a**ψ*_{α,p}*(x*_{1}*, F*_{1}*(x)),* *· · · , ∇**a**ψ*_{α,p}*(x*_{n}*, F*_{n}*(x))*

)_{T}

*,*

*∇**b**ψ**α,p**(x, F (x)) :=*

(

*∇**b**ψ**α,p**(x*1*, F*1*(x)),* *· · · , ∇**b**ψ**α,p**(x**n**, F**n**(x))*

)_{T}

*.* (35)

**Proof. The second part of the conclusions is direct by Corollary 3.1 and the ﬁrst part.**

From the deﬁnition of *∇**a**ψ*_{α,p}*(x, F (x)),∇**b**ψ*_{α,p}*(x, F (x)) and Ψ*_{p}*(x), the ﬁrst part of the*
conclusions is equivalent to proving that the following inequality

(*∇**a**ψ*_{α,p}*(a, b) +∇**b**ψ*_{α,p}*(a, b))*^{2} *≥ 2*^{(}2*− 2*^{1}^{p}^{)}^{2}*ψ*_{p}*(a, b)* (36)
*holds for all (a, b)∈ IR*^{2}*. When (a, b) = (0, 0), the inequality (36) clearly holds. Suppose*
*(a, b)̸= (0, 0). Then, it follows from equation (15) that*

(*∇**a**ψ*_{α,p}*(a, b) +∇**b**ψ*_{α,p}*(a, b))*^{2}

=

{

*α(a + b)(ab)*_{+}*+ (ϕ*_{p}*(a, b))*

(*sgn(a)· |a|*^{p−1}*+ sgn(b)· |b|*^{p−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)}_{2}

*= α*^{2}*(a + b)*^{2}*(ab)*^{2}_{+}*+ (ϕ*_{p}*(a, b))*^{2}

(*sgn(a)· |a|*^{p}^{−1}*+ sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)_{2}

*+2α(a + b)(ab)*_{+}*(ϕ*_{p}*(a, b))*

(*sgn(a)· |a|*^{p}^{−1}*+ sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)

*.* (37)

*Now, we claim that for all (a, b)̸= (0, 0) ∈ IR*^{2},
*2α(a + b)(ab)*_{+}*(ϕ*_{p}*(a, b))*

(*sgn(a)· |a|*^{p}^{−1}*+ sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)

*≥ 0.* (38)

*If ab≤ 0, then (ab)*+ *= 0 and the inequality (36) is clear. If a, b > 0, then by noting that*

(*sgn(a)· |a|*^{p}^{−1}*+ sgn(b)· |b|*^{p}^{−1}

*∥(a, b)∥*^{p}^{p}^{−1}*− 2*

)

*≤ 0, ∀(a, b) ̸= (0, 0) ∈ IR*^{2} (39)
*and ϕ*_{p}*(a, b)* *≤ 0, the inequality (38) also holds. If a, b < 0, then ϕ**p**(a, b)* *≥ 0, which*
together with (39) then yields the inequality (38). Thus, we prove that the inequality
*(38) holds for all (a, b)* *̸= (0, 0). Using Lemma 3.3 and equations (38)–(39), we readily*
*obtain the inequality (36) holds for all (a, b)̸= (0, 0). The proof is thus complete.* *2*

**4** **A descent algorithm and convergence results**

In this section, we propose a derivative-free descent algorithm based on the function Ψ* _{α,p}*.
By Lemma 3.1 (d), it is easy to verify that ¯

*d :=−∇*

*b*

*ψ*

*α,p*

*(x, F (x)) is a descent direction*for monotone nonlinear complementarity problems, i.e., the following result holds.

**Lemma 4.1 Let Ψ**_{α,p}*be deﬁned as in (12). If the mapping F is monotone, then ¯d :=*

*−∇**b**ψ*_{α,p}*(x, F (x)) is a descent direction of Ψ*_{α,p}*at any x∈ IR*^{n}*, i.e.,* *∇Ψ**α,p**(x)*^{T}*d < 0.*¯
However, we observe that ¯*d does not involve any information of* *∇**a**ψ*_{α,p}*(x, F (x)) and is*
*lack of a certain symmetry, for which we can not ﬁnd a constant c > 0 such that*

*∥ ¯d∥ ≥ cψ**α,p**(x, F (x)).*

This sets a big obstacle to establish the convergence rate of the derivative-free algorithm
based on ¯*d. In view of this, we follow the similar line as [25] to adopt the search direction*
of the following form:

*d*^{k}*(ρ) :=−∇**b**ψ*_{α,p}*(x*^{k}*, F (x** ^{k}*))

*− ρ∇*

*a*

*ψ*

_{α,p}*(x*

^{k}*, F (x*

^{k}*)),*(40)

*where ρ is a parameter such that ρ*

*∈ (0, 1) and ∇*

*a*

*ψ*

_{α,p}*(x, F (x)),*

*∇*

*b*

*ψ*

_{α,p}*(x, F (x)) are*

*deﬁned as in (35). Although d*

^{k}*(ρ) for any ρ*

*∈ (0, 1) is not necessarily a descent direction*of Ψ

_{α,p}*at the iterate x*

^{k}*, Lemma 4.1 implies that it is a descent one if ρ∈ (0, ¯ρ*

*k*) where

¯

*ρ**k*:= 1 if *∇**a**ψ**α,p**(x*^{k}*, F (x** ^{k}*))

^{T}*∇Ψ*

*α,p*

*(x*

*)*

^{k}*≥ 0,*and otherwise

¯

*ρ** _{k}* := min

{

*1,−∇**b**ψ*_{α,p}*(x*^{k}*, F (x** ^{k}*))

^{T}*∇Ψ*

*α,p*

*(x*

*)*

^{k}*∇**a**ψ*_{α,p}*(x*^{k}*, F (x** ^{k}*))

^{T}*∇Ψ*

*α,p*

*(x*

*)*

^{k}}

*.*

Clearly, ¯*ρ*_{k}*∈ (0, 1) except that x*^{k}*is a solution of the NCP. Thus, d** ^{k}*is a descent direction
of Ψ

_{α,p}*at x*

^{k}*for monotone NCPs only if ρ is chosen suﬃciently small. Similar to [25],*

*we also determine an appropriate ρ*

*k*by the backtracking search of Armijo-type instead of the value of ¯

*ρ*

*, in our algorithm described as below.*

_{k}**Algorithm 4.1**

*(Step 0) Given real numbers p > 1 and α* *≥ 0 and a starting point x*^{0} *∈ IR** ^{n}*. Choose

*the parameters σ∈ (0, 1), β ∈ (0, 1), γ ∈ (0, 1) and ε ≥ 0. Set k := 0.*

(Step 1) If Ψ_{α,p}*(x** ^{k}*)

*≤ ε, then stop.*

*(Step 2) Let m*_{k}*be the smallest nonnegative integer m satisfying*

Ψ_{α,p}*(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*≤ (1 − σβ*

*)Ψ*

^{2m}

_{α,p}*(x*

^{k}*),*(41) where

*d*^{k}*(γ** ^{m}*) :=

*−∇*

*b*

*ψ*

_{α,p}*(x*

^{k}*, F (x*

*))*

^{k}*− γ*

^{m}*∇*

*a*

*ψ*

_{α,p}*(x*

^{k}*, F (x*

^{k}*)).*

*(Step 3) Set x*^{k+1}*:= x*^{k}*+ β*^{m}^{k}*d*^{k}*(γ*^{m}^{k}*), k := k + 1 and go to Step 1.*

We see that Algorithm 4.1 does not involve the computation of *∇Ψ**α,p* and *∇F ,*
and hence it is a derivative-free algorithm. In what follows, we establish the convergence
results for Algorithm 4.1, and particularly, analyze its convergence rate under the strongly
*monotone assumption of F . To this end, we assume that the parameter ε in Algorithm*
4.1 equals to zero and Algorithm 4.1 generates an inﬁnite sequence *{x*^{k}*}.*

**Proposition 4.1 Suppose that F is monotone. Then Algorithm 4.1 is well-deﬁned for***any starting point x*^{0}*. Furthermore, if x*^{∗}*is an accumulation point of the sequence* *{x*^{k}*}*
*generated by Algorithm 4.1, then x*^{∗}*is a solution of the NCP.*

**Proof. We ﬁrst prove that Algorithm 4.1 is well-deﬁned. From the construction of the**
algorithm, it suﬃces to show that Step 2 is well-deﬁned. Assume to the contrary that
*there is no nonnegative integer m satisfying (41). Then, for any integer m≥ 0,*

Ψ_{α,p}*(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*− Ψ*

*α,p*

*(x*

^{k}*) >−σβ*

*Ψ*

^{2m}

_{α,p}*(x*

^{k}*).*

*Dividing the above inequality by β*^{m}*and passing to the limit m→ +∞, we obtain that*

*m*lim*→+∞*

Ψ*α,p**(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*− Ψ*

*α,p*

*(x*

*)*

^{k}*β*^{m}*≥ 0.* (42)

Since Ψ* _{α,p}* is continuously diﬀerentiable, we have that Ψ

*is locally Lipschitz continuous*

_{α,p}*at x*

^{k}*, which in turn implies that there exists L > 0 such that*

*∥Ψ**α,p**(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*− Ψ*

*α,p*

*(x*

^{k}*+ β*

^{m}*d*

*(0))*

^{k}*∥ ≤ Lβ*

^{m}*∥d*

^{k}*(γ*

*)*

^{m}*− d*

*(0)*

^{k}*∥*

*for all suﬃciently large m. Consequently,*

*m*lim*→+∞*

Ψ_{α,p}*(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*− Ψ*

*α,p*

*(x*

*)*

^{k}*β*

^{m}= lim

*m**→+∞*

Ψ_{α,p}*(x*^{k}*+ β*^{m}*d** ^{k}*(0))

*− Ψ*

*α,p*

*(x*

*)*

^{k}*β*

^{m}+ lim

*m→+∞*

Ψ*α,p**(x*^{k}*+ β*^{m}*d*^{k}*(γ** ^{m}*))

*− Ψ*

*α,p*

*(x*

^{k}*+ β*

^{m}*d*

*(0))*

^{k}*β*

^{m}*≤ ∇Ψ**α,p**(x** ^{k}*)

^{T}*d*

^{k}*(0).*

This together with (42) yields that *∇Ψ**α,p**(x** ^{k}*)

^{T}*d*

*(0)*

^{k}*≥ 0. However, by Lemma 4.1,*

*∇Ψ**α,p**(x** ^{k}*)

^{T}*d*

^{k}*(0) < 0 which leads to a contradiction. Hence, Algorithm 4.1 is well-*deﬁned.

*Next we prove that any accumulation point x** ^{∗}* of

*{x*

^{k}*} is a solution of the NCP. Let*

*{x*

^{k}*}*

*k*

*∈K*

*be a subsequence converging to x*

*. Notice that Ψ*

^{∗}*is continuously diﬀerentiable*

_{α,p}