to appear in Computational Optimization and Applications, 2010

**A new class of penalized NCP-functions and its properties**

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

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

Zheng-Hai Huang ^{2}
Department of Mathematics

Tianjin University Tianjin 300072, China E-mail: huangzhenghai@tju.edu.cn

Chin-Yu She

Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677 E-mail: bbgmmshe@hotmail.com

April 17, 2009

(ﬁrst revised on September 15, 2009) (second revised on December 18, 2009)

**Abstract. In this paper, we consider a class of penalized NCP-functions, which includes**
several existing well-known NCP-functions as special cases. The merit function induced
by this class of NCP-functions is shown to have bounded level sets and provide error
bounds under mild conditions. A derivative free algorithm is also proposed, its global

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

2This author’s work is partly supported by by the National Natural Science Foundation of China (Grant No. 10871144).

convergence is proved and numerical performance compared with those based on some existing NCP-functions is reported.

**Key Words. NCP-function, penalized, bounded level sets, error bounds.**

**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 [6, 14].

Many methods have been proposed to solve the NCP; see [3, 5, 14, 16, 17, 18, 20, 23, 26, 30, 31]. For more details, please refers to the excellent monograph [9]. One of the most powerful and popular methods is to reformulate the NCP as a system of nonlinear equations [24, 25, 31], or as an unconstrained minimization problem [7, 10, 11, 12, 19, 21, 27, 30]. 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 (1). 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)

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

*ϕ*_{FB}*(a, b) =√*

*a*^{2}*+ b*^{2} *− (a + b), ∀(a, b) ∈ IR*^{2}*.* (3)
*Through this NCP-function ϕ*_{FB}, the NCP (1) can be reformulated as a system of nons-
mooth 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 the unconstrained minimization problem:

*x*min*∈IR** ^{n}*Ψ

_{FB}

*(x).*(7)

There are several generalizations of the FB function in the literature. For example,
*Kanzow and Kleinmichel [22] extend ϕ*_{FB} function to

*ϕ*_{θ}*(a, b) :=*√

*(a− b)*^{2}*+ θab− (a + b), θ ∈ (0, 4).*

Chen, Chen, and Kanzow [2] study a penalized FB function

*ϕ*_{λ}*(a, b) := λϕ*_{FB}*(a, b) + (1− λ)a*+*b*_{+}*,* *λ* *∈ (0, 1).*

Some other types of penalized FB functions are also investigated by Sun and Qi in [28].

Recently, Chen and Pan [3, 5] consider the following generalization of the FB function:

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

*|a|** ^{p}*+

*|b|*

*. Another further generalization is proposed by Hu, Huang and Chen in [15]:*

^{p}*ϕ**θ,p**(a, b) :=*√^{p}

*θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)(|a − b|*

*)*

^{p}*− (a + b),*(9)

*where p > 1, θ∈ (0, 1].*

*All the aforementioned functions are NCP-functions. The corresponding function ψ** _{θ}*,

*ψ*

_{λ}*, ψ*

_{p}*, and ψ*

_{θ,p}*is square of ϕ*

_{θ}*, ϕ*

_{λ}*, ϕ*

_{p}*, and ϕ*

*, respectively, and naturally induces a merit function Ψ*

_{θ,p}*θ*, Ψ

*λ*, Ψ

*p*, and Ψ

*θ,p*

*like what ψ*

_{FB}function does. Along this track, in this paper, we study the following merit function Ψ

*: IR*

_{α,θ,p}

^{n}*→ IR*+ for the NCP:

Ψ_{α,θ,p}*(x) :=*

∑*n*
*i=1*

*ψ*_{α,θ,p}*(x*_{i}*, F*_{i}*(x)),* (10)

*where ψ** _{α,θ,p}* : IR

^{2}

*→ IR*+ is an NCP-function deﬁned by

*ψ*

_{α,θ,p}*(a, b) :=*

*α*

2(max*{0, ab})*^{2} *+ ψ*_{θ,p}*(a, b)* (11)

*with α≥ 0 being a real parameter. Note that ψ**α,θ,p* *includes all the above functions ψ*_{FB},
*ψ*_{p}*, ψ*_{θ}*, ψ*_{θ,p}*(and ψ*_{7} *in [28]) as special cases. Although ψ** _{α,θ,p}* is obtained by penalizing

*the function ψ*

_{θ,p}*considered in [15], more favorable properties of ψ*

*are explored in this work. In particular, Ψ*

_{α,θ,p}*α,θ,p*has property of bounded level sets and provides a global error bound for the NCP under mild condition which were not studied in [15]. Thus, this paper can be viewed as a follow-up of [15]. On the other hand, as remarked in [2], penalized Fischer-Burmeister (FB) function not only possesses stronger properties than FB function but also gives extremely promising numerical performance, which is another motivation of our considering this generalization of several NCP-functions.

This paper is organized as follows. In Section 2, we review some deﬁnitions and
preliminary results to be used in the subsequent analysis. In Section 3, we show some
properties of the proposed merit function. In Section 4, we propose a derivative free
algorithm based on this merit function Ψ* _{α,θ,p}*, show its global convergence, and report
some numerical results. In Section 5, we make concluding remarks.

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 use* *∥x∥**p*

*to denote the p-norm of x and denote* *∥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*
will frequently mention two merit functions. One is 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)),* (12)

*where ϕ*_{NR} : IR^{2} *→ IR denotes the minimum NCP-function min{a, b}. Another one is*
Ψ* _{θ,p}* : IR

^{n}*→ IR*+

*induced by ψ*

*:*

_{θ,p}Ψ*θ,p**(x) :=* 1
2

∑*n*
*i=1*

*ϕ*^{2}_{θ,p}*(x**i* *, F**i**(x)).* (13)

*Unless otherwise stated, in the sequel, we always suppose that p is a ﬁxed real number*
*in (1,∞).*

**2** **Preliminaries**

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

*A matrix is said to be P -matrix if each of its principal minors is positive, and is called*
*P*_{0}*-matrix if each of its principal minors is nonnegative. Obviously, P -matrix is a gen-*
*eralization of positive deﬁnite matrix, while P*_{0}-matrix is a generalization of positive
*semideﬁnite matrix. Such concepts of P -matrix and P*_{0}-function can be further extended
*to nonlinear mapping, which we call them P -function and P*_{0}-function.

**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 if***⟨x − y, F (x) − F (y)⟩ ≥ µ∥x − y∥*^{2} *for some µ > 0 and 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 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}*.*

*It is well-known that every monotone function is an P*_{0} function and every strongly
*monotone function is a uniform P -function. For a continuously diﬀerentiable function F ,*
if its (transpose) Jacobian *∇F (x) is an P -matrix then F is an P -function (the converse*
may not be true), whereas the (transpose) Jacobian *∇F (x) is an P*0-matrix if and only
*if F is an P*_{0}*-function. For more detailed properties of various monotone and P (P*_{0})-
function, please refer to [9].

**3** **Properties of the New NCP-Function**

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

*has bounded level sets and provides a global error bound, respectively. To this end, we present some technical lemmas which are needed for subsequent analysis.*

_{α,θ,p}**Lemma 3.1 For p > 1, a > 0, b > 0, we have a**^{p}*+ b*^{p}*≤ (a + b)*^{p}

**Proof. We present two diﬀerent ways to prove this lemma.**

*(1) For any p > 1, p = n + m, where n = [p] (the greatest integer less than or equal to*
*p) and m = p− n, applying binomial theorem gives*

*(a + b)*^{p}*= (a + b)*^{n}*(a + b)*^{m}

*≥ (a*^{n}*+ b*^{n}*)(a + b)*^{m}

*= a*^{n}*(a + b)*^{m}*+ b*^{n}*(a + b)*^{m}

*≥ a*^{n}*a*^{m}*+ b*^{n}*b*^{m}

*= a*^{p}*+ b*^{p}*.*

*(2) Let f (t) = (t + 1)*^{p}*− (t*^{p}*+ 1). It is easy to verify that f is increasing on [0,∞) when*
*p > 1. Hence, f (a/b)≥ f(0) = 0 which yields (a + b)*^{p}*≥ a*^{p}*+ b** ^{p}*.

*2*

**Lemma 3.2 The function ψ**_{α,θ,p}*deﬁned by (11) has the following favorable properties:*

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

**(b) ψ***α,θ,p* *is continuously diﬀerentiable everywhere. Moreover, if (a, b)̸= (0, 0),*

*∇**a**ψ*_{α,θ,p}*(a, b)*

*= αb(ab)*_{+}+

(*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

*ϕ*_{θ,p}*(a, b),*

*∇**b**ψ*_{α,θ,p}*(a, b)*

*= αa(ab)*_{+}+

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

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

*ϕ*_{θ,p}*(a, b),*
(14)
*and otherwise,* *∇**a**ψ*_{α,θ,p}*(0, 0) =∇**b**ψ*_{α,θ,p}*(0, 0) = 0.*

**(c) For p***≥ 2, the gradient of ψ**α,θ,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,θ,p**(a, b)− ∇ψ**α,θ,p**(c, d)∥ ≤ L∥(a, b) − (c, d)∥.*

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

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

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

**Proof. (a) It is clear that ψ**_{α,θ,p}*(a, b)* *≥ 0 for all (a, b) ∈ IR*^{2} *from the deﬁnition of ψ** _{α,θ,p}*.
Then by [15, Prop. 2.1], we have

*ψ*_{α,θ,p}*(a, b) = 0⇐⇒* *α*

2(max*{0, ab})*^{2} *= 0 and ψ*_{θ,p}*(a, b) = 0⇐⇒ a ≥ 0, b ≥ 0, ab = 0.*

*Hence, ψ** _{α,θ,p}* is an NCP-function.

*(b) First, direct calculations give the partial derivatives of ψ*_{α,θ,p}*. Then, using αb(ab)*_{+} *→*
*(0, 0) and αa(ab)*+ *→ (0, 0) as (a, b) → (0, 0), we have* ^{α}_{2}(max*{0, ab})*^{2} is continuously
*diﬀerentiable everywhere. By [15, Prop. 2.5], it is known that ψ** _{θ,p}* is continuously diﬀer-
entiable everywhere. In view of the expression of

*∇ψ*

*α,θ,p*

*(a, b), ψ*

*is also continuously diﬀerentiable everywhere.*

_{α,θ,p}*(c) First, we claim that a(ab)*_{+} *for any a, b∈ IR is Lipschitz continuous on any nonempty*
*bounded set S. For any (a, b)* *∈ S and (c, d) ∈ S, without loss of generality, we may*
*assume that a*^{2}*+ b*^{2} *≤ k and c*^{2}*+ d*^{2} *≤ k which imply |a| ≤ k + 1, |b| ≤ k + 1, |c| ≤ k + 1*
and *|d| ≤ k + 1. Then,*

*a(ab)*^{+}*− c(cd)*+

= 1 2

*a*^{2}*b + a|ab| − c*^{2}*d− c|cd|*

= 1 2

*a*^{2}*b− a*^{2}*d + a*^{2}*d− c*^{2}*d + a|ab| − c|ab| + c|ab| − c|cd|*

*≤* 1
2

(

*|a*^{2}*b− a*^{2}*d| + |a*^{2}*d− c*^{2}*d| +**a|ab| − c|ab|*+*c|ab| − c|cd|*)

= 1 2

(

*a*^{2}*|b − d| + |a + c||d||a − c| + |ab||a − c| + |c||ab − cd|*

)

*≤* 1
2

[

*k|b − d| + (|a| + |c|)|d||a − c| + k|a − c| + (k + 1)|ab − ad + ad − cd|*

]

*≤* 1
2

[

*k|b − d| + 2(k + 1)*^{2}*|a − c| + k|a − c| + (k + 1)*^{2}(*|b − d| + |a − c|)*
]

= 1 2

{ [*2(k + 1)*^{2}*+ k + (k + 1)*^{2}]

*|a − c| +*[

*k + (k + 1)*^{2}]

*|b − d|*

}

*≤ l*(

*|a − c| + |b − d|*)

*≤* *√*

*2l∥(a, b) − (c, d)∥,*

*where l = 2(k + 1)*^{2} *+ k + (k + 1)*^{2}*. Hence, the mapping a(ab)*_{+} is Lipschitz continuous
*on any nonempty bounded set S and so is αa(ab)*_{+}*. Similarly, αb(ab)*_{+} is Lipschitz
*continuous on any nonempty bounded set S. All of these imply the gradient function*
of the function ^{α}_{2}(max*{0, ab})*^{2} *is Lipschitz continuous on any bounded set S. On the*

*other hand, by [15, Theorem 2.1], the gradient function of the function ψ*_{θ,p}*with p≥ 2,*
*θ* *∈ (0, 1] is Lipschitz continuous. Thus, the gradient of ψ**α,θ,p* is Lipschitz continuous on
*any nonempty bounded set S.*

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

*∇**a**ψ**α,θ,p**(a, b)· ∇**b**ψ**α,θ,p**(a, b)* (15)

*= cdϕ*^{2}_{θ,p}*(a, b) + α*^{2}*ab(ab)*_{+}^{2}*+ αa(ab)*_{+}*cϕ*_{θ,p}*(a, b) + αb(ab)*_{+}*dϕ*_{θ,p}*(a, b),*
where

*c =*

(*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)
*,*
*d =*

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

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)
*.*
*From the proof of [15, Prop. 2.5 ], we know ab(ab)*^{2}_{+} *≥ 0 and*

(*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*)

*≤ 0,* (16)

it suﬃces to show that the last two terms of (15) are nonnegative. For this purpose, we claim that

*αa(ab)*_{+}

(*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p−1)/p}*− 1*

)

*ϕ*_{θ,p}*(a, b)≥ 0* (17)
*for all (a, b)* *̸= (0, 0). If a ≤ 0 and b ≤ 0, then ϕ**θ,p**(a, b)* *≥ 0, which together with the*
*second inequality in (16) implies that (17) holds. If a* *≤ 0 and b ≥ 0, then (ab)*+ = 0,
*which says that (17) holds. If a > 0 and b > 0, then* *|a|** ^{p}* +

*|b|*

^{p}*≥ |a − b|*

*. Thus,*

^{p}*ϕ*

_{θ,p}*(a, b)*

*≤ ϕ*

*p*

*(a, b)≤ 0, which together with the second inequality in (16) yields (17). If*

*a > 0 and b≤ 0, then (ab)*+= 0, and hence (17) holds. Similarly, we also have

*αb(ab)*_{+}

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

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

*ϕ*_{θ,p}*(a, b)≥ 0*

*for all (a, b)* *̸= (0, 0). Consequently, ∇**a**ψ*_{α,θ,p}*(a, b)* *· ∇**b**ψ*_{α,θ,p}*(a, b)* *≥ 0. Besides, by the*
*proof of [15, Prop. 2.5], we know c = 0 if and only if b = 0 and a > 0; d = 0 if and*
*only if a = 0 and b > 0. This together with (15) says* *∇**a**ψ*_{α,θ,p}*(a, b)· ∇**b**ψ*_{α,θ,p}*(a, b) = 0*
if and only if *{ψ**θ,p**(a, b) = 0 and α*^{2}*ab(ab)*_{+}^{2} = 0*} or {c = 0} or {d = 0} if and only*
if *{ψ**θ,p**(a, b) = 0 and ab* *≤ 0} or {c = 0} or {d = 0} if and only if ψ**θ,p**(a, b) = 0 and*

*α*

2(max*{0, ab})*^{2} *= 0 if and only if ψ*_{α,θ,p}*(a, b) = 0.*

*(e) If ψ*_{α,θ,p}*(a, b) = 0, then* ^{α}_{2}(max*{0, ab})*^{2} *= 0 and ψ*_{θ,p}*(a, b) = 0, which imply ab* *≤ 0*
*and ϕ*_{θ,p}*(a, b) = 0. Hence,* *∇**a**ψ*_{α,θ,p}*(a, b) = 0 and* *∇**b**ψ*_{α,θ,p}*(a, b) = 0. Now, it remains to*
show that *∇**a**ψ*_{α,θ,p}*(a, b) = 0 implying ψ*_{α,θ,p}*(a, b) = 0. Suppose that* *∇**a**ψ*_{α,θ,p}*(a, b) = 0.*

Then,

*αb(ab)*+ =*−*

*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

*ϕ**θ,p**(a, b).* (18)
We will argue that the equality (18) implies(

*b = 0, a≥ 0*)
or(

*b > 0, a = 0*)

. To see this, we let

*c = αb(ab)*_{+}*,*

*d =* *−*

*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)

*ϕ*_{θ,p}*(a, b),*
*e =*

*θsgn(a)· |a|*^{p}* ^{−1}*+ (1

*− θ)sgn(a − b)|a − b|*

^{p}

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 1*

)
*.*

It is not hard to observe that (

*e* *≤ 0*)
and (

*e = 0 implies b = 0*)

which are helpful for the following discussions.

*Case 1: b = 0 and a < 0. Then, c = 0 but d̸= 0 which violates (18).*

*Case 2: b < 0 and a* *≥ 0. Then, we have e < 0, and hence c = 0 but d ̸= 0, which*
violates (18).

*Case 3: b < 0 and a < 0. Then, we have e < 0 and ϕ**θ,p**(a, b) > 0, which lead to c* *≤ 0*
*but d > 0. This contradicts to (18) too.*

*Case 4: b > 0 and a > 0. Then, we have e < 0 and ϕ*_{θ,p}*(a, b) < 0, which imply c≥ 0 but*
*d < 0. This contradicts to (18) too.*

*Case 5: b > 0 and a < 0. Similar arguments as in Case 2 cause a contradiction.*

Thus, (18) implies (

*b = 0, a* *≥ 0*)
or (

*b > 0, a = 0*)

, and each of which always yields
*ψ**α,θ,p**(a, b) = 0. By symmetry,* *∇**b**ψ**α,θ,p**(a, b) = 0 also implies ψ**α,θ,p**(a, b) = 0.*

*(f) If a→ −∞ or b → −∞, from [15, Prop. 2.4], we know |ϕ**θ,p**(a, b)| → ∞. In addition,*
the fact ^{α}_{2}(max*{0, ab})*^{2} *≥ 0 gives ψ**α,θ,p**(a, b)* *→ ∞. If ab → ∞, since α > 0, we have*

*α*

2(max*{0, ab})*^{2} *→ ∞. This together with ψ**θ,p**(a, b)≥ 0 says ψ**α,θ,p**(a, b)→ ∞.* *2*

By Lemma 3.2(a), we immediately have the following theorem.

**Theorem 3.1 Let Ψ**_{α,θ,p}*be deﬁned as in (10). Then Ψ*_{α,θ,p}*(x)≥ 0 for all x ∈ IR*^{n}*and*
Ψ_{α,θ,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,θ,p}*if and only if x solves the NCP.*

**Proof.** *Since ψ*_{θ,p}*is an NCP-function, from [15, Prop. 2.5], we have that x solving*
the NCP *⇐⇒ x ≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩ = 0 ⇐⇒ x ≥ 0, F (x) ≥ 0, x**i**F*_{i}*(x) = 0 for*
*all i* *∈ {1, 2, · · · , n} ⇐⇒ Ψ**α,θ,p**(x) = 0. Besides, Ψ*_{α,θ,p}*(x) is nonnegative. Thus, if x*
*solves the NCP, then x is a global minimizer of Ψ**α,θ,p*. Next, we claim that if the NCP
*has at least one solution, then x is a global minimizer of Ψ** _{α,θ,p}* =

*⇒ x solves the NCP.*

*Suppose x does not solve the NCP. From x solves the NCP* *⇐⇒ Ψ**α,θ,p**(x) = 0 and*
Ψ_{α,θ,p}*(x) is nonnegative, it is clear Ψ*_{α,θ,p}*(x) > 0. However, by assumption, the NCP has*
*a solution, say y, which makes that Ψ**α,θ,p**(y) = 0. Then, we get a contradiction that*
Ψ_{α,θ,p}*(x) > 0 = Ψ*_{α,θ,p}*(y) and x is a global minimizer of Ψ** _{α,θ,p}*. Thus, we complete the
proof.

*2*

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

*x*min*∈IR** ^{n}*Ψ

_{α,θ,p}*(x).*(19)

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

*is a global minimum. Using Lemma 3.2(d) and the same proof techniques as in [21, Theorem 3.5], we can establish that each stationary point of Ψ*

_{α,θ,p}*α,θ,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 (19) if and only if x*^{∗}*is a stationary point of Ψ**α,θ,p**.*

**Theorem 3.3 The function Ψ**_{α,θ,p}*has bounded level sets* *L(Ψ**α,θ,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.*

**Proof. From [2], if F is a monotone function with a strictly feasible point, then the**
following condition holds: for every sequence *{x*^{k}*} such that ∥x*^{k}*∥ → ∞, (−x** ^{k}*)

_{+}

*<*

*∞, and (−F (x** ^{k}*))

_{+}

*<*

*∞, we have max*

1*≤i≤n*

{*(x*^{k}* _{i}*)

_{+}

*F*

_{i}*(x*

*)*

^{k}_{+}}

*→ ∞. Suppose that there*
*exists an unbounded sequence x*^{k}*⊆ L(Ψ**α,θ,p**, c) for some c* *∈ IR. Since Ψ**α,θ,p**(x** ^{k}*)

*≤ c,*

*there is no index i such that x*

^{k}

_{i}*→ −∞ or F*

*i*

*(x*

*)*

^{k}*→ −∞ by Lemma 3.2(f). Hence,*

1≤i≤nmax

{*(x*^{k}* _{i}*)

_{+}

*F*

_{i}*(x*

*)*

^{k}_{+}}

*→ ∞. Also, there is an index j, and at least a subsequence x*^{k}*j*

such that {

*(x*^{k}* _{j}*)

_{+}

*F*

_{j}*(x*

*)*

^{k}_{+}}

*→ ∞. However, this implies that Ψ**α,θ,p**(x** ^{k}*) is unbounded by
Lemma 3.2(f), contracting to the assumption on level sets. Another part of the proof is
similar to the proof of [5, Prop. 3.5].

*2*

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

_{NR}and Ψ

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

_{α,θ,p}**Lemma 3.3 Let ϕ*** _{θ,p}* : IR

^{2}

*→ IR be deﬁned as in (9). Then for any p > 1 and all*

*θ*

*∈ (0, 1] we have*

(2*− 2*^{1}* ^{p}*)

*| min{a, b}| ≤ |ϕ*

*θ,p*

*(a, b)| ≤ (2 + 2*

^{1}

*)*

^{p}*| min{a, b}|.*(20)

**Proof. Without loss of generality, we assume 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 need to discuss two subcases:*

(i)*|a|** ^{p}*+

*|b|*

^{p}*≥ |a − b|*

*. In this subcase, we have*

^{p}*|ϕ**θ,p**(a, b)| ≥ |*√^{p}

*θ(|a − b|** ^{p}*) + (1

*− θ)(|a − b|*

*)*

^{p}*− (a + b)|*

= *|*√^{p}

(*|a − b|** ^{p}*)

*− (a + b)|*

= *|(|a − b| − (a + b)|*

= *|a − b − (a + b)|*

= *|2b|*

= 2*| min{a, b}|*

*≥ (2 − 2*^{1}* ^{p}*)

*| min{a, b}|*(21)

On the other hand, since *|a|** ^{p}*+

*|b|*

^{p}*≥ |a − b|*

*and by [4, Lemma 3.2], we have*

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

*| min{a, b}|.*(22)

(ii) *|a|** ^{p}* +

*|b|*

^{p}*<|a − b|*

*. Since*

^{p}*|a|*

*+*

^{p}*|b|*

^{p}*<|a − b|*

*and by [4, Lemma 3.2], we have*

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

*| min{a, b}|.*(23) On the other hand, by the discussion of Case(1),

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

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

*Case(2): a + b > 0. If ab=0, then (20) clearly holds. Thus, we proceed the arguments by*discussing two subcases:

*(i) ab < 0. In this subcases, we have a > 0, b < 0,|a| > |b|. By Lemma 3.1, |a|** ^{p}*+

*|b|*

^{p}*≤*

*|a − b|** ^{p}*. Then,

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

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

On the other hand,

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

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

*(ii) ab > 0. In this subcases, we have a* *≥ b > 0, |a|** ^{p}*+

*|b|*

^{p}*≥ |a − b|*

*. By Lemma 3.1,*

^{p}*ϕ*

_{θ,p}*(a, b)*

*≤ ϕ*

*p*

*(a, b)≤ 0 . Notice that ϕ*

*θ,p*

*(a, b)≥ |a − b| − (a + b) = −2b = −2 min{a, b},*and hence we obtain that

*|ϕ**θ,p**(a, b)| ≤ 2| min{a, b}| ≤ (2 + 2*^{p}^{1})*| min{a, b}|.* (27)
*On the other hand, since ϕ*_{θ,p}*(a, b)* *≤ ϕ**p**(a, b)≤ 0 , and by [4, Lemma 3.2], and hence we*
obtain that

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

*| min{a, b}|.*(28) All the aforementioned inequalities (21)-(28) imply that (20) holds.

*2*

**Proposition 3.1 Let Ψ**_{θ,p}*, Ψ*_{NR} *and Ψ*_{α,θ,p}*be deﬁned as in (13), (12) and (10), respec-*
*tively. Let S be an arbitrary bounded set. Then, for any p > 1, we have*

(2*− 2*^{1}* ^{p}*)

^{2}Ψ

_{NR}

*(x)≤ Ψ*

*θ,p*

*(x)≤ (2 + 2*

^{1}

*)*

^{p}^{2}Ψ

_{NR}

*(x)*

*for all x∈ IR*

*(29)*

^{n}*and*

(2*− 2*^{1}* ^{p}*)

^{2}Ψ

_{NR}

*(x)≤ Ψ*

*α,θ,p*

*(x)≤ (αB*

^{2}+ (2 + 2

^{1}

*)*

^{p}^{2})Ψ

_{NR}

*(x)*

*for all x∈ S,*(30)

*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 (29) is direct by Lemma 3.3 and the deﬁnitions of Ψ*** _{θ,p}* and
Ψ

_{NR}. In addition, from Lemma 3.3 and the deﬁnition of Ψ

*, it follows that*

_{α,θ,p}Ψ_{α,θ,p}*(x)≥*(

2*− 2*^{1}* ^{p}*)2

Ψ_{NR}*(x)* *for all x∈ IR*^{n}*.*

It remains to prove the inequality on the right hand side of (30). From the proof of [4,
*Prop. 3.1], we know for each i,*

*(x**i**F**i**(x))*+*≤ B| min{x**i**, F**i**(x)}| for all x ∈ S.* (31)
*By Lemma 3.3 and (31), for all i = 1, . . . , n and x∈ S,*

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

{

*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 deﬁnitions of Ψ** _{α,θ,p}* and Ψ

_{NR}.

*2*

From Proposition 3.1, we immediately obtain the following result.

**Corollary 3.1 Let Ψ**_{θ,p}*and Ψ*_{α,θ,p}*be deﬁned by (13) and (10), 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,θ,p**(x)≤ Ψ**θ,p**(x)≤* (2 + 2^{1}* ^{p}*)

^{2}(2

*− 2*

^{1}

*)*

^{p}^{2}

Ψ_{α,θ,p}*(x)*

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

Since Ψ_{θ,p}*, Ψ*_{NR} and Ψ* _{α,θ,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,θ,p}*provides a global error bound without the Lipschitz continuity of F when α > 0.*

**Theorem 3.4 Let Ψ**_{α,θ,p}*be deﬁned as in (10). 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,θ,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,θ,p}*(x),*

√

Ψ_{α,θ,p}*(x)*
})^{1}

2

*for all x∈ S;*

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

**Proof. By the proof of [4, Theorem 3.4], we have**
*µ∥x − x*^{∗}*∥*^{2} *≤ max*

1*≤i≤n**τ*_{i}*{(x**i**F*_{i}*(x))*_{+}+ (*−F**i**(x))*_{+}+ (*−x**i*)_{+}*},* (32)
*where τ** _{i}* := max

*{1, x*

^{∗}*i*

*, F*

_{i}*(x*

*)*

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

^{2},

(*−a)*+2

+ (*−b)*+2 *≤ [ϕ**θ,p**(a, b)]*^{2}*.* (33)
*To see this, without loss of generality, we assume a≥ b and discuss three cases:*

*(i) If a≥ b ≥ 0, then (33) holds obviously.*

*(ii) If a* *≥ 0 ≥ b, then |a|** ^{p}* +

*|b|*

^{p}*≤ |a − b|*

^{p}*by Lemma 3.1, which implies ϕ*

_{θ,p}*(a, b)*

*≥*

*∥(a, b)∥**p**− (a + b) ≥ −b ≥ 0. Hence, (−a)*+

2+ (*−b)*+

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

+ (*−b)*+2

*= a*^{2}*+ b*^{2} *≤ [ϕ**θ,p**(a, b)]*^{2}. Hence, (33) follows.

*Suppose that α > 0. Using the inequality (33), 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}_{+}*+ [ϕ*_{θ,p}*(a, b)]*^{2}]

*≤ τ*
[*α*

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

2*[ϕ*_{θ,p}*(a, b)]*^{2}
]

*= τ ψ*_{α,θ,p}*(a, b),* (34)

*where τ := max*
{6

*α, 6*
}

*> 0. Combining (34) with (32) and letting ˆτ = max*

1*≤i≤n**τ** _{i}*, we get

*µ∥x − x*

^{∗}*∥*

^{2}

*≤ max*

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

*≤ ˆττ** ^{1/2}*max

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

*≤ ˆττ** ^{1/2}*
{

_{n}∑

*i=1*

*{ψ**α,θ,p**(x**i**, F**i**(x))*
}*1/2*

= *τ τ*ˆ * ^{1/2}*Ψ

_{α,θ,p}*(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 (31) holds.

*Combining with equations (32)–(33), 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)}| + 2(ψ**θ,p**(x*_{i}*, F*_{i}*(x)))** ^{1/2}*]

*≤ ˆτ max*

1≤i≤n

*[√*2 ˆ*B(ψ*_{θ,p}*(x*_{i}*, F*_{i}*(x)))*^{1/2}*+ 2(ψ*_{θ,p}*(x*_{i}*, F*_{i}*(x)))** ^{1/2}*
]

*≤ (√*

2 ˆ*B + 2)ˆτ (Ψ*_{θ,p}*(x))*^{1/2}

= (*√*

2 ˆ*B + 2)ˆτ (Ψ**α,θ,p**(x))*^{1/2}

*≤ (√*

2 ˆ*B + 2)ˆτ (max*
{

Ψ*α,θ,p**(x),*

√

Ψ*α,θ,p**(x)*
}

)
where ˆ*B = B/(2− 2*^{1}* ^{p}*), ˆ

*τ = max*

1*≤i≤n**τ** _{i}*and the second inequality is from Lemma 3.3. Letting

*κ*2 :=

[
(*√*

2 ˆ*B + 2)ˆτ /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 we suspect is useful in analysis of convergence rate.

**Lemma 3.4 For all (a, b)**̸= (0, 0) and p > 1, we have the following inequality:

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)2

*≥*(

2*− 2*^{p}^{1})2

*∀θ ∈ (0, 1].*

**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): Since θ* *∈ (0, 1] and p > 1, it follows that θ*^{1/p}*≤ 1. Now, by the proof of [4,*
Lemma 3.3], we have

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

*[θ(|a|*

*+*

^{p}*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}= *θ[|a|*^{p}* ^{−1}*+

*|b|*

^{p}*]*

^{−1}*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*≤* *θ[|a|*^{p}* ^{−1}*+

*|b|*

^{p}*]*

^{−1}*[θ(|a|*

*+*

^{p}*|b|*

*)]*

^{p}

^{(p}

^{−1)/p}= *θ** ^{1/p}*[

*|a|*

^{p}*+*

^{−1}*|b|*

^{p}*] [(*

^{−1}*|a|*

*+*

^{p}*|b|*

*)]*

^{p}

^{(p}

^{−1)/p}*≤ 2*^{1/p}*for p > 1.*

Therefore, 2*−* *θ[|a|*^{p}* ^{−1}*+

*|b|*

^{p}*]*

^{−1}*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*≥ 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 −* *θ[|a|*^{p}* ^{−1}*+

*|b|*

^{p}*]*

^{−1}*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*≤ 2 +* *θ[|a|*^{p}* ^{−1}*+

*|b|*

^{p}*]*

^{−1}*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*for p > 1,*from which the result follows immediately.

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

*|b|*

^{p}*]*

^{−1}*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*≥* *θ[|a|*^{p}^{−1}*− |b|*^{p}* ^{−1}*]

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*for p > 1.*

Thus, 2*−2*^{1}^{p}*≤ 2−* *θ[|a|*^{p}^{−1}*− |b|*^{p}* ^{−1}*]

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*for p > 1 and the desired result*is also satisﬁed.

*2*

**Proposition 3.2 Let ψ**_{α,θ,p}*be given as in (11). Then, for all x∈ IR*^{n}*and p > 1,*
*
∇**a**ψ*_{α,θ,p}*(x, F (x)) +∇**b**ψ*_{α,θ,p}*(x, F (x))*
^{2} *≥ 2*(

2*− 2*^{1}* ^{p}*)2

Ψ_{θ,p}*(x)* *∀θ ∈ (0, 1].*

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

^{4}

(

*αB*^{2}+ (2 + 2^{1}* ^{p}*)

^{2}

)Ψ*α,θ,p**(x)* *∀θ ∈ (0, 1],*

*where B is deﬁned as in Proposition 3.1 and*

*∇**a**ψ*_{α,θ,p}*(x, F (x)) :=*

(

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

*,*

*∇**b**ψ*_{α,θ,p}*(x, F (x)) :=*

(

*∇**b**ψ*_{α,θ,p}*(x*_{1}*, F*_{1}*(x)),* *· · · , ∇**b**ψ*_{α,θ,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. Thus, it remains to show the ﬁrst part. From the deﬁnitions of *∇**a**ψ*_{α,θ,p}*(x, F (x)),*

*∇**b**ψ*_{α,θ,p}*(x, F (x)) and Ψ*_{θ,p}*(x), showing the ﬁrst part is equivalent to proving that the*
following inequality

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

2*− 2*^{1}* ^{p}*)2

*ψ**θ,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 (14) that*

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

= {

*α(a + b)(ab)*+*+ (ϕ**θ,p**(a, b))*

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)}2

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

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

*[θ(|a|** ^{p}* +

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)2

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

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)

*.* (37)
*Now, we claim that for all (a, b)̸= (0, 0) ∈ IR*^{2},

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

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)

*≥ 0.* (38)
*If ab≤ 0, then (ab)*+ *= 0 and the inequality (38) is clear. If a, b > 0, then by the proof*
of Lemma 3.4, we have

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

*[θ(|a|** ^{p}*+

*|b|*

*) + (1*

^{p}*− θ)|a − b|*

*)]*

^{p}

^{(p}

^{−1)/p}*− 2*)

*≤ 0, ∀(a, b) ̸= (0, 0) ∈ IR*^{2} (39)

*and ϕ*_{θ,p}*(a, b)≤ 0, which imply the inequality (38) also holds. If a, b < 0, then ϕ**θ,p**(a, b)≥*
0, which together with (39) yields the inequality (38). Thus, we obtain that the inequality
*(38) holds for all (a, b)̸= (0, 0). Now using Lemma 3.4 and equations (37)–(38), we readily*
*obtain the inequality (36) holds for all (a, b)̸= (0, 0). The proof is thus complete.* *2*

**4** **Algorithm and Numerical Experiments**

In this section, we investigate a derivative free algorithm for complementarity problems based on the new family of NCP-function and its related merit function. In addition, we prove the global convergence of the algorithm.

**Algorithm 4.1 (A Derivative Free Algorithm)**

**Step 0 Given real numbers α > 0, p > 1, θ***∈ (0, 1] and x*^{0} *∈ IR*^{n}*. Choose σ∈ (0, 1) and*
*ρ, γ* *∈ (0, 1). Set k := 0.*

**Step 1 If Ψ**_{α,θ,p}*(x*^{k}*) = 0, stop, otherwise go to Step 2.*

**Step 2 Find the smallest nonnegative integer m**_{k}*such that*

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

*≤ (1 − σρ*

^{2m}*)Ψ*

^{k}

_{α,θ,p}*(x*

^{k}*),*(40)

*where d*

_{k}*(γ*

^{m}*) :=*

^{k}*−∂Ψ*

_{α,θ,p}*(x*

^{k}*, F (x*

*))*

^{k}*∂b* *− γ*^{m}^{k}*∂Ψ*_{α,θ,p}*(x*^{k}*, F (x** ^{k}*))

*∂a* *.*

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

**Proposition 4.1 Let x**^{k}*∈ IR*^{n}*and F be a monotone function. Then the search direction*
*deﬁned in Algorithm 4.1 satisﬁes the descent condition* *∇Ψ**α,θ,p**(x** ^{k}*)

^{T}*d*

_{k}*< 0 as long as x*

^{k}*is not a solution of the NCP. Moreover, if F is strongly monotone with modulus µ > 0,*

*then*

*∇Ψ*

*α,θ,p*

*(x*

*)*

^{k}

^{T}*d*

_{k}*<−µ∥d*

*k*

*∥*

^{2}

*.*

**Proof. The proof is similar to the one given in [5, Lemma 4.1].** *2*

**Proposition 4.2 Suppose that F is strongly monotone. Then the sequence***{x*^{k}*} gener-*
*ated by Algorithm 4.1 has at least one accumulation point and any accumulation point is*
*a solution of the NCP.*