* World Scientific Publishing Co. & Operational Research Society of Singapore*c

**ON SOME NCP-FUNCTIONS BASED ON THE GENERALIZED**
**FISCHER–BURMEISTER FUNCTION**

JEIN-SHAN CHEN
*Department of Mathematics*
*National Taiwan Normal University*

*Taipei, Taiwan 11677*
*jschen@math.ntnu.edu.tw*

Received 4 June 2005 Revised 18 November 2005 Second Revised 11 January 2006

In this paper, we study several NCP-functions for the nonlinear complementarity prob-
lem (NCP) which are indeed based on the generalized Fischer–Burmeister function,
*φ**p*(*a, b) = (a, b)**p**− (a + b). It is well known that the NCP can be reformulated as*
an equivalent unconstrained minimization by means of merit functions involving NCP-
functions. Thus, we aim to investigate some important properties of these NCP-functions
that will be used in solving and analyzing the reformulation of the NCP.

*Keywords: NCP-function; complementarity; merit function; bounded level sets;*

stationary point.

**1. Introduction**

The nonlinear complementarity problem (NCP) (Harker and Pang, 1990; Pang,
1994) is to ﬁnd a point*x ∈ R** ^{n}* such that

*x ≥ 0, F (x) ≥ 0, x, F (x) = 0,* (1.1)

where*·, · is the Euclidean inner product and F = (F*1*, F*2*, . . . , F** _{n}*)

^{T}maps fromR

*toR*

^{n}*. We assume that*

^{n}*F is continuously diﬀerentiable throughout this paper. The*NCP has attracted much attention due to its various applications in operations research, economics, and engineering (Ferris and Pang, 1997; Harker and Pang, 1990; Pang, 1994).

There have been many methods proposed for solving the NCP (Harker and Pang, 1990; Pang, 1994). Among which, one of the most popular and powerful approaches that has been studied intensively recently is to reformulate the NCP as a system of nonlinear equations (Mangasario 1976) or as an unconstrained minimization prob- lem (Facchinei and Soares, 1997; Fisher, 1992; Kanzow, 1996). Such a function that can constitute an equivalent unconstrained minimization problem for the NCP is

401

*called a merit function. In other words, a merit function is a function whose global*
minima are coincident with the solutions of the original NCP. For constructing a
merit function, the class of functions, so-called NCP-functions and deﬁned as below,
serves an important role.

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

*φ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0.* (1.2)
Many NCP-functions and merit functions have been explored during the past two
*decades (De Luca et al., 1996; Kanzow et al., 1997; San and Qi, 1999; Tseng,*
1996). Among which, a popular NCP-function intensively studied recently is the
well-known Fischer–Burmeister NCP-function (Fisher, 1992, 1997) deﬁned as

*φ*_{FB}(*a, b) =*

*a*^{2}+*b*^{2}*− (a + b).* (1.3)

With the above characterization of *φ*_{FB}, the NCP is equivalent to a system of
nonsmooth equations:

Φ_{FB}(*x) =*

*φ*_{FB}(*x*1 *, F*1(*x))*
...
*φ*_{FB}(*x**n* *, F**n*(*x))*

* = 0.* (1.4)

For each NCP-function, there is a natural merit function, Ψ_{FB} :R^{n}*→ R*+ given by
Ψ_{FB}(*x) :=* 1

2*Φ*FB(*x)*^{2}=1
2

*n*
*i=1*

*φ*FB(*x**i**, F**i*(*x))*^{2}*,* (1.5)

from which the NCP can be recast as an unconstrained minimization:

*x∈R*min* ^{n}*Ψ

_{FB}(

*x).*(1.6)

In this paper, we are particularly interested in the generalized Fischer–

Burmeister function, i.e.,*φ** _{p}*:R

^{2}

*→ R given by*

*φ**p*(*a, b) := (a, b)**p**− (a + b),* (1.7)
where *p is a positive integer greater than one and (a, b)** _{p}* =

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

*|b|*

*means the*

^{p}*p-norm of (a, b). Notice that φ*

*p*reduces to the well known Fischer–Burmeister function

*φ*

_{FB}when

*p = 2 and its related properties were recently presented in (Chen*and Pan, 2006; Chen, 2006). Corresponding to

*φ*

*, we deﬁne*

_{p}*ψ*

*:R*

_{p}^{2}

*→ R*+ by

*ψ**p*(*a, b) :=*1

2*|φ**p*(*a, b)|*^{2}*.* (1.8)

Then both*φ** _{p}* and

*ψ*

*are NCP-functions and yield a merit function Ψ*

_{p}*(*

_{p}*x) :=*

^{n}*i=1*

*ψ** _{p}*(

*x*

_{i}*, F*

*(*

_{i}*x)) =*1 2

*n*
*i=1*

*φ** _{p}*(

*x*

_{i}*, F*

*(*

_{i}*x))*

^{2}

*,*(1.9)

from which the NCP can be reformulated as an unconstrained minimization:

*x∈R*min* ^{n}*Ψ

*(*

_{p}*x).*(1.10)

However, there has some limitations for the (generalized) Fischer–Burmeister func-
tions and some of its variants when dealing with monotone complementarity prob-
lem. In particular, its natural merit function Ψ* _{p}* does not guarantee bounded level

*sets for this class of problem which is an important class (see page 4 of Chen et al.,*2000). Some modiﬁcations to the Fischer–Burmeister have been proposed to con-

*quer the above problem, see (Kanzow et al., 1997; Sun and Qi, 1999). In this paper,*we extend these modiﬁcations to the generalized Fischer–Burmeister function

*φ*

*p*. More speciﬁcally, we study the following NCP-functions:

*φ*1(*a, b) := φ** _{p}*(

*a, b) − αa*+

*b*+

*,*

*α > 0,*

*φ*2(

*a, b) := φ*

*p*(

*a, b) − α(ab)*+

*,*

*α > 0,*

*φ*3(

*a, b) :=*

[*φ**p*(*a, b)]*^{2}+*α(a*+*b*+)^{2}*, α > 0,*
*φ*4(*a, b) :=*

[*φ**p*(*a, b)]*^{2}+*α[(ab)*+]^{2}*, α > 0,*

(1.11)

The function *φ*1 is called penalized Fischer–Burmeister function when *p = 2*
*and was studied in (Chen et al., 2000). The functionsφ*2*, φ*3*, φ*4generalize the merit
functions of*p = 2, which were discussed in Sun and Qi (1999) and Yamada et al.*

(2000). Note that for*i = 1, 2, 3, 4, we have*

*φ**i*(*a, b) ≡ φ**p*(*a, b)* (1.12)

for all (*a, b) ∈ N** _{−}* (this notation is used in Sun and Qi, 1999) where

*N**−*:=*{(a, b)| ab ≤ 0}.* (1.13)

Thus,*φ**i* where*i = 1, 2, 3, 4 are only diﬀerent in the ﬁrst or third quadrant.*

Similarly, for each*φ** _{i}* there is an associated

*ψ*

*:R*

_{i}^{2}

*→ R*+ given by

*ψ*

*i*(

*a, b) :=*1

2*|φ**i*(*a, b)|*^{2}*, i = 1, 2, 3, 4,* (1.14)
which is also an NCP-function for every *i. Moreover, for φ ∈ {φ*1*, φ*2*, φ*3*, φ*4*}, we*
can deﬁne

Φ(*x) =*

*φ(x*1 *, F*1(*x))*
...
*φ(x**n* *, F**n*(*x))*

* ,* (1.15)

from which the NCP is equivalent to the unconstrained minimization:

*x∈R*min* ^{n}*Ψ(

*x)*(1.16)

where

Ψ(*x) :=* 1

2*Φ(x)*^{2}= 1
2

*n*
*i=1*

*φ(x**i* *, F**i*(*x))*^{2} (1.17)

is the natural merit function corresponding to*φ ∈ {φ*1*, φ*2*, φ*3*, φ*4*}.*

The paper is organized as follows. In Section 2, we review some background
deﬁnitions including monotonicity, *P*0-function, semismoothness, etc. and known
results about Ψ* _{p}* and its related properties. In Section 3, we show that all
(

*φ*

*i*)

^{2}

*, i ∈ {1, 2, 3, 4} are continuously diﬀerentiable and investigate properties of*the merit function Ψ constructed via

*φ*

*with*

_{i}*i ∈ {1, 2, 3, 4}. In particular, it pro-*vides bounded level sets for a monotone NCP with a strictly feasible point. In addition, we give conditions under which a stationary point of Ψ is a solution of the NCP. In general, the analytic techniques used in this paper are similar to

*those in Chen et al. (2000), Ficchinei and Soares (1997), Sun and Qi (1999) since*the work is somewhat considered the extensions of NCP-functions studied in those literatures.

Throughout this paper,R* ^{n}* denotes the space of

*n-dimensional real column vec-*tors and

*denotes transpose. For any diﬀerentiable function*

^{T}*f : R*

^{n}*→ R, ∇f(x)*denotes the gradient of

*f at x. For any diﬀerentiable mapping F = (F*1

*, . . . , F*

*m*)

*: R*

^{T}

^{n}*→ R*

*,*

^{m}*∇F (x) = [∇F*1(

*x) · · · ∇F*

*(*

_{m}*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. In this*whole paper, we assume

*p is a positive integer greater than one.*

**2. Preliminaries**

In this section, we recall some background concepts and materials which will play
an important role in the subsequent analysis. Let*F : R*^{n}*→ R** ^{n}*. Then,

(1) *F is monotone if x − y, F (x) − F (y) ≥ 0, for all x, y ∈ R** ^{n}*.

(2) *F is strictly monotone if x − y, F (x) − F (y) > 0, for all x, y ∈ R** ^{n}* and

*x = y.*

(3) *F is strongly monotone with modulus µ > 0 if x−y, F (x)−F (y) ≥ µx−y*^{2},
for all*x, y ∈ R** ^{n}*.

(4) *F is a P*0-function if max_{1 ≤ i ≤ n}

*x**i**= y**i*

(*x**i**− y**i*)(*F**i*(*x) − F**i*(*y)) ≥ 0, for all x, y ∈ R** ^{n}*
and

*x = y.*

(5) *F is a P -function if max**1≤i≤n*(*x**i**− y**i*)(*F**i*(*x) − F**i*(*y)) > 0, for all x, y ∈ R** ^{n}*and

*x = y.*

(6) *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)) ≥ µx − y*

^{2}, for all

*x, y ∈ R*

*. (7)*

^{n}*F is a R*0-function if for every sequence

*{x*

^{k}*} satisfying {x*

^{k}*} → ∞, lim inf*

*k→∞*min

_{i}*x*

^{k}

_{i}*x*^{k}*≥ 0, and*
lim inf_{k→∞}^{min}_{x}^{i}^{F}^{i}_{k}^{(x}^{k}^{)} *≥ 0, there exists an index j such that {x*^{k}_{j}*} → ∞ and*
*{F** _{j}*(

*x*

*)*

^{k}*} → ∞.*

It is clear that strongly monotone functions are strictly monotone, and strictly
monotone functions are monotone. Moreover, *F is a P*0-function if *F is mono-*
tone and *F is a uniform P -function with modulus µ > 0 if F is strongly mono-*
tone with modulus *µ > 0. In addition, when F is continuously diﬀerentiable, we*
have the following: (i)*F is monotone if and only if ∇F (x) is positive semi-deﬁnite*
for all *x ∈ R** ^{n}*. (ii)

*F is strictly monotone if ∇F (x) is positive deﬁnite for all*

*x ∈ R** ^{n}*. (iii)

*F is strongly monotone if and only if ∇F (x) is uniformly positive*deﬁnite. An

*R*0-function can be viewed as a generalization of a uniform

*P -function*since every uniform

*P -function is an R*0-function (see, Chen and Harker, 1997, Proposition 3.11).

A matrix*M ∈ R** ^{n×n}*is a

*P*0-matrix if every of its principal minors is nonnegative, and it is a

*P -matrix if every of its principal minors is positive. In addition, it is said*to be a

*R*0-matrix if the following system has only zero solution:

*x ≥ 0,*

*M*_{i}*x = 0* if *x*_{i}*> 0,*
*M**i**x ≥ 0* if *x**i*= 0*,*

It is obvious that every *P -matrix is also a P*0-matrix and it is known that the
Jacobian of every continuously diﬀerentiable*P*0-function is a*P*0-matrix. For more
properties about*P -matrix and P*0-matrix, please refer to Facchinei and Pang (2003).

It is also known that*F is an R*0-function if and only if*M is an R*0-matrix when*F*
is an aﬃne function (see, Chen and Harker, 1997, Proposition 3.10).

Next, we recall the deﬁnition of semismoothness. First, we introduce that
*F is strictly continuous (also called “locally Lipschitz continuous”) at x ∈ R** ^{n}*
(Rockafellar and Wets, 1998, Chapter 9) if there exist scalars

*κ > 0 and δ > 0 such*that

*F (y) − F (z) ≤ κy − z ∀y, z ∈ R** ^{n}* with

*y − x ≤ δ, z − x ≤ δ;*

and*F is strictly continuous if F is strictly continuous at every x ∈ R** ^{n}*. If

*δ can be*taken to be

*∞, then F is Lipschitz continuous with Lipschitz constant κ. We say*

*F is directionally diﬀerentiable at x ∈ R*

*if*

^{n}*F** ^{}*(

*x; h) := lim*

*t→0*^{+}

*F (x + th) − F (x)*

*t* exists*∀h ∈ R** ^{n}*;

and *F is directionally diﬀerentiable if F is directionally diﬀerentiable at every*
*x ∈ R** ^{n}*.

Assume *F : R*^{n}*→ R** ^{m}* is strictly continuous. We say

*F is semismooth at x if*

*F is directionally diﬀerentiable at x and, for any V ∈ ∂F (x + h) (the generalized*Jacobian), we have

*F (x + h) − F (x) − V h = o(h).*

We say*F is ρ-order semismooth at x (0 < ρ < ∞) if F is semismooth at x and, for*
any*V ∈ ∂F (x + h), we have*

*F (x + h) − F (x) − V h = O(h** ^{1+ρ}*)

*.*

We say *F is semismooth (respectively, ρ-order semismooth) if F is semismooth*
(respectively, *ρ-order semismooth) at every x ∈ R** ^{k}*. We say

*F is strongly semis-*

*mooth if it is 1-order semismooth. Convex functions and piecewise continuously dif-*ferentiable functions are examples of semismooth functions. Examples of strongly

semismooth functions include piecewise linear functions and*LC*^{1}*functions meaning*
smooth functions with gradients being locally Lipschitz continuous (strictly contin-
uous) (Facchinei and Soares, 2003; Qi, 1994). The composition of two (respectively,
*ρ-order) semismooth functions is also a (respectively, ρ-order) semismooth function.*

The property of semismoothness plays an important role in nonsmooth Newton methods (Qi, 1993; Qi and Sun, 1993) as well as in some smoothing methods mentioned in the Section 1. For extensive discussions of semismooth functions, see Fischer (1997), Miﬄin (1977), and Qi and Sun (1993).

To end this section, we collect some useful properties of*φ**p**, ψ**p*deﬁned as in (1.7)
and (1.8), respectively, that will be used in the subsequent analysis. All the proofs
can be found in Chen and Pan (2006).

**Property 2.1** *(Chen and Pan, 2006, Proposition 3.1, Lemma 3.1). Letφ**p* :R^{2}*→*
*R be deﬁned as (1.7). Then*

(a) *φ*_{p}*is an NCP-function, i.e. it satisﬁes (1.2).*

(b) *φ**p* *is sub-additive, i.e.φ**p*(*w + w** ^{}*)

*≤ φ*

*p*(

*w) + φ(w*

^{}*) for allw, w*

^{}*∈ R*

^{2}. (c)

*φ*

*p*

*is positive homogeneous, i.e.φ*

*p*(

*αw) = αφ*

*p*(

*w) for all w ∈ R*

^{2}

*andα ≥ 0.*

(d) *φ*_{p}*is convex, i.e. φ** _{p}*(

*αw+(1−α)w*

*)*

^{}*≤ αφ*

*(*

_{p}*w)+(1−α)φ*

*(*

_{p}*w*

^{}*) for allw, w*

^{}*∈ R*

^{2}

*andα ≥ 0.*

(e) *φ*_{p}*is Lipschitz continuous withL*1= 1 +*√*

*2, i.e.|φ** _{p}*(

*w)−φ*

*(*

_{p}*w*

*)*

^{}*| ≤ L*1

*w−w*

^{}*;*

*or withL*2= 1+2^{(1−1/p)}*, i.e.|φ**p*(*w)−φ**p*(*w** ^{}*)

*| ≤ L*2

*w−w*

^{}*p*

*for allw, w*

^{}*∈ R*

^{2}. (f)

*φ*

_{p}*is semismooth.*

*(g) If{(a*^{k}*, b** ^{k}*)

*} ⊆ R*

^{2}

*with (a*

^{k}*→ −∞) or (b*

^{k}*→ −∞) or (a*

^{k}*→ ∞ and b*

^{k}*→ ∞),*

*then we have*

*|φ*

*(*

_{p}*a*

^{k}*, b*

*)*

^{k}*| → ∞ for k → ∞.*

**Property 2.2** *(Chen and Pan, 2006, Proposition 3.2). Let* *φ*_{p}*, ψ*_{p}*be deﬁned as*
(1*.7) and (1.8), respectively. Then*

(a) *ψ**p* *is an NCP-function, i.e. it satisﬁes (1.2).*

(b) *ψ**p*(*a, b) ≥ 0 for all (a, b) ∈ R*^{2}.

(c) *ψ*_{p}*is continuously diﬀerentiable everywhere.*

(d) *∇**a**ψ**p*(*a, b)·∇**b**ψ**p*(*a, b) ≥ 0 for all (a, b) ∈ R*^{2}*. The equality holds⇔ φ**p*(*a, b) = 0.*

(e) *∇*_{a}*ψ** _{p}*(

*a, b) = 0 ⇔ ∇*

_{b}*ψ*

*(*

_{p}*a, b) = 0 ⇔ φ*

*(*

_{p}*a, b) = 0.*

From these properties, it was proved in Chen and Pan (2006) that Ψ* _{p}*(

*x) ≥ 0 for*all

*x ∈ R*

*and Ψ*

^{n}*(*

_{p}*x) = 0 if and only if x solves the NCP (1.1), where Ψ*

*p*:R

^{n}*→ R*is deﬁned as (1.9). Moreover, suppose that the NCP has at least one solution. Then

*x is a global minimizer of Ψ*

*if and only if*

_{p}*x solves the NCP. In addition, it was*also shown in Chen and Pan (2006) that if

*F is either monotone or P*0-function, then every stationary point of Ψ

*is a global minima of (1.10); and therefore solves the original NCP. We will investigate the analogous results for the merit func- tion Ψ which is based on*

_{p}*φ*

*studied in this paper. On the other hand, as men- tioned the natural merit function induced from the generalized Fischer–Burmeister*

_{i}(which behaves like the Fischer–Burmeister function) does not guarantee bounded
level sets under the assumption of *F being monotone. Instead, there needs that F*
is strongly monotone or uniform *P -function to ensure that the property is held.*

Another main purpose of this work is to obtain same results for the merit function
Ψ studied in this paper under the weaker assumption that *F is monotone only*
(see Section 4).

**3. Properties of** **φ and ψ**

In this section, we investigate properties of *φ ∈ {φ*1*, φ*2*, φ*3*, φ*4*} and ψ ∈*
*{ψ*1*, ψ*2*, ψ*3*, ψ*4*} deﬁned as in (1.11) and (1.14), respectively. These include strong*
semismoothness of*φ and continuous diﬀerentiability of ψ. First, we denote*

*N**φ*:=*{(a, b)| a ≥ 0, b ≥ 0, ab = 0}.* (3.1)
*This notation is adopted from Chen et al. (2000) and it is easy to see that (a, b) ∈ N** _{φ}*
if and only if (

*a, b) satisﬁes (1.2). Now we are ready to show the favorable properties*of

*φ and ψ.*

**Proposition 3.1.** *Letφ ∈ {φ*1*, φ*2*, φ*3*, φ*4*} be deﬁned as in (1.11). Then*
(a) *φ(a, b) = 0 ⇔ (a, b) ∈ N**φ**⇔ (a, b) satisﬁes (1.2).*

(b) *φ is strongly semismooth.*

*(c) Let* *{a*^{k}*}, {b*^{k}*} ⊆ R be any two sequences such that either a** ^{k}*+

*b*

*+*

^{k}*→ ∞ or a*

^{k}*→*

*−∞ or b*^{k}*→ −∞. Then |φ(a*^{k}*, b** ^{k}*)

*| → ∞ for k → ∞.*

**Proof. (a) It is enough to prove the ﬁrst equivalence. Suppose***φ(a, b) = 0, for i =*
2*, 3, 4, φ**i*(*a, b) = 0 yields φ**p*(*a, b) = 0 which says (a, b) ∈ N**φ*by Property 2.1(a). For
*i = 1, φ*1(*a, b) = 0 implies φ** _{p}*(

*a, b) = αa*+

*b*+. Since

*α could be any arbitrary positive*number, the above leads to

*φ*

*p*(

*a, b) = a*+

*b*+ = 0 which which says (

*a, b) ∈ N*

*φ*by Property 2.1(a) again. On the other hand, suppose (

*a, b) ∈ N*

*φ*then

*φ*

*p*(

*a, b) = 0 by*by Property 2.1(a). Since

*a ≥ 0, b ≥ 0, we obtain a*+

*b*+=

*ab = 0. Hence we see that*all

*φ*

*i*(

*a, b) = 0, i = 1, 2, 3, 4.*

(b) The veriﬁcation of strong semismoothness of*φ is a routine work which can*
*be done as in Yamada et al. (2000) of Lemma 1. We omit it.*

(c) This follows from Property 2.1(g) and deﬁnition of (*·)*+.

**Proposition 3.2.** *Let Φ be deﬁned as in (1.15) with φ ∈ {φ*1*, φ*2*, φ*3*, φ*4*}. Then*
*(a) Φ is semismooth.*

*(b) Φ is strongly semismooth if* *F**i* *isLC*^{1} *function.*

**Proof. By using Proposition 3.1(b) and the fact that every***LC*^{1}function is strongly
semismooth, the results follow.

The following is a technical lemma which describes the generalized gradients of
all*φ**i**, i = 1, 2, 3, 4 deﬁned as in Eq. (1.11). It will be used for proving Propotion 3.3.*

**Lemma 3.1.** *Letφ*1*, φ*2*, φ*3*, φ*4 *be deﬁned as (1.11).*

*(a) The generalized gradient* *∂φ*1(*a, b) of φ*1 *at a point (a, b) is equal to the set of*
*all (v**a**, v**b**) such that*

(*v*_{a}*, v** _{b}*) =

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1,* *b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*− α(b*+*∂a*+*, a*+*∂b*+)*,*
*if (a, b) = (0, 0) and p is even,*

sgn(*a) · a*^{p−1}

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

*(a, b)*^{p−1}*p* *− 1*

*− α(b*+*∂a*+*, a*+*∂b*+)*,*
*if (a, b) = (0, 0) and p is odd,*
(*ξ − 1, ζ − 1),* *if (a, b) = (0, 0),*

(3.2)

*where (ξ, ζ) is any vector satisfying (ξ, ζ)**p* *≤ 1 and*

*∂z*+=

1*,* *if* *z > 0,*
[0*, 1], if z = 0,*
0*,* *if* *z < 0.*

*(b) The generalized gradient* *∂φ*2(*a, b) of φ*2 *at a point (a, b) is equal to the set of*
*all (v*_{a}*, v*_{b}*) such that*

(*v*_{a}*, v** _{b}*) =

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1,* *b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*− α(b, a),*

*if (a, b) = (0, 0), ab > 0 and p is even,*

(*v*_{a}*, v** _{b}*) =

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1,* *b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*− α(b, a) · [0, 1],*

*if (a, b) = (0, 0), ab = 0 and p is even,*

(*v**a**, v**b*) =

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1,* *b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*,*

*if (a, b) = (0, 0), ab < 0 and p is even,*

(*v**a**, v**b*) =

sgn(*a) · a*^{p−1}

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

*(a, b)*^{p−1}*p* *− 1*

*− α(b, a),*

*if (a, b) = (0, 0), ab > 0 and p is odd,* (3.3)
(*v*_{a}*, v** _{b}*) =

sgn(*a) · a*^{p−1}

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

*(a, b)*^{p−1}*p* *− 1*

*− α(b, a) · [0, 1],*

*if (a, b) = (0, 0), ab = 0 and p is odd,*

(*v**a**, v**b*) =

sgn(*a) · a*^{p−1}

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

*(a, b)*^{p−1}*p* *− 1*

*,*

*if (a, b) = (0, 0), ab < 0 and p is odd,*
(*v**a**, v**b*) = (*ξ − 1, ζ − 1) − α(b, a) · [0, 1],*

*if (a, b) = (0, 0),*

*where (ξ, ζ) is any vector satisfying (ξ, ζ)**p**≤ 1.*

(c) *φ*3 *is continuously diﬀerentiable everywhere except at (0, 0) with*

*∇*_{a}*φ*3(*a, b) =*

*φ** _{p}*(

*a, b) ·*

*a*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(a*+)(*b*+)^{2}

*φ*3(*a, b)* *,*

*if (a, b) = (0, 0), and p is even,*
*φ**p*(*a, b) ·*

*sgn(a)·a*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(a*+)(*b*+)^{2}

*φ*3(*a, b)* *,*

*if (a, b) = (0, 0), and p is odd,*

(3.4)

*∇*_{b}*φ*3(*a, b) =*

*φ** _{p}*(

*a, b) ·*

*b*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(a*+)^{2}(*b*+)

*φ*3(*a, b)* *,*

*if (a, b) = (0, 0), and p is even,*
*φ**p*(*a, b) ·*

*sgn(b)·b*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(a*+)^{2}(*b*+)

*φ*3(*a, b)* *,*

*if (a, b) = (0, 0), and p is odd,*

(3.5)

*and∂φ*3(0*, 0) = (v*_{a}*, v*_{b}*) where (v*_{a}*, v** _{b}*)

*∈ (−∞, ∞).*

(d) *φ*4 *is continuously diﬀerentiable everywhere except at (0, 0) with*

*∇**a**φ*4(*a, b) =*

*φ** _{p}*(

*a, b) ·*

*a*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(ab)*+*· b*

*φ*4(*a, b)* *,*

*if (a, b) = (0, 0), and p is even,*
*φ**p*(*a, b) ·*

*sgn(a)·a*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(ab)*+*· b*

*φ*4(*a, b)* *,*

*if (a, b) = (0, 0), and p is odd,*

(3.6)

*∇**b**φ*4(*a, b) =*

*φ**p*(*a, b) ·*

*b*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(ab)*+*· a*

*φ*4(*a, b)* *,*

*if (a, b) = (0, 0), and p is even,*
*φ** _{p}*(

*a, b) ·*

*sgn(b)·b*^{p−1}

*(a,b)*^{p−1}*p* *− 1*

+*α(ab)*+*· a*

*φ*4(*a, b)* *,*

*if (a, b) = (0, 0), and p is odd,*

(3.7)

*and* *∂φ*4(0*, 0) = (v*_{a}*, v*_{b}*) where (v*_{a}*, v** _{b}*)

*∈ (−∞, ∞).*

**Proof. (a) First, we note that***φ**p* is continuously diﬀerentiable everywhere except
at (0*, 0) (see Chen and Pan, 2006). Hence, by the Corollary to Proposition 2.2.1*
in Clarke (1983), *φ**p* is strictly diﬀerentiable everywhere except at the origin. Let
*φ*+(*a, b) := a*+*b*+. Then*φ*+ is strictly diﬀerentiable at the origin as proved in Chen
*et al. (2000) of Proposition 2.1. Bothφ*1and*φ*+are strongly semismooth functions,
we know that they are locally Lipschitz (strictly continuous) functions. Thus, the
Corollary 2 to Proposition 2.3.3 in Clarke (1983) yields

*∂φ*1(*a, b) = ∂φ** _{p}*(

*a, b) − α · ∂φ*+(

*a, b).*

On the other hand, the generalized gradient of *φ** _{p}* can be veriﬁed as below (see
Chen, 2004):

*∂φ**p*(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1,* *b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*, if (a, b) = (0, 0) and p is even,*

*∂φ**p*(*a, b) =*

sgn(*a) · a*^{p−1}

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

*(a, b)*^{p−1}*p* *− 1*

*, if (a, b) = (0, 0) and p is odd,*

*∂φ** _{p}*(

*a, b) = (ξ − 1, ζ − 1), if (a, b) = (0, 0),*(3.8) where (

*ξ, ζ) is any vector satisfying (ξ, ζ)*

_{p}*≤ 1. In addition, it was already shown*

*in Chen et al. (2000) Proposition 2.1 that*

*∂φ*+(*a, b) = (b*+*∂a*+*, a*+*∂b*+)*.*
Thus, the desired results follow.

(b) Following the same arguments as in part(a) and using the fact that

*∂(ab)*+=

(*b, a),* if *ab > 0,*
(0*, 0),* if *ab < 0,*
(*b, a) · [0, 1], if ab = 0,*
the desired results hold.

(c) It is known that (*φ**p*)^{2}and (*a*+*b*+)^{2}are continuously diﬀerentiable. Then the
desired results follow by direct computations using the chain rule and the fact that

*∂(√*
*z) =*

1
2*√*

*z,* if *z > 0,*
(*−∞, ∞), if z = 0.*

(d) Same arguments as part(c).

**Proposition 3.3.** *Letψ ∈ {ψ*1*, ψ*2*, ψ*3*, ψ*4*} be deﬁned as in (1.14). Then*
(a) *ψ(a, b) = 0 ⇔ (a, b) ∈ N*_{φ}*⇔ (a, b) satisﬁes (1.2).*

(b) *ψ is continuously diﬀerentiable on R*^{2}*.*
(c) *∇*_{a}*ψ(a, b) · ∇*_{b}*ψ(a, b) ≥ 0 for all (a, b) ∈ R*^{2}*.*

(d) *ψ(a, b) = 0 ⇔ ∇ψ(a, b) = 0 ⇔ ∇**a**ψ(a, b) = 0 ⇔ ∇**b**ψ(a, b) = 0.*

**Proof. (a) The** proof is straightforward by the same arguments as in
Propositon 3.1(a).

(b) The ideas for the proof are indeed borrowed from Facchinei and Soares (1997) of Propositon 3.4.

For*i = 1 and p is even, ψ*1(*a, b) =* ^{1}_{2}(*φ*1(*a, b))*^{2}. By the chain rule (see Clarke,
1983, Theorem 2.2.4) we obtain*∂ψ*1(*a, b) = ∂φ*1(*a, b)*^{T}*φ*1(*a, b). We will show that*

*∂φ*1(*a, b)*^{T}*φ*1(*a, b) is single-valued for all (a, b) ∈ R*^{2} because the zero of *φ*1 cancels
the multi-valued portion of*∂φ*1(*a, b)** ^{T}*. To see this, we discuss several cases as below.

(i) If *a > 0, b > 0, then (b*+*∂a*+*, a*+*∂b*+) = (*b, a) which is single-valued. Hence,*
by (3.2), it is easy to see that *∂φ*1(*a, b)*^{T}*φ*1(*a, b) is single-valued.*

(ii) If *a > 0, b < 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, a) which is single-valued. Hence,*
by (3.2),*∂φ*1(*a, b)*^{T}*φ*1(*a, b) is single-valued.*

(iii) If *a > 0, b = 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, a · [0, 1]) which is multi-valued.*

However, under this case, we observe that *φ*1(*a, b) = (a, b)*_{p}*− (a + b) −*
*αa*+*b*+= 0. Hence,*∂φ*1(*a, b)*^{T}*φ*1(*a, b) is still single-valued.*

(iv) If *a < 0, b > 0 or a < 0, b < 0, or a < 0, b = 0, then (b*+*∂a*+*, a*+*∂b*+)
all equals (0*, 0) which is single-valued. Hence, by (3.2), ∂φ*1(*a, b)*^{T}*φ*1(*a, b) is*
single-valued.

(v) If *a = 0, b > 0, then (b*+*∂a*+*, a*+*∂b*+) = (*b · [0, 1], 0) which is multi-valued.*

However, under this case, we observe that *φ*1(*a, b) = (a, b)**p* *− (a + b) −*
*αa*+*b*+= 0. Hence,*∂φ*1(*a, b)*^{T}*φ*1(*a, b) is still single-valued.*

(vi) If *a = 0, b < 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, 0) which is single-valued. Hence,*
by (3.2),*∂φ*1(*a, b)*^{T}*φ*1(*a, b) is single-valued.*

(vii) If *a = 0, b = 0 then φ*1(*a, b) = 0. Hence, ∂φ*1(*a, b)*^{T}*φ*1(*a, b) is single-valued.*

Thus, by applying the corollary to Theorem 2.2.4 in Clarke (1983), the above
facts yield that *ψ*1 is continuously diﬀerentiable everywhere. For *p is odd,*
going over the same cases, the proof follows.

For *i = 2, ψ*2(*a, b) =* ^{1}_{2}(*φ*2(*a, b))*^{2}. We discuss the following cases: (i) (*a, b) =*
(0*, 0) and ab > 0, (ii) (a, b) = (0, 0) and ab = 0, (iii) (a, b) = (0, 0) and ab < 0,*
(iv) (*a, b) = (0, 0). From (3.3), we know that ∂φ*2(*a, b) becomes multi-valued*
when *ab = 0 or (a, b) = (0, 0). However, φ*2(*a, b) = 0 under these two cases*
which implies that *∂φ*2(*a, b)*^{T}*φ*2(*a, b) is still single-valued. Hence, ψ*2 is continu-
ously diﬀerentiable everywhere by the Corollary to Theorem 2.2.4 in Clarke (1983)
again.

For *i = 3, 4, from (3.4)–(3.8), it is trivial that ∂φ*3(*a, b), ∂φ*4(*a, b) are single-*
valued when (*a, b) = (0, 0). When (a, b) = (0, 0), we observe that φ*3(*a, b) =*
*φ*4(*a, b) = 0. Hence, ∂φ*3(*a, b)*^{T}*φ*3(*a, b) and ∂φ*4(*a, b)*^{T}*φ*4(*a, b) are still single-valued,*
which yield that*ψ*3*, ψ*4are continuously diﬀerentiable everywhere by the same rea-
son as above.

(c) For *i = 1, ψ*1 = ^{1}_{2}(*φ*1)^{2}, we employ and go over the cases discussed as in
part (b).

(i) If *a > 0, b > 0, then (b*+*∂a*+*, a*+*∂b*+) = (*b, a). Hence, from (3.3), we*
obtain that

*∇*_{a}*ψ*1(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αb*

*φ*1(*a, b),*

*∇**b**ψ*1(*a, b) =*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*1(*a, b),*

for both*p are even and odd. Then, ∇**a**ψ*1(*a, b) · ∇**b**ψ*1(*a, b) equals*
*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αb*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*^{2}1(*a, b).*

Since,_{(a,b)}^{a}^{p−1}*p−1*
*p*

* ≤ 1,*_{(a,b)}^{b}^{p−1}*p−1*
*p*

* ≤ 1, and αa > 0, αb > 0, we know*
*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αb*

*< 0 and*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*< 0,*

which implies that*∇**a**ψ*1(*a, b) · ∇**b**ψ*1(*a, b) ≥ 0.*

(ii) If*a > 0, b < 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, a). Hence, from (3.2), we have*

*∇**a**ψ*1(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

*∇**b**ψ*1(*a, b) =*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*1(*a, b),*

for*p is even; and*

*∇**a**ψ*1(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

*∇**b**ψ*1(*a, b) =*

*−b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*1(*a, b),*

for*p is odd. Again, since* _{(a,b)}^{a}^{p−1}*p−1*
*p*

* ≤ 1,*_{(a,b)}^{b}^{p−1}*p−1*
*p*

* ≤ 1, and αa > 0, it can*
be easily veriﬁed that *∇*_{a}*ψ*1(*a, b) · ∇*_{b}*ψ*1(*a, b) ≥ 0.*

(iii) If *a > 0, b = 0, then φ*1(*a, b) = 0 which says ∇**a**ψ*1(*a, b) = 0 = ∇**b**ψ*1(*a, b).*

Hence,*∇*_{a}*ψ*1(*a, b) · ∇*_{b}*ψ*1(*a, b) = 0.*

(iv) If*a < 0, b > 0 or a < 0, b < 0, or a < 0, b = 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, 0).*

Hence, from (3.3), we have

*∇*_{a}*ψ*1(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

*∇**b**ψ*1(*a, b) =*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

for*p is even; and*

*∇**a**ψ*1(*a, b) =*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

*∇**b**ψ*1(*a, b) =*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

for *p is odd. Again, by* _{(a,b)}^{a}^{p−1}*p−1*
*p*

* ≤ 1, and* _{(a,b)}^{b}^{p−1}*p−1*
*p*

* ≤ 1, the desired*
inequality holds.

(v) If *a = 0, b > 0, then φ*1(*a, b) = 0 which says ∇**a**ψ*1(*a, b) = 0 = ∇**b**ψ*1(*a, b).*

Hence,*∇*_{a}*ψ*1(*a, b) · ∇*_{b}*ψ*1(*a, b) = 0.*

(vi) If*a = 0, b < 0, then (b*+*∂a*+*, a*+*∂b*+) = (0*, 0). Hence, from (3.2), we have*

*∇**a**ψ*1(*a, b) = −φ*1(*a, b), ∇**b**ψ*1(*a, b) =*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

for*p is even; and*

*∇**a**ψ*1(*a, b) = −φ*1(*a, b), ∇**b**ψ*1(*a, b) =*

*−b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*φ*1(*a, b),*

for *p is odd. By the same reasons as in previous discussions, we obtain that*

*∇**a**ψ*1(*a, b) · ∇**b**ψ*1(*a, b) ≥ 0.*

(vii) If *a = 0, b = 0, then φ*1(*a, b) = 0. Hence, ∇*_{a}*ψ*1(*a, b) = 0 = ∇*_{b}*ψ*1(*a, b) and*

*∇**a**ψ*1(*a, b) · ∇**b**ψ*1(*a, b) = 0.*

For*i = 2, ψ*2= ^{1}_{2}(*φ*2)^{2}, we discuss discuss four cases as in part (b).

(i) If (*a, b) = (0, 0) and ab > 0, from (3.3), we have*

*∇**a**ψ*2(*a, b) =*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αb*

*φ*2(*a, b),*

*∇*_{b}*ψ*2(*a, b) =*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*2(*a, b),*

for*p is even; and*

*∇**a**ψ*2(*a, b) =*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αb*

*φ*2(*a, b),*

*∇*_{b}*ψ*2(*a, b) =*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1 − αa*

*φ*2(*a, b),*

for *p is odd. By the same reasons as in previous discussions, it can be easily*
veriﬁed that*∇**a**ψ*1(*a, b) · ∇**b**ψ*1(*a, b) ≥ 0.*

(ii) If (*a, b) = (0, 0) and ab = 0, then φ*2(*a, b) = 0. Hence, ∇*_{a}*ψ*2(*a, b) = 0 =*

*∇**b**ψ*2(*a, b) and ∇**a**ψ*2(*a, b) · ∇**b**ψ*2(*a, b) = 0.*

(iii) If (*a, b) = (0, 0) and ab < 0, the arguments are the same as case*
(iv) for*i = 1 except that φ*1 is replaced by*φ*2.

(iv) If (*a, b) = (0, 0), then φ*2(*a, b) = 0. Hence, ∇*_{a}*ψ*2(*a, b) = 0 = ∇*_{b}*ψ*2(*a, b) and*

*∇**a**ψ*2(*a, b) · ∇**b**ψ*2(*a, b) = 0.*

For*i = 3, ψ*3= ^{1}_{2}(*φ*3)^{2}, we have two cases as below.

(i) If (*a, b) = (0, 0), from (3.4)–(3.5), we have*

*∇**a**ψ*3(*a, b) = φ**p*(*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(a*+)(*b*+)^{2}*,*

*∇**b**ψ*3(*a, b) = φ**p*(*a, b)*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(a*+)^{2}(*b*+)*,*

for*p is even; and*

*∇**a**ψ*3(*a, b) = φ**p*(*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(a*+)(*b*+)^{2}*,*

*∇**b**ψ*3(*a, b) = φ**p*(*a, b)*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(a*+)^{2}(*b*+)*,*
for*p is odd. Thus, ∇*_{a}*ψ*3(*a, b) · ∇*_{b}*ψ*3(*a, b) equals*

*φ*^{2}* _{p}*(

*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α*^{2}(*a*+)^{3}(*b*+)^{3}

+*φ**p*(*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(a*+)^{2}(*b*+)

+*φ** _{p}*(

*a, b)*

*b*^{p−1}

*(a, b)*^{p−1}_{p}*− 1*

*α(a*+)(*b*+)^{2}
or

*φ*^{2}* _{p}*(

*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}_{p}*− 1*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}_{p}*− 1*

+*α*^{2}(*a*+)^{3}(*b*+)^{3}

+*φ** _{p}*(

*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(a*+)^{2}(*b*+)

+*φ**p*(*a, b)*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(a*+)(*b*+)^{2}*.*

Note that in the above expressions, it is trivial that the ﬁrst and second terms are nonnegative. We also notice that

(*a*+)(*b*+) =

*ab, if a > 0, b > 0*
0*,* else*.*

Therefore, we only need to consider the subcase of*a > 0, b > 0 for the third*
and fourth terms. In fact, summing up the third and fourth term under this
subcase gives

*αab · φ**p*(*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*a +*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*b*

=*αab · φ**p*(*a, b)*

*a** ^{p}*+

*b*

^{p}*(a, b)*^{p−1}*p* *− (a + b)*

=*αab · φ** _{p}*(

*a, b)[(a, b)*

_{p}*− (a + b)]*

=*αab · φ*^{2}* _{p}*(

*a, b)*

*≥ 0.*

Thus, we proved*∇**a**ψ*2(*a, b) · ∇**b**ψ*2(*a, b) ≥ 0.*

For*i = 4, ψ*4= ^{1}_{2}(*φ*4)^{2}, we also have two cases as below.

(i) If (*a, b) = (0, 0), from (3.6) and (3.7), we have*

*∇**a**ψ*4(*a, b) = φ**p*(*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(ab)*+*· b,*

*∇*_{b}*ψ*4(*a, b) = φ** _{p}*(

*a, b)*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(ab)*+*· a,*

for*p is even; and*

*∇**a**ψ*4(*a, b) = φ**p*(*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(ab)*+*· b,*

*∇*_{b}*ψ*4(*a, b) = φ** _{p}*(

*a, b)*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α(ab)*+*· a,*

for*p is odd. Thus, ∇*_{a}*ψ*4(*a, b) · ∇*_{b}*ψ*4(*a, b) equals*

*φ*^{2}* _{p}*(

*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α*^{2}(*ab)*^{2}+*· (ab)*

+*φ** _{p}*(

*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(ab)*+*· a*

+*φ** _{p}*(

*a, b)*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(ab)*+*· b*

or
*φ*^{2}* _{p}*(

*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

+*α*^{2}(*ab)*^{2}+*· (ab)*

+*φ**p*(*a, b)*

sgn(*a) · a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*α(ab)*+*· a + φ**p*(*a, b)*

sgn(*b) · b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*× α(ab)*+*· b.*

The ﬁrst and second terms are non-negative by the same reasons in previous discussions. We notice that

(*ab)*+=

*ab, if ab > 0*
0*,* else*.*

Thus, we only need to consider the subcase of*ab > 0 for the third and fourth*
terms. In fact, summing up the third and fourth term under this subcase gives

*α(ab)*+*· φ** _{p}*(

*a, b)*

*a*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*a +*

*b*^{p−1}

*(a, b)*^{p−1}*p* *− 1*

*b*

=*α(ab)*+*· φ**p*(*a, b)*

*a** ^{p}*+

*b*

^{p}*(a, b)*^{p−1}*p* *− (a + b)*

=*α(ab)*+*· φ** _{p}*(

*a, b)[(a, b)*

_{p}*− (a + b)]*

=*α(ab)*+*· φ*^{2}* _{p}*(

*a, b)*

*≥ 0.*

The arguments hold as well for *p is odd. Hence, we proved ∇*_{a}*ψ*2(*a, b) ·*

*∇**b**ψ*2(*a, b) ≥ 0.*

(d) Going over exactly the same cases for each*i discussed as in part (c) where*

*∇*_{a}*ψ(a, b) and ∇*_{b}*ψ(a, b) are formed, it is not hard to verify that the desired result*
is satisﬁed. We omit the details.

Based on the properties of *ψ stated as in Proposition 3.3 and using the same*
*proof techniques developed in (De Luca et al., 1996; Kanzow and Kleinmichel, 1998;*

*Kanzow et al., 1997), we have the following condition for a stationary point to be*
a solution of the NCP. We omit the details.

**Proposition 3.4.** *Assume that* *x*^{∗}*∈ R*^{n}*is a stationary point of Ψ deﬁned as*
(1*.15) − (1.17) (except for Ψ induced from ψ*2*) such that the Jacobian∇F (x*^{∗}*) is a*
*P*0*-matrix. Then* *x*^{∗}*is a solution of the NCP.*

As pointed out in Proposition 3.4, if Ψ is induced from*ψ*2then Proposition 3.4
does not necessary hold for such a Ψ. The reason is that there needs *∇**a**ψ(a, b) ·*

*∇*_{b}*ψ(a, b) > 0 when ψ(a, b) = 0 in the proof. However, this is not always true*
(we proved that *∇**a**ψ(a, b) · ∇**b**ψ(a, b) ≥ 0) for ψ*2. A counterexample for *p = 2*
was given in Sun and Qi (1999, pp. 206–207). Hence, due to this reason, the merit
function induced from*ψ*2may not be recommended even though it is continuously
diﬀerentiable.

**4. Bounded Level Sets**

As mentioned earlier, the merit function Ψ* _{p}* deﬁned as in Eq. (1.9) does not guar-
antee bounded level sets for monotone NCP. In fact, it needs that

*F is either*