A smoothing Newton method based on the generalized Fischer–Burmeister function for MCPs

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Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

A smoothing Newton method based on the generalized Fischer–Burmeister function for MCPs

Jein-Shan Chen

^a,∗,1

, Shaohua Pan

^b

, Tzu-Ching Lin

^a

aDepartment of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

bSchool of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China

a r t i c l e i n f o

Article history:

Received 5 June 2009 Accepted 12 January 2010

Keywords:

Mixed complementarity problem The generalized FB function Smoothing approximation Convergence rate

a b s t r a c t

We present a smooth approximation for the generalized Fischer–Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > ^{1), and} establish some favorable properties for it — for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p∈(¹,²]has better numerical performance than the one with p>^2.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The mixed complementarity problem (MCP) arises in many applications including the fields of economics, engineering, and operations research [1–4] and has attracted much attention in last decade [5–10]. A collection of nonlinear mixed complementarity problems called MCPLIB can be found in [11] and the excellent book [12] is a good source for seeking theoretical backgrounds and numerical methods.

Given a mapping F

: [

l

,

^u

] →

_Rⁿ with F

= (

^F1

, . . . ,

^Fn

)

^{T, where l}

= (

^l1

, . . . ,

^ln

)

^{T and u}

= (

^u1

, . . . ,

^un

)

^{T with} l_i

∈

_R

∪ {−∞}

and u_i

∈

_R

∪ {+∞}

being given lower and upper bounds satisfying l_i

<

^ui for i

=

1

,

²

, . . . ,

^{n. The} MCP is to find a vector x^∗

∈ [

l

,

^u

]

such that each component x^∗_i satisfies exactly one of the following implications:

x^∗_i

=

l_i

H⇒

F_i

(

^x∗

) ≥

⁰

,

x^∗_i

∈ (

^li

,

^ui

) H⇒

^Fi

(

^x∗

) =

⁰

,

x^∗_i

=

u_i

H⇒

F_i

(

^x∗

) ≤

⁰

.

⁽¹⁾

It is not hard to see that, when l_i

= −∞

and u_i

= +∞

for all i

=

1

,

²

, . . . ,

^{n, the MCP(1)}is equivalent to solving the nonlinear system of equations

F

(

^x

) =

⁰

;

(2)

whereas when l_i

=

0 and u_i

= +∞

for all i

=

1

,

²

, . . . ,

n, it reduces to the nonlinear complementarity problems (NCP) which is to find a point x

∈

_Rⁿsuch that

x

≥

0

,

^F

(

^x

) ≥

⁰

, h

x

,

^F

(

^x

)i =

⁰

.

⁽³⁾

∗Corresponding author. Tel.: +886 2 29325417; fax: +886 2 29332342.

E-mail addresses:jschen@math.ntnu.edu.tw(J.-S. Chen),shhpan@scut.edu.cn(S. Pan),cashplayer35@yahoo.com.tw(T.-C. Lin).

1 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

0362-546X/$ – see front matter©2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.na.2010.01.012

In fact, from Theorem 2 of [13], the MCP(1)is also equivalent to the famous variational inequality problem (VIP) which is to find a vector x^∗

∈ [

l

,

^u

]

such that

h

F

(

^x∗

),

^x

−

x^∗

i ≥

0

∀

x

∈ [

l

,

^u

] .

⁽⁴⁾

In the rest of this paper, we assume the mapping F to be continuously differentiable.

It is well-known that NCP functions play an important role in the design of algorithms for the MCP(1). Specifically,

φ :

R

×

_R

→

R is called an NCP function if

φ(

^a

,

^b

) =

⁰

⇐⇒

a

≥

0

,

^b

≥

0

,

^ab

=

0

.

⁽⁵⁾

With such a function, the MCP(1)can be reformulated as a nonsmooth systemΦ

(

^x

) =

0, and consequently nonsmooth Newton methods or smoothing Newton methods can be applied for solving the systemΦ

(

^x

) =

0. Among others, the latter is based on a smooth approximation of

φ

. In the past two decades, many smooth approximations and Newton-type methods using smoothing NCP functions for complementarity problems have been proposed (see, e.g., [14–18,8,19]). Most of these methods focus on the Chen–Mangasarian class of smooth approximations of the minimum NCP function or the smoothing function of the Fischer–Burmeister (FB) NCP function. It is worthwhile to mention that the smoothing Newton method developed by Chen et al. [19] has global and superlinear (even quadratic) convergence by solving only one linear system of equations at each iteration.

Recently, an extension of the FB NCP function was considered in [20–22] by two of the authors. Specifically, they define the generalized FB function as

φ

p

(

^a

,

^b

) := k(

^a

,

^b

)k

p

− (

^a

+

b

) ∀

^a

,

^b

∈

_R

,

⁽⁶⁾

where p is an arbitrary fixed real number from the interval

(

¹

, +∞)

^and

k (

^a

,

^b

)k

p denotes the p-norm of

(

^a

,

^b

)

^, i.e.,

k (

^a

,

^b

)k

p

= √

^p

|

a

|

^p

+ |

b

|

^p. In other words, in the function

φ

p, they replace the 2-norm of

(

^a

,

^b

)

involved in the FB function by a more general p-norm. The function

φ

pis still an NCP-function — that is, it satisfies the equivalence in(5). Moreover, it turns out that

φ

ppossesses all favorable properties of the FB function; see [20–22]. For example,

φ

pis strongly semismooth and its square is a continuously differentiable NCP function. In particular, numerical results in [23] for all MCPLIB problems indicate that the least-square semismooth Newton method with p close to 1 has better performance than the case of p

=

2.

Thus, it is natural to ask whether the smoothing Newton method based on

φ

phas similar a numerical performance.

In this paper, we are concerned with the smoothing Newton method [19,28] based on the generalized FB function, motivated by the inexpensive computation work of the method at each iteration, and the fact that there are no corresponding numerical experiments to verify the effectiveness of this algorithm. We investigate the influence of the parameter p on the numerical performance of the smoothing method for solving the MCPLIB test problems. Specifically, in Section3, we present a smoothing function of the generalized FB function, and studied some of its favorable properties, including the Jacobian consistency property; in Section4, we describe the iterative steps of the smoothing algorithm and provide the corresponding conditions for the global convergence and local superlinear (or quadratic) convergence; in Section5, we report the numerical results of the smoothing algorithm for solving the MCPLIB test problems.

Throughout this paper, Rⁿdenotes the space of n-dimensional real column vectors and e_imeans a unit vector with ith component being 1 and the others being 0. For a differentiable mapping F , F⁰

(

^x

)

^and

∇

F

(

^x

)

to denote the Jacobian of F at x and the transposed Jacobian of F , respectively. Given an index setI, the notation

[

F⁰

(

^x

)]

IIdenotes the submatrix consisting of the ith row and the jth column with i

∈

_Iand j

∈

_I.

2. Preliminary

In this section, we review some basic concepts and results that will be used in subsequent analysis. We start with introducing the concept of generalized Jacobian of a mapping. Let G

:

_Rⁿ

→

_R^mbe a locally Lipschitz continuous mapping.

Then, G is almost everywhere differentiable by Rademacher’s Theorem (see [24]). In this case, the generalized Jacobian

∂

^G

(

^x

)

of G at x (in the Clarke sense) is defined as the convex hull of the B-subdifferential

∂

BG

(

^x

) := 

^V

∈

_R^m×n

|∃{

x^k

} ⊆

D_G

: {

x^k

} →

x and G⁰

(

^xk

) →

^V

,

where D_Gis the set of differentiable points of G. In other words,

∂

^G

(

^x

) =

^conv

∂

BG

(

^x

)

^{. If m}

=

1, we call

∂

^G

(

^x

)

the generalized gradient of G at x. The calculation of

∂

^G

(

^x

)

is usually difficult in practice, and Qi [25] proposed so-called C -subdifferential of G:

∂

CG

(

^x

)

^T

:= ∂

^G1

(

^x

) × · · · × ∂

^Gm

(

^x

)

⁽⁷⁾

which is easier to compute than the generalized Jacobian

∂

^G

(

^x

)

. Here, the right-hand side of(7)denotes the set of matrices in R^n×mwhose i-th column is given by the generalized gradient of the i-th component function G_i. In fact, by Proposition 2.6.2 of [24],

∂

^G

(

^x

)

^T

⊆ ∂

CG

(

^x

)

^T. We assume that the reader is familiar with the concepts of (strongly) semismooth functions, and refer to [26,27] for details.

We also need the definitions of P-functions and P-matrices in the subsequent sections.

Definition 2.1. Let F

= (

^F1

, . . . ,

^Fn

)

^{Twith F}i

:

_Rⁿ

→

_{R for i}

=

1

,

²

, . . . ,

^{n. Then,}

(a) the mapping F is called a P₀-function if, for every x and y in Rⁿwith x

6=

y, there is an index i

∈ {

1

,

²

, . . . ,

ⁿ

}

such that x_i

6=

y_i and

(

^xi

−

y_i

)(

^Fi

(

^x

) −

^Fi

(

^y

)) ≥

⁰

;

(b) the mapping F is called a P-function if, for every x and y in Rⁿwith x

6=

y, there is an index i

∈ {

1

,

²

, . . . ,

ⁿ

}

such that x_i

6=

y_i and

(

^xi

−

y_i

)(

^Fi

(

^x

) −

^Fi

(

^y

)) >

⁰

.

(c) the mapping F is called a uniform P-function if there exists a positive constant

µ >

0 such that, for every x and y in Rⁿ, there is an index i

∈ {

1

,

²

, . . . ,

ⁿ

}

such that

(

^xi

−

y_i

)(

^Fi

(

^x

) −

^Fi

(

^y

)) ≥ µk

^x

−

y

k

²

.

Definition 2.2. A matrix M

∈

_R^n×nis called an

(a) P₀-matrix if each of its principal minors is nonnegative.

(b) P-matrix if each of its principal minors is positive.

FromDefinitions 2.1and2.2, it is not hard to see that a continuously differentiable mapping F is a P₀-function if and only if

∇

F

(

^x

)

^{is P}0-matrix for all x

∈

_Rⁿ. For the P₀-matrix, we also have the following important property.

Lemma 2.1 ([12]). A matrix M

∈

_R^n×nis a P₀-matrix if and only if for every nonzero vector x, there exists an index i such that x_i

6=

0 and x_i

(

^Mx

)

i

≥

0.

Next we recall some favorable properties of

φ

pwhose proofs can be found in [20–22].

Lemma 2.2. Let

φ

p

:

_R

×

_R

→

R be defined by(6). Then, the following results hold.

(a)

φ

pis a strongly semismooth NCP-function.

(b) Given any point

(

^a

,

^b

) ∈

R², each element in the generalized gradient

∂φ

p

(

^a

,

^b

)

has the representation

(ξ −

¹

, ζ −

¹

)

^where, if

(

^a

,

^b

) 6= (

⁰

,

⁰

)

^,

(ξ, ζ ) =

^sign

(

^a

) · |

^a

|

^p−1

k (

^a

,

^b

)k

^pp⁻¹

,

^sign

(

^b

) · |

^b

|

^p−1

k (

^a

,

^b

)k

^pp⁻¹

! ,

and otherwise

(ξ, ζ )

is an arbitrary vector in R²satisfying

| ξ|

^p−p1

+ | ζ |

^p−p1

≤

1.

(c) The square of

φ

pis a continuously differentiable NCP function.

(d) If

{ (

^ak

,

^bk

)} ⊆

R²satisfies

(

^ak

→ −∞ )

^or

(

^bk

→ −∞ )

^or

(

^ak

→ ∞

and b^k

→ ∞ )

, then we have

| φ

p

(

^ak

,

^bk

)| → ∞

^as k

→ ∞

.

The following lemma establishes another property of

φ

p, which plays a key role in the nonsmooth system reformulation of the MCP(1)with the generalized FB function.

Lemma 2.3. Let

φ

p

:

_R

×

_R

→

R be defined by(6). Then, the following limits hold.

(a) lim

li→−∞

φ

p x_i

−

l_i

, φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

))
= −φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

))

^. (b) lim

ui→∞

φ

p x_i

−

l_i

, φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

))
= φ

p

(

^xi

−

l_i

,

^Fi

(

^x

))

^. (c) lim

li→−∞ lim

ui→∞

φ

p x_i

−

l_i

, φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

))
= −

^Fi

(

^x

)

^.

Proof. Let

{

a^k

} ⊆

R be any sequence converging to

+∞

as k

→ ∞

and b

∈

R be any fixed real number. We will prove lim

k→∞

φ

p

(

^ak

,

^b

) = −

b, and part (a) then follows by continuity arguments. Without loss of generality, assume that a^k

>

^{0 for} each k. Then,

φ

p

(

^ak

,

^b

) =

^{ak 1}

+ (|

^b

| /

^ak

)

^p

1/p

−

a^k

−

b

=

a^k

"

1

+

¹ p

 |

b

|

a^k



p

+

¹

−

p 2p²

 |

b

|

a^k



2p

+ · · · + (

¹

−

p

) · · · (

¹

−

pn

+

p

)

n

!

pⁿ

 |

b

|

a^k



np

+

o

 |

b

|

a^k



pn

 #

−

a^k

−

b

=

¹ p

|

b

|

^p

(

^ak

)

^p−1

+

¹

−

p 2p²

|

b

|

^2p

(

^ak

)

^2p−1

+ · · · + (

¹

−

p

) · · · (

¹

−

pn

+

p

)

n

!

pⁿ

|

b

|

^np

(

^ak

)

^np−1

+ (

^ak

)|

^b

|

^np

(

^ak

)

^np

o

|

b

| /

^ak

pn

|

b

| /

^ak

pn

−

b where the second equality is using the Taylor expansion of the function

(

¹

+

t

)

^1/p and the notation o

(

^t

)

^means lim_t→0o

(

^t

)/

^t

=

0. Since a^k

→ +∞

as k

→ ∞

, we have _(a^|_k^b₎^|_np^np−1

→

0 for all n. This together with the last equation implies lim_k→∞

φ

p

(

^ak

,

^b

) = −

b. This proves part (a). Part (b) and (c) are direct by part (a) and the continuity of

φ

FB. 

To close this section, we summarize the monotonicity of two scalar-valued functions that will be used in the subsequent section. Since the proof is direct, we omit it here.

Lemma 2.4. For any fixed 0

≤ µ

1

< µ

2, the following functions f₁

(

^t

) := (

^t

+ µ

1

)

^−p−p1

− (

^t

+ µ

2

)

^−p−p1

(

^t

>

⁰

)

and

f₂

(

^t

) := (

^t

+ µ

2

)

^p−p1

− (

^t

+ µ

1

)

^p−p1

(

^t

≥

0

)

are decreasing on

(

⁰

, +∞)

, and furthermore, f₂

(

^t

) ≤

^f2

(

⁰

) = µ

⁽2^p−1)/p

− µ

⁽1^p−1)/p

.

3. The smoothing function and its properties

For convenience, in the rest of this paper, we adopt the following notations of index sets:

I_l

:= {

i

∈ {

1

,

²

, . . . ,

ⁿ

} | −∞ <

^li

<

^ui

= +∞} ,

I_u

:= {

i

∈ {

1

,

²

, . . . ,

ⁿ

} | −∞ =

l_i

<

^ui

< +∞} ,

I_lu

:= {

i

∈ {

1

,

²

, . . . ,

ⁿ

} | −∞ <

^li

<

^ui

< +∞} ,

I_f

:= {

i

∈ {

1

,

²

, . . . ,

ⁿ

} | −∞ =

l_i

<

^ui

= +∞} .

(8)

With the generalized FB function, we define a operatorΦp

:

_Rⁿ

→

_Rⁿcomponentwise as

Φp,i

(

^x

) :=



 

 

φ

p

(

^xi

−

l_i

,

^Fi

(

^x

))

^{if i}

∈

I_l

,

− φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

))

^{if i}

∈

I_u

, φ

p

(

^xi

−

l_i

, φ

p

(

^ui

−

x_i

, −

^Fi

(

^x

)))

^{if i}

∈

I_lu

,

−

F_i

(

^x

)

^{if i}

∈

I_f

,

(9)

where the minus sign for i

∈

I_uand i

∈

I_fis motivated byLemma 2.3. In fact, all results of this paper would be true without the minus sign. Using the equivalence in(5), it is not difficult to verify that the following result holds.

Proposition 3.1. x^∗

∈

_Rⁿis a solution of the MCP(1)if and only if x^∗solves the nonlinear system of equationsΦp

(

^x

) =

^0.

We want to point out that, unlike for the nonlinear complementarity problem, when writing the generalized FB function

φ

pas

φ

p

(

^a

,

^b

) = (

^a

+

b

) − k(

^a

,

^b

)k

p

,

the conclusion ofProposition 3.1does not necessarily hold since, if I_l

= {

1

,

²

, . . . ,

ⁿ

}

, then_x

¯ =

l satisfiesΦp

(¯

^x

) =

^{0, but F}

(¯

^x

) ≥

0 does not necessarily hold. Similar phenomenon also appears when replacing

φ

pby the minimum NCP function.

Since

φ

pis not differentiable at the origin, the systemΦp

(

^x

) =

0 is nonsmooth. In this paper, we will find a solution of nonsmooth systemΦp

(

^x

) =

0 by solving a sequence of smooth approximationsΨp

(

^x

, ε) =

^{0, where}

ε >

0 is a smoothing parameter and the operatorΨp

:

_Rⁿ

×

_R++

→

_Rⁿis defined componentwise as

Ψp,i

(

^x

, ε) :=



 

 

ψ

p

(

^xi

−

l_i

,

^Fi

(

^x

), ε)

^{if i}

∈

I_l

,

− ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε)

^{if i}

∈

I_u

, ψ

p x_i

−

l_i

, ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε), ε

^{if i}

∈

I_lu

,

−

F_i

(

^x

)

^{if i}

∈

I_f

,

(10)

with

ψ

p

(

^a

,

^b

, ε) := p

^p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

− (

^a

+

b

).

⁽¹¹⁾

In what follows, we concentrate on the favorable properties of the smoothing function

ψ

pand the operatorΨp. First, let us state the favorable properties of

ψ

p.

Lemma 3.1. Let

ψ

p

:

_R³

→

R be defined by(11). Then, the following result holds.

(a) For any fixed

ε >

^0,

ψ

p

(

^a

,

^b

, ε)

is continuously differentiable at all

(

^a

,

^b

) ∈

R²with

−

2

< ∂ψ

p

(

^a

,

^b

, ε)

∂

^a

<

⁰

, −

2

< ∂ψ

p

(

^a

,

^b

, ε)

∂

^b

<

⁰

.

⁽¹²⁾

(b) For any fixed

(

^a

,

^b

) ∈

R²,

ψ

p

(

^a

,

^b

, ε)

is continuously differentiable, strictly increasing and convex with respect to

ε >

^0.

Moreover, for any 0

< ε

1

≤ ε

2,

0

≤ ψ

p

(

^a

,

^b

, ε

2

) − ψ

p

(

^a

,

^b

, ε

1

) ≤ (ε

2

− ε

1

).

⁽¹³⁾

In particular,

| ψ

p

(

^a

,

^b

, ε) − φ

p

(

^a

,

^b

)| ≤ ε

^{for all}

ε ≥

^0.

(c) For any fixed

(

^a

,

^b

) ∈

R², let

ψ

p⁰

(

^a

,

^b

) :=



limε↓0

∂ψp(a,b,ε)

∂a

,

^lim

ε↓0

∂ψp(a,b,ε)

∂b

 .

^Then,

lim

h=(h1,h2)→(0,0)

φ

p

(

^a

+

h₁

,

^b

+

h₂

) − φ

p

(

^a

,

^b

) − ψ

p⁰

(

^a

+

h₁

,

^b

+

h₂

)

^Th

k

h

k =

0

.

(d) For any given

ε >

^{0, if p}

≥

2, then

ψ

p

(

^a

,

^b

, ε) =

⁰

H⇒

a

>

⁰

,

^b

>

⁰

,

^2ab

≤ ε

², and whenever p

>

^1,

ψ

p

(

^a

,

^b

, ε) =

0

H⇒

a

>

⁰

,

^b

>

⁰

,

^min

{

a

,

^b

} ≤

^√_p ^ε

2^p−2.

Proof. (a) Using an elementary calculation, we immediately obtain that

∂ψ

p

(

^a

,

^b

, ε)

∂

^a

=

^sign

(

^a

)|

^a

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−₁

−

1

,

∂ψ

p

(

^a

,

^b

, ε)

∂

^b

=

^sign

(

^b

)|

^b

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−1

−

1

.

⁽¹⁴⁾

For any fixed

ε >

^{0, since∂ψp(}_∂^a_a^{,b,ε)and∂ψp(}_∂^a_b^,b,ε)are continuous at all

(

^a

,

^b

) ∈

R², it follows that

ψ

p

(

^a

,

^b

, ε)

is continuously differentiable at all

(

^a

,

^b

) ∈

R². Noting that

    

sign

(

^a

)|

^a

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−1

    

<

^{1 and}

    

sign

(

^b

)|

^b

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−1

    

<

¹

,

we readily get the inequality(12).

(b) For any

ε >

0, an elementary calculation yields that

∂ψ

p

(

^a

,

^b

, ε)

∂ε = ε

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−1

>

⁰

,

∂

²

ψ

p

(

^a

,

^b

, ε)

∂ε

²

= (

^p

−

1

)ε

^p−2

√

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

p−1



1

− ε

^p

|

a

|

^p

+ |

b

|

^p

+ ε

^p



≥

0

.

Therefore, for any fixed

(

^a

,

^b

) ∈

R²,

ψ

p

(

^a

,

^b

, ε)

is continuously differentiable, strictly increasing and convex with respect to

ε >

0. By the mean-value theorem, for any 0

< ε

1

≤ ε

2, there exists some

ε

0

∈ (ε

1

, ε

2

)

^{such that}

ψ

p

(

^a

,

^b

, ε

2

) − ψ

p

(

^a

,

^b

, ε

1

) = ∂ψ

p

∂ε (

^a

,

^b

, ε

0

)(ε

2

− ε

1

).

Since^∂ψ_∂ε^p

(

^a

,

^b

, ε

0

) ≤

1 by the proof of part (a), inequality(13)holds for all 0

< ε

1

≤ ε

2. Letting

ε

1

↓

0, the desired result then follows.

(c) Using the formula(14), it is easy to calculate that

limε↓0

∂ψ

p

(

^a

,

^b

, ε)

∂

^a

=







sign

(

^a

)|

^a

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

p−1

−

1 if

(

^a

,

^b

) 6= (

⁰

,

⁰

),

−

1 if

(

^a

,

^b

) = (

⁰

,

⁰

);

limε↓0

∂ψ

p

(

^a

,

^b

, ε)

∂

^b

=







sign

(

^b

)|

^b

|

^p−1

√

p

|

a

|

^p

+ |

b

|

^p

p−1

−

1 if

(

^a

,

^b

) 6= (

⁰

,

⁰

),

−

1 if

(

^a

,

^b

) = (

⁰

,

⁰

).

(15)

From this, we see that

ψ

p⁰

(

^a

,

^b

) = 

_∂φ

p(a,b)

∂a

,

^∂φp_∂⁽_b^a,b)



at

(

^a

,

^b

) 6= (

⁰

,

⁰

)

. Therefore, we only need to check the case

(

^a

,

^b

) = (

⁰

,

⁰

)

. The desired result follows by

φ

p

(

^h1

,

^h2

) − φ

p

(

⁰

,

⁰

) − ψ

p⁰

(

^h1

,

^h2

)

^Th

= p

^p

|

h₁

|

^p

+ |

h₂

|

^p

− |

h₁

|

^p

+ |

h₂

|

^p

( √

^p

|

h₁

|

^p

+ |

h₂

|

^p

)

^p−1

= p

^p

|

h₁

|

^p

+ |

h₂

|

^p

− p

^p

|

h₁

|

^p

+ |

h₂

|

^p

=

0

.

(d) From the definition of

ψ

p

(

^a

,

^b

, ε)

^{, clearly,}

ψ

p

(

^a

,

^b

, ε) =

0 implies a

+

b

≥

0, and hence a

≥

0 or b

≥

0. Note that, whenever a

≥

0

,

^b

≤

0 or a

≤

0

,

^b

≥

0, there holds that

p

p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

> p

^p

|

a

|

^p

+ |

b

|

^p

≥

max

{|

a

| , |

^b

|} ≥

a

+

b

,

i.e.,

ψ

p

(

^a

,

^b

, ε) >

0. Hence, for any given

ε >

^0,

ψ

p

(

^a

,

^b

, ε) =

0 implies a

>

^{0 and b}

>

^0.

(i) If p

≥

2, using the nonincreasing of p-norm with respect to p leads to

ψ

p

(

^a

,

^b

, ε) =

⁰

⇐⇒

a

+

b

= p

^p

|

a

|

^p

+ |

b

|

^p

+ ε

^p

≤ p

|

a

|

²

+ |

b

|

²

+ ε

²

H⇒ (

^a

+

b

)

²

≤

a²

+

b²

+ ε

²

H⇒

2ab

≤ ε

²

.

(ii) For p

>

1, without loss of generality, we assume 0

<

^a

≤

b. For any fixed a

≥

0, consider f

(

^t

) = (

^t

+

a

)

^p

−

t^p

−

a^p

− ε

^p

(

^t

≥

0

)

. It is easy to verify that the function f is strictly increasing on

[

0

, +∞)

^{. Since}

ψ

p

(

^a

,

^b

, ε) =

0, we have f

(

^b

) =

^{0 which} says f

(

^a

) = (

^2p

−

2

)

^ap

− ε

^p

≤

f

(

^b

) =

0. From this inequality, we get min

{

a

,

^b

} =

a

≤

^√_p ^ε

2^p−2.  UsingLemma 3.1and the expression ofΨp, we readily obtain the following result.

Proposition 3.2. LetΨpbe defined by(10). Then, the following results hold.

(a) For any fixed

ε >

^0,Ψp

(

^x

, ε)

is continuously differentiable on Rⁿwith

∇

_xΨp

(

^x

, ε) =

^Da

(

^x

, ε) + ∇

^F

(

^x

)

^Db

(

^x

, ε),

where D_a

(

^x

, ε)

^{and D}b

(

^x

, ε)

^{are n}

×

n diagonal matrices with the diagonal elements

(

^Da

)

ii

(

^x

, ε)

^and

(

^Db

)

ii

(

^x

, ε)

^{defined as} follows:

(a1) For i

∈

I_l,

(

^Da

)

ii

(

^x

, ε) =

^sign

(

^xi

−

l_i

)|

^xi

−

l_i

|

^p−1

k (

^xi

−

l_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

−

1

,

(

^Db

)

ii

(

^x

, ε) =

^sign

(

^Fi

(

^x

))|

^Fi

(

^x

)|

^p−1

k (

^xi

−

l_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

−

1

.

(a2) For i

∈

I_u,

(

^Da

)

ii

(

^x

, ε) =

^sign

(

^ui

−

x_i

)|

^ui

−

x_i

|

^p−1

k (

^ui

−

x_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

−

1

,

(

^Db

)

ii

(

^x

, ε) = −

sign

(

^Fi

(

^x

))|

^Fi

(

^x

)|

^p−1

k (

^ui

−

x_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

−

1

.

(a3) For i

∈

I_lu,

(

^Da

)

ii

(

^x

, ε) =

^ai

(

^x

, ε) +

^bi

(

^x

, ε)

^ci

(

^x

, ε)

^and

(

^Db

)

ii

(

^x

, ε) =

^bi

(

^x

, ε)

^di

(

^x

, ε)

with

a_i

(

^x

, ε) =

^sign

(

^xi

−

l_i

)|

^xi

−

l_i

|

^p−1

(

^xi

−

l_i

, ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε), ε)

p−1 p

−

1

,

b_i

(

^x

, ε) =

^sign

(ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε))|ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε)|

^p−1

(

^xi

−

l_i

, ψ

p

(

^ui

−

x_i

, −

^Fi

(

^x

), ε), ε)

p−1 p

−

1

,

c_i

(

^x

, ε) = −

^sign

(

^ui

−

x_i

)|

^ui

−

x_i

|

^p−1

k (

^ui

−

x_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

+

1

,

d_i

(

^x

, ε) =

^sign

(

^Fi

(

^x

))|

^Fi

(

^x

)|

^p−1

k (

^ui

−

x_i

,

^Fi

(

^x

), ε)k

^pp⁻¹

+

1

.

(a4) For i

∈

I_f,

(

^Da

)

ii

(

^x

, ε) =

^{0 and}

(

^Db

)

ii

(

^x

, ε) = −

^1.

Moreover,

−

2

< (

^Da

)

ii

(

^x

, ε) <

^{0 and}

−

2

< (

^Db

)

ii

(

^x

, ε) <

0 for all i

∈

I_l

∪

I_u, and

−

6

< (

^Da

)

ii

(

^x

, ε) <

^{0 and}

−

4

< (

^Db

)

ii

(

^x

, ε) <

^{0 for i}

∈

I_lu. (b) For any given

ε

1

>

^{0 and}

ε

2

>

^{0, we have}

k

_Ψp

(

^x

, ε

2

) −

Ψp

(

^x

, ε

1

)k ≤ √

n

 √

_p 2

+

1



| ε

2

− ε

1

| , ∀

^x

∈

_Rⁿ

.

Particularly, for any given

ε >

^0,

k

_Ψp

(

^x

, ε) −

Φp

(

^x

)k ≤ √

n

 √

_p

2

+

1

 ε, ∀

^x

∈

_Rⁿ

.

The Jacobian consistency property plays a crucial role in the analysis of local fast convergence of the smoothing algorithm [19]. To show that the smoothing operatorΨpsatisfies the Jacobian consistency property, we need the following characterization of the generalized Jacobian

∂

CΦp

(

^x

)

, which is direct byLemma 2.2(b).

Proposition 3.3. For any given x

∈

_Rⁿ,

∂

CΦp

(

^x

)

^T

= {

D_a

(

^x

) + ∇

^F

(

^x

)

^Db

(

^x

)}

^{, where D}a

(

^x

),

^Db

(

^x

)

^{are n}

×

n diagonal matrices whose diagonal elements are given as below:

(a) For i

∈

I_l, if

(

^xi

−

l_i

,

^Fi

(

^x

)) 6= (

⁰

,

⁰

)

^{, then}

(

^Da

)

ii

(

^x

) =

^sign

(

^xi

−

l_i

) · |

^xi

−

l_i

|

^p−1

k (

^xi

−

l_i

,

^Fi

(

^x

))k

^pp⁻¹

−

1

, (

^Db

)

ii

(

^x

) =

^sign

(

^Fi

(

^x

)) · |

^Fi

(

^x

)|

^p−1

k (

^xi

−

l_i

,

^Fi

(

^x

))k

^pp⁻¹

−

1

;