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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}

a*Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan*

b*School 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}### ] →

_{R}

^{n}*with F*

### = (

*1*

^{F}### , . . . ,

^{F}*n*

### )

^{T}

^{, where l}### = (

*1*

^{l}### , . . . ,

^{l}*n*

### )

^{T}

^{and u}### = (

*1*

^{u}### , . . . ,

^{u}*n*

### )

^{T}^{with}

*l*

_{i}### ∈

_{R}

### ∪ {−∞}

*and u*

_{i}### ∈

_{R}

### ∪ {+∞}

*being given lower and upper bounds satisfying l*

_{i}### <

^{u}*i*

*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}### ∈ (

^{l}*i*

### ,

^{u}*i*

### ) H⇒

^{F}*i*

### (

^{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*

### ∈

_{R}

*such that*

^{n}*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*

### |

*. In other words, in the function*

^{p}### φ

*p*, they replace the 2-norm of

### (

^{a}### ,

^{b}### )

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

### φ

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

### φ

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

### φ

*p*is 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

*p*has 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, R^{n}*denotes the space of n-dimensional real column vectors and e*_{i}*means a unit vector with ith*
*component being 1 and the others being 0. For a differentiable mapping F , F*^{0}

### (

^{x}### )

^{and}

### ∇

*F*

### (

^{x}### )

*to denote the Jacobian of F at x*

*and the transposed Jacobian of F , respectively. Given an index set*I, the notation

### [

*F*

^{0}

### (

^{x}### )]

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

### ∈

_{I}

*and 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*

### :

_{R}

^{n}### →

_{R}

*be a locally Lipschitz continuous mapping.*

^{m}*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*

### ∂

*B*

*G*

### (

^{x}### ) :=

^{V}### ∈

_{R}

^{m}^{×}

^{n}### |∃{

*x*

^{k}### } ⊆

*D*

_{G}### : {

*x*

^{k}### } →

*x and G*

^{0}

### (

^{x}

^{k}### ) →

^{V}### ,

*where D*_{G}*is the set of differentiable points of G. In other words,*

### ∂

^{G}### (

^{x}### ) =

^{conv}

### ∂

*B*

*G*

### (

^{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:*

### ∂

*C*

*G*

### (

^{x}### )

^{T}### := ∂

*1*

^{G}### (

^{x}### ) × · · · × ∂

^{G}*m*

### (

^{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}^{×}

^{m}*whose i-th column is given by the generalized gradient of the i-th component function G*

*. In fact, by Proposition 2.6.2 of [24],*

_{i}### ∂

^{G}### (

^{x}### )

^{T}### ⊆ ∂

*C*

*G*

### (

^{x}### )

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

^{T}*We also need the definitions of P-functions and P-matrices in the subsequent sections.*

**Definition 2.1. Let F**

### = (

*1*

^{F}### , . . . ,

^{F}*n*

### )

^{T}

^{with F}*i*

### :

_{R}

^{n}### →

_{R for i}### =

1### ,

^{2}

### , . . . ,

^{n. Then,}*(a) the mapping F is called a P*_{0}*-function if, for every x and y in R*^{n}*with x*

### 6=

*y, there is an index i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### }

such that*x*

_{i}### 6=

*y*

*and*

_{i}### (

^{x}*i*

### −

*y*

_{i}### )(

^{F}*i*

### (

^{x}### ) −

^{F}*i*

### (

^{y}### )) ≥

^{0}

### ;

*(b) the mapping F is called a P-function if, for every x and y in R*^{n}*with x*

### 6=

*y, there is an index i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### }

such that*x*

_{i}### 6=

*y*

*and*

_{i}### (

^{x}*i*

### −

*y*

_{i}### )(

^{F}*i*

### (

^{x}### ) −

^{F}*i*

### (

^{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 R*

*,*

^{n}*there is an index i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### }

such that### (

^{x}*i*

### −

*y*

_{i}### )(

^{F}*i*

### (

^{x}### ) −

^{F}*i*

### (

^{y}### )) ≥ µk

^{x}### −

*y*

### k

^{2}

### .

**Definition 2.2. A matrix M**### ∈

_{R}

^{n}^{×}

*is called an*

^{n}*(a) P*_{0}-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 P_{0}-function if and only
if

### ∇

*F*

### (

^{x}### )

*0*

^{is P}*-matrix for all x*

### ∈

_{R}

^{n}*. For the P*

_{0}-matrix, we also have the following important property.

**Lemma 2.1 ([12]). A matrix M**

### ∈

_{R}

^{n}^{×}

^{n}*is a P*

_{0}

*-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

*p*whose 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

*p*

*is a strongly semismooth NCP-function.*

*(b) Given any point*

### (

^{a}### ,

^{b}### ) ∈

R^{2}

*, 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

^{p}*p*

^{−}

^{1}

### ,

^{sign}

### (

^{b}### ) · |

^{b}### |

^{p}^{−}

^{1}

### k (

^{a}### ,

^{b}### )k

^{p}*p*

^{−}

^{1}

### ! ,

*and otherwise*

### (ξ, ζ )

*is an arbitrary vector in R*

^{2}

*satisfying*

### | ξ|

^{p}^{−}

^{p}^{1}

### + | ζ |

^{p}^{−}

^{p}^{1}

### ≤

*1.*

*(c) The square of*

### φ

*p*

*is a continuously differentiable NCP function.*

*(d) If*

### { (

^{a}

^{k}### ,

^{b}

^{k}### )} ⊆

R^{2}

*satisfies*

### (

^{a}

^{k}### → −∞ )

^{or}### (

^{b}

^{k}### → −∞ )

^{or}### (

^{a}

^{k}### → ∞

*and b*

^{k}### → ∞ )

*, then we have*

### | φ

*p*

### (

^{a}

^{k}### ,

^{b}

^{k}### )| → ∞

^{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

*l**i*→−∞

### φ

*p*

*x*

_{i}### −

*l*

_{i}### , φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### )) = −φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ))

*(b) lim*

^{.}*u**i*→∞

### φ

*p*

*x*

_{i}### −

*l*

_{i}### , φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### )) = φ

*p*

### (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ))

*(c) lim*

^{.}*l**i*→−∞ lim

*u**i*→∞

### φ

*p*

*x*

_{i}### −

*l*

_{i}### , φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### )) = −

^{F}*i*

### (

^{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*

### (

^{a}

^{k}### ,

^{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*

### (

^{a}

^{k}### ,

^{b}### ) =

^{a}

^{k}^{1}

### + (|

^{b}### | /

^{a}

^{k}### )

*1/*

^{p}*p*

### −

*a*

^{k}### −

*b*

### =

*a*

^{k}### "

1

### +

^{1}

*p*

### |

*b*

### |

*a*

^{k}*p*

### +

^{1}

### −

*p*

*2p*

^{2}

### |

*b*

### |

*a*

^{k}*2p*

### + · · · + (

^{1}

### −

*p*

### ) · · · (

^{1}

### −

*pn*

### +

*p*

### )

*n*

### !

*p*

^{n}### |

*b*

### |

*a*

^{k}*np*

### +

*o*

### |

*b*

### |

*a*

^{k}*pn*

### #

### −

*a*

^{k}### −

*b*

### =

^{1}

*p*

### |

*b*

### |

^{p}### (

^{a}

^{k}### )

^{p}^{−}

^{1}

### +

^{1}

### −

*p*

*2p*

^{2}

### |

*b*

### |

^{2p}### (

^{a}

^{k}### )

^{2p}^{−}

^{1}

### + · · · + (

^{1}

### −

*p*

### ) · · · (

^{1}

### −

*pn*

### +

*p*

### )

*n*

### !

*p*

^{n}### |

*b*

### |

^{np}### (

^{a}

^{k}### )

^{np}^{−}

^{1}

### + (

^{a}

^{k}### )|

^{b}### |

^{np}### (

^{a}

^{k}### )

^{np}*o*

### |

*b*

### | /

^{a}

^{k}*pn*

### |

*b*

### | /

^{a}

^{k}*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

*→0*

_{t}*o*

### (

^{t}### )/

^{t}### =

*0. Since a*

^{k}### → +∞

*as k*

### → ∞

, we have_{(}

_{a}^{|}

_{k}

^{b}_{)}

^{|}

_{np}*−1*

^{np}### →

*0 for all n. This together with the last equation*implies lim

*→∞*

_{k}### φ

*p*

### (

^{a}

^{k}### ,

^{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*

*f*

_{1}

### (

^{t}### ) := (

^{t}### + µ

1### )

^{−}

^{p}^{−}

^{p}^{1}

### − (

^{t}### + µ

2### )

^{−}

^{p}^{−}

^{p}^{1}

### (

^{t}### >

^{0}

### )

*and*

*f*_{2}

### (

^{t}### ) := (

^{t}### + µ

2### )

^{p}^{−}

^{p}^{1}

### − (

^{t}### + µ

1### )

^{p}^{−}

^{p}^{1}

### (

^{t}### ≥

0### )

*are decreasing on*

### (

^{0}

### , +∞)

*, and furthermore, f*

_{2}

### (

^{t}### ) ≤

*2*

^{f}### (

^{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}### } | −∞ <

^{l}*i*

### <

^{u}*i*

### = +∞} ,

*I*

_{u}### := {

*i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### } | −∞ =

*l*

_{i}### <

^{u}*i*

### < +∞} ,

*I*

_{lu}### := {

*i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### } | −∞ <

^{l}*i*

### <

^{u}*i*

### < +∞} ,

*I*

_{f}### := {

*i*

### ∈ {

1### ,

^{2}

### , . . . ,

^{n}### } | −∞ =

*l*

_{i}### <

^{u}*i*

### = +∞} .

(8)

With the generalized FB function, we define a operatorΦ*p*

### :

_{R}

^{n}### →

_{R}

*componentwise as*

^{n}Φ*p*,*i*

### (

^{x}### ) :=

###

###

###

### φ

*p*

### (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ))

^{if i}### ∈

*I*

_{l}### ,

### − φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ))

^{if i}### ∈

*I*

_{u}### , φ

*p*

### (

^{x}*i*

### −

*l*

_{i}### , φ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### )))

^{if i}### ∈

*I*

_{lu}### ,

### −

*F*

_{i}### (

^{x}### )

^{if i}### ∈

*I*

_{f}### ,

(9)

*where the minus sign for i*

### ∈

*I*

_{u}*and i*

### ∈

*I*

*is 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.*

_{f}**Proposition 3.1. x**^{∗}

### ∈

_{R}

^{n}*is a solution of the MCP*(1)

*if and only if x*

^{∗}

*solves the nonlinear system of equations*Φ

*p*

### (

^{x}### ) =

^{0.}We want to point out that, unlike for the nonlinear complementarity problem, when writing the generalized FB function

### φ

*p*as

### φ

*p*

### (

^{a}### ,

^{b}### ) = (

^{a}### +

*b*

### ) − k(

^{a}### ,

^{b}### )k

*p*

### ,

the conclusion ofProposition 3.1*does not necessarily hold since, if I*

_{l}### = {

1### ,

^{2}

### , . . . ,

^{n}### }

, then

_{x}### ¯ =

*l satisfies*Φ

*p*

### (¯

^{x}### ) =

^{0, but F}### (¯

^{x}### ) ≥

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

*p*by the minimum NCP function.

Since

### φ

*p*is not differentiable at the origin, the systemΦ

*p*

### (

^{x}### ) =

0 is nonsmooth. In this paper, we will find a solution of nonsmooth systemΦ*p*

### (

^{x}### ) =

0 by solving a sequence of smooth approximationsΨ*p*

### (

^{x}### , ε) =

^{0, where}

### ε >

0 is a smoothing parameter and the operatorΨ*p*

### :

_{R}

^{n}### ×

_{R}

_{++}

### →

_{R}

*is defined componentwise as*

^{n}Ψ*p*,*i*

### (

^{x}### , ε) :=

###

###

###

### ψ

*p*

### (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)

^{if i}### ∈

*I*

_{l}### ,

### − ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ), ε)

^{if i}### ∈

*I*

_{u}### , ψ

*p*

*x*

_{i}### −

*l*

_{i}### , ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{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

*p*and the operatorΨ

*p*. First, let us state the favorable properties of

### ψ

*p*.

**Lemma 3.1. Let**

### ψ

*p*

### :

_{R}

^{3}

### →

*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}### ) ∈

R^{2}

*with*

### −

2### < ∂ψ

*p*

### (

^{a}### ,

^{b}### , ε)

### ∂

^{a}### <

^{0}

### , −

2### < ∂ψ

*p*

### (

^{a}### ,

^{b}### , ε)

### ∂

^{b}### <

^{0}

### .

^{(12)}

*(b) For any fixed*

### (

^{a}### ,

^{b}### ) ∈

R^{2}

*,*

### 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}### ) ∈

R^{2}

*, let*

### ψ

*p*

^{0}

### (

^{a}### ,

^{b}### ) :=

limε↓0∂ψ*p*(*a*,*b*,ε)

∂*a*

### ,

^{lim}

ε↓0

∂ψ*p*(*a*,*b*,ε)

∂*b*

### .

^{Then,}lim

*h*=(*h*1,*h*2)→(0,0)

### φ

*p*

### (

^{a}### +

*h*

_{1}

### ,

^{b}### +

*h*

_{2}

### ) − φ

*p*

### (

^{a}### ,

^{b}### ) − ψ

*p*

^{0}

### (

^{a}### +

*h*

_{1}

### ,

^{b}### +

*h*

_{2}

### )

^{T}

^{h}### 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}

^{∂ψ}

^{p}^{(}

_{∂}

^{a}

_{a}^{,}

^{b}^{,ε)}

^{and}

^{∂ψ}

^{p}^{(}

_{∂}

^{a}

_{b}^{,}

^{b}^{,ε)}are continuous at all

### (

^{a}### ,

^{b}### ) ∈

R^{2}, it follows that

### ψ

*p*

### (

^{a}### ,

^{b}### , ε)

is continuously differentiable at all### (

^{a}### ,

^{b}### ) ∈

R^{2}. 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}### ) ∈

R^{2},

### 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

*p*

^{0}

### (

^{a}### ,

^{b}### ) =

_{∂φ}

*p*(*a*,*b*)

∂*a*

### ,

^{∂φ}

^{p}_{∂}

^{(}

_{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*

### (

*1*

^{h}### ,

*2*

^{h}### ) − φ

*p*

### (

^{0}

### ,

^{0}

### ) − ψ

*p*

^{0}

### (

*1*

^{h}### ,

*2*

^{h}### )

^{T}

^{h}### = p

^{p}### |

*h*

_{1}

### |

^{p}### + |

*h*

_{2}

### |

^{p}### − |

*h*

_{1}

### |

^{p}### + |

*h*

_{2}

### |

^{p}### ( √

^{p}### |

*h*

_{1}

### |

^{p}### + |

*h*

_{2}

### |

^{p}### )

^{p}^{−}

^{1}

### = p

^{p}### |

*h*

_{1}

### |

^{p}### + |

*h*

_{2}

### |

^{p}### − p

^{p}### |

*h*

_{1}

### |

^{p}### + |

*h*

_{2}

### |

^{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}

### ≤

*a*

^{2}

### +

*b*

^{2}

### + 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}### ) = (

^{2}

^{p}### −

2### )

^{a}

^{p}### − ε

^{p}### ≤

*f*

### (

^{b}### ) =

0. From this inequality, we get min### {

*a*

### ,

^{b}### } =

*a*

### ≤

^{√}

_{p}^{ε}

2* ^{p}*−2.
UsingLemma 3.1and the expression ofΨ

*p*, we readily obtain the following result.

* Proposition 3.2. Let*Ψ

*p*

*be defined by*(10). Then, the following results hold.

*(a) For any fixed*

### ε >

*Ψ*

^{0,}*p*

### (

^{x}### , ε)

*is continuously differentiable on R*

^{n}*with*

### ∇

_{x}_{Ψ}

_{p}### (

^{x}### , ε) =

^{D}*a*

### (

^{x}### , ε) + ∇

^{F}### (

^{x}### )

^{D}*b*

### (

^{x}### , ε),

*where D*_{a}

### (

^{x}### , ε)

^{and D}*b*

### (

^{x}### , ε)

^{are n}### ×

*n diagonal matrices with the diagonal elements*

### (

^{D}*a*

### )

*ii*

### (

^{x}### , ε)

^{and}### (

^{D}*b*

### )

*ii*

### (

^{x}### , ε)

^{defined as}*follows:*

*(a1) For i*

### ∈

*I*

_{l}*,*

### (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) =

^{sign}

### (

^{x}*i*

### −

*l*

_{i}### )|

^{x}*i*

### −

*l*

_{i}### |

^{p}^{−}

^{1}

### k (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### −

1### ,

### (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) =

^{sign}

### (

^{F}*i*

### (

^{x}### ))|

^{F}*i*

### (

^{x}### )|

^{p}^{−}

^{1}

### k (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### −

1### .

*(a2) For i*

### ∈

*I*

_{u}*,*

### (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) =

^{sign}

### (

^{u}*i*

### −

*x*

_{i}### )|

^{u}*i*

### −

*x*

_{i}### |

^{p}^{−}

^{1}

### k (

^{u}*i*

### −

*x*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### −

1### ,

### (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) = −

sign### (

^{F}*i*

### (

^{x}### ))|

^{F}*i*

### (

^{x}### )|

^{p}^{−}

^{1}

### k (

^{u}*i*

### −

*x*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### −

1### .

*(a3) For i*

### ∈

*I*

_{lu}*,*

### (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) =

^{a}*i*

### (

^{x}### , ε) +

^{b}*i*

### (

^{x}### , ε)

^{c}*i*

### (

^{x}### , ε)

^{and}### (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) =

^{b}*i*

### (

^{x}### , ε)

^{d}*i*

### (

^{x}### , ε)

*with*

*a*_{i}

### (

^{x}### , ε) =

^{sign}

### (

^{x}*i*

### −

*l*

_{i}### )|

^{x}*i*

### −

*l*

_{i}### |

^{p}^{−}

^{1}

### (

^{x}*i*

### −

*l*

_{i}### , ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ), ε), ε)

*p*−1
*p*

### −

1### ,

*b*_{i}

### (

^{x}### , ε) =

^{sign}

### (ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ), ε))|ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ), ε)|

^{p}^{−}

^{1}

### (

^{x}*i*

### −

*l*

_{i}### , ψ

*p*

### (

^{u}*i*

### −

*x*

_{i}### , −

^{F}*i*

### (

^{x}### ), ε), ε)

*p*−1
*p*

### −

1### ,

*c*_{i}

### (

^{x}### , ε) = −

^{sign}

### (

^{u}*i*

### −

*x*

_{i}### )|

^{u}*i*

### −

*x*

_{i}### |

^{p}^{−}

^{1}

### k (

^{u}*i*

### −

*x*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### +

1### ,

*d*_{i}

### (

^{x}### , ε) =

^{sign}

### (

^{F}*i*

### (

^{x}### ))|

^{F}*i*

### (

^{x}### )|

^{p}^{−}

^{1}

### k (

^{u}*i*

### −

*x*

_{i}### ,

^{F}*i*

### (

^{x}### ), ε)k

^{p}*p*

^{−}

^{1}

### +

1### .

*(a4) For i*

### ∈

*I*

_{f}*,*

### (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) =

^{0 and}### (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) = −

^{1.}*Moreover,*

### −

2### < (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) <

^{0 and}### −

2### < (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) <

*0 for all i*

### ∈

*I*

_{l}### ∪

*I*

_{u}*, and*

### −

6### < (

^{D}*a*

### )

*ii*

### (

^{x}### , ε) <

^{0 and}### −

4### < (

^{D}*b*

### )

*ii*

### (

^{x}### , ε) <

^{0 for i}### ∈

*I*

_{lu}*.*

*(b) For any given*

### ε

1### >

^{0 and}### ε

2### >

^{0, we have}### k

_{Ψ}

_{p}### (

^{x}### , ε

2### ) −

Ψ*p*

### (

^{x}### , ε

1### )k ≤ √

*n*

### √

*2*

_{p}### +

1### | ε

2### − ε

1### | , ∀

^{x}### ∈

_{R}

^{n}### .

*Particularly, for any given*

### ε >

^{0,}### k

_{Ψ}

_{p}### (

^{x}### , ε) −

Φ*p*

### (

^{x}### )k ≤ √

*n*

### √

_{p}2

### +

1### ε, ∀

^{x}### ∈

_{R}

^{n}### .

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 operatorΨ*p*satisfies the Jacobian consistency property, we need the following
characterization of the generalized Jacobian

### ∂

*C*Φ

*p*

### (

^{x}### )

, which is direct byLemma 2.2(b).**Proposition 3.3. For any given x**

### ∈

_{R}

^{n}*,*

### ∂

*C*Φ

*p*

### (

^{x}### )

^{T}### = {

*D*

_{a}### (

^{x}### ) + ∇

^{F}### (

^{x}### )

^{D}*b*

### (

^{x}### )}

^{, where D}*a*

### (

^{x}### ),

^{D}*b*

### (

^{x}### )

^{are n}### ×

*n diagonal matrices*

*whose diagonal elements are given as below:*

*(a) For i*

### ∈

*I*

_{l}*, if*

### (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### )) 6= (

^{0}

### ,

^{0}

### )

^{, then}### (

^{D}*a*

### )

*ii*

### (

^{x}### ) =

^{sign}

### (

^{x}*i*

### −

*l*

_{i}### ) · |

^{x}*i*

### −

*l*

_{i}### |

^{p}^{−}

^{1}

### k (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ))k

^{p}*p*

^{−}

^{1}

### −

1### , (

^{D}*b*

### )

*ii*

### (

^{x}### ) =

^{sign}

### (

^{F}*i*

### (

^{x}### )) · |

^{F}*i*

### (

^{x}### )|

^{p}^{−}

^{1}

### k (

^{x}*i*

### −

*l*

_{i}### ,

^{F}*i*

### (

^{x}### ))k

^{p}*p*

^{−}

^{1}