Nonlinear Analysis: Theory, Methods and Applications, vol. 72, pp. 3739-3758, 2010

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

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

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

Shaohua Pan^{2}

School of Mathematical Sciences South China University of Technology

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

Tzu-Ching Lin

Department of Mathematics National Taiwan Normal University

Taipei, Taiwan 11677

E-mail: cashplayer35@yahoo.com.tw

June 4, 2009

(revised on December 11, 2009)

**Abstract. 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 [13] to the solution set. In particular, we
*investigate the inﬂuence of p on the numerical performance of the algorithm by solving*

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

2The author’s work is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001).

*all MCPLIP test problems, and conclude that the smoothing algorithm with p* *∈ (1, 2]*

*has better numerical performance than the one with p > 2.*

**Key Words. Mixed complementarity problem, the generalized FB function, smoothing**
approximation, convergence rate.

**1** **Introduction**

The mixed complementarity problem (MCP) arises in many applications including the ﬁelds of economics, engineering, and operations research [14, 19, 20, 23] and has attracted much attention in last decade [2, 3, 18, 24, 25, 26]. A collection of nonlinear mixed com- plementarity problems called MCPLIB can be found in [16] and the excellent book [17]

is a good source for seeking theoretical backgrounds and numerical methods.

*Given a mapping F : [l, u]→ IR*^{n}*with F = (F*_{1}*, . . . , F** _{n}*)

^{T}*, where l = (l*

_{1}

*, . . . , l*

*)*

_{n}*and*

^{T}*u = (u*

_{1}

*, . . . , u*

*)*

_{n}

^{T}*with l*

_{i}*∈ IR ∪ {−∞} and u*

*i*

*∈ IR ∪ {+∞} 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}* =

*⇒ F*

*i*

*(x*

*)*

^{∗}*≥ 0,*

*x*

^{∗}

_{i}*∈ (l*

*i*

*, u*

*i*) =

*⇒ F*

*i*

*(x*

^{∗}*) = 0,*

*x*

^{∗}

_{i}*= u*

*=*

_{i}*⇒ 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∈ IR*

*such that*

^{n}*x≥ 0, F (x) ≥ 0, ⟨x, F (x)⟩ = 0.* (3)

In fact, from Theorem 2 of [15], the MCP (1) is also equivalent to the famous variational
*inequality problem (VIP) which is to ﬁnd a vector x*^{∗}*∈ [l, u] such that*

*⟨F (x*^{∗}*), x− x*^{∗}*⟩ ≥ 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 : IR× IR → IR 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 system Φ(x) = 0,*
and consequently nonsmooth Newton methods or smoothing Newton methods can be ap-
*plied for solving the system Φ(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., [4, 5, 11, 21, 22, 24, 12]). 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. [12] 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 [6, 7, 8] by two of the authors. Speciﬁcally, they deﬁne the generalized FB function as

*ϕ*_{p}*(a, b) :=∥(a, b)∥**p**− (a + b)* *∀a, b ∈ IR,* (6)
*where p is an arbitrary ﬁxed real number from the interval (1, +∞) and ∥(a, b)∥**p* denotes
*the p-norm of (a, b), i.e.,∥(a, b)∥**p* = ^{p}

√*|a|** ^{p}*+

*|b|*

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

*, they*

_{p}*replace the 2-norm of (a, b) involved in the FB function by a more general p-norm. The*

*function ϕ*

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

_{p}*it turns out that ϕ*

*possesses all favorable properties of the FB function; see [6, 7, 8]. For*

_{p}*example, ϕ*

*is strongly semismooth and its square is a continuously diﬀerentiable NCP function. Particularly, numerical results in [9] for all MCPLIB problems indicate that*

_{p}*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*

*has similar numerical performance.*

_{p}In this paper, we are concerned with the smoothing Newton method [12] 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 Section 3, we present a smoothing function of the generalized
FB function, and studied some of its favorable properties, including the Jacobian consis-
tency property; in Section 4, 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 Section 5, we report the numerical results of the smoothing
algorithm for solving the MCPLIB test problems.

Throughout this paper, IR^{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*^{′}*(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*^{′}*(x)]*_{II}*denotes 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 : IR*^{n}*→ IR*^{m}*be a locally Lipschitz continuous mapping. Then, G is almost everywhere*
*diﬀerentiable by Rademacher’s Theorem (see [10]). 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* *∈ IR*^{m}^{×n}*| ∃{x*^{k}*} ⊆ D**G* :*{x*^{k}*} → x and G*^{′}*(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 [30] proposed so-called C-subdiﬀerential of G:*

*∂*_{C}*G(x)*^{T}*:= ∂G*_{1}*(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 IR^{n}^{×m}*whose i-th column is given by the gen-*
*eralized gradient of the i-th component function G**i*. In fact, by Proposition 2.6.2 of
*[10], ∂G(x)*^{T}*⊆ ∂**C**G(x)** ^{T}*. We assume that the reader is familiar with the concepts of
(strongly) semismooth functions, and refer to [28, 29] for details.

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

* Definition 2.1 Let F = (F*1

*, . . . , F*

*n*)

^{T}*with F*

*i*: IR

^{n}*→ IR for i = 1, 2, . . . , n. Then,*

**(a) the mapping F is called a P**_{0}

*-function if, for every x and y in IR*

^{n}*with x̸= y, there*

*is an index i∈ {1, 2, . . . , n} such that*

*x*_{i}*̸= y**i* *and (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 IR**^{n}*with x* *̸= y, there*
*is an index i∈ {1, 2, . . . , n} such that*

*x*_{i}*̸= y**i* *and (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 IR*^{n}*, there is an index i∈ {1, 2, . . . , n} such that*

*(x*_{i}*− y**i**)(F*_{i}*(x)− F**i**(y))≥ µ∥x − y∥*^{2}*.*

**Definition 2.2 A matrix M***∈ IR*^{n}^{×n}*is called an*

**(a) P**_{0}*-matrix if each of its principal minors is nonnegative.*

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

From Deﬁnitions 2.1 and 2.2, it is not hard to see that a continuously diﬀerentiable
*mapping F is a P*_{0}-function if and only if *∇F (x) is P*0*-matrix for all x* *∈ IR** ^{n}*. For the

*P*

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

**Lemma 2.1 [17] A matrix M***∈ IR*^{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}*̸= 0 and x**i**(M x)*_{i}*≥ 0.*

*Next we recall some favorable properties of ϕ** _{p}* whose proofs can be found in [6, 7, 8].

**Lemma 2.2 Let ϕ*** _{p}* : IR

*× IR → IR be deﬁned by (6). Then, the following results hold.*

**(a) ϕ**_{p}*is a strongly semismooth NCP-function.*

**(b) Given any point (a, b)***∈ IR*^{2}*, each element in the generalized gradient ∂ϕ*_{p}*(a, b) has*
*the representation (ξ− 1, ζ − 1) where, if (a, b) ̸= (0, 0),*

*(ξ, ζ) =*

(*sign(a)· |a|*^{p−1}

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

*∥(a, b)∥*^{p}^{p}^{−1}

)

*,*

*and otherwise (ξ, ζ) is an arbitrary vector in IR*^{2} *satisfying* *|ξ|*^{p}^{−1}* ^{p}* +

*|ζ|*

^{p}

^{−1}

^{p}*≤ 1.*

**(c) The square of ϕ**_{p}*is a continuously diﬀerentiable NCP function.*

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

*} ⊆ IR*

^{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*: IR*× IR → IR 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}*} ⊆ IR be any sequence converging to +∞ as k → ∞ and b ∈ IR 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}*)

^{p}^{)}

^{1/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*

_{t}

_{→0}*o(t)/t = 0. Since a*

^{k}*→ +∞ as k → ∞, we have*

*|b|*

^{np}*(a** ^{k}*)

^{np}

^{−1}*→ 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}.

*2*

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 here omit it.

* Lemma 2.4 For any ﬁxed 0≤ µ*1

*< µ*

_{2}

*, the following functions*

*f*1

*(t) := (t + µ*1)

^{−}

^{p−1}

^{p}*− (t + µ*2)

^{−}

^{p−1}

^{p}*(t > 0)*

*and*

*f*_{2}*(t) := (t + µ*_{2})^{p}^{−1}^{p}*− (t + µ*1)^{p}^{−1}^{p}*(t≥ 0)*

*are decreasing on (0, +∞), and furthermore, f*2*(t)≤ f*2*(0) = µ*^{(p}_{2} ^{−1)/p}*− µ** ^{(p}*1

^{−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 operator Φ* _{p}*: IR

^{n}*→IR*

*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**f* is motivated by Lemma 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**^{∗}*∈ IR*^{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)− ∥(a, b)∥**p**, the conclusion of*
*Proposition 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*

*by the minimum NCP function.*

_{p}*Since ϕ** _{p}* is not diﬀerentiable at the origin, the system Φ

_{p}*(x) = 0 is nonsmooth. In*this paper, we will ﬁnd a solution of nonsmooth system Φ

_{p}*(x) = 0 by solving a sequence*of smooth approximations Ψ

*p*

*(x, ε) = 0, where ε > 0 is a smoothing parameter and the*operator Ψ

*: IR*

_{p}

^{n}*× IR*++

*→ IR*

*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}

√*|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 operator Ψ*p**. First, let us state the favorable properties of ψ**p*.

**Lemma 3.1 Let ψ*** _{p}* : IR

^{3}

*→ IR 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)∈ IR*^{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)**∈ IR^{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)**∈ IR^{2}*, let ψ*_{p}^{0}*(a, b) :=*

(

lim*ε**↓0*

*∂ψ*_{p}*(a, b, ε)*

*∂a* *, lim*

*ε**↓0*

*∂ψ*_{p}*(a, b, ε)*

*∂b*

)

*. Then,*

*h=(h*1lim*,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*

*∥h∥* *= 0.*

**(d) For any given ε > 0, if p**≥ 2, then ψ*p**(a, b, ε) = 0 =⇒ a > 0, b > 0, 2ab ≤ ε*^{2}*, and*
*whenever p > 1, ψ*_{p}*(a, b, ε) = 0 =⇒ a > 0, b > 0, min{a, b} ≤* *ε*

*√**p*

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, b, ε)*

*∂a* and *∂ψ*_{p}*(a, b, ε)*

*∂b* *are continuous at all (a, b)* *∈ IR*^{2},
*it follows that ψ**p**(a, b, ε) is continuously diﬀerentiable at all (a, b)∈ IR*^{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)* *∈ IR*^{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

lim*ε**↓0*

*∂ψ*_{p}*(a, b, ε)*

*∂a* =

*sign(a)|a|*^{p}^{−1}

(√*p*

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

*|b|*

^{p}^{)}

^{p}

^{−1}*− 1 if (a, b) ̸= (0, 0),*

*−1* *if (a, b) = (0, 0);*

lim*ε**↓0*

*∂ψ*_{p}*(a, b, ε)*

*∂b* =

*sign(b)|b|*^{p}^{−1}

(

*p*

√*|a|** ^{p}*+

*|b|*

^{p}^{)}

^{p}

^{−1}*− 1 if (a, b) ̸= (0, 0),*

*−1* *if (a, b) = (0, 0).*

(15)

*From this, we see that ψ*_{p}^{0}*(a, b) =*^{(}^{∂ϕ}^{p}_{∂a}^{(a,b)}*,*^{∂ϕ}^{p}_{∂b}^{(a,b)}^{)} *at (a, b)̸= (0, 0). Therefore, we only*
*need to check the case (a, b) = (0, 0). The desired result follows by*

*ϕ*_{p}*(h*_{1}*, h*_{2})*− ϕ**p**(0, 0)− ψ**p*^{0}*(h*_{1}*, h*_{2})^{T}*h*

= ^{p}

√*|h*1*|** ^{p}*+

*|h*2

*|*

^{p}*−*

*|h*1

*|*

*+*

^{p}*|h*2

*|*

*(*

^{p}

^{p}√*|h*1*|** ^{p}* +

*|h*2

*|*

*)*

^{p}

^{p}

^{−1}= ^{p}

√*|h*1*|** ^{p}*+

*|h*2

*|*

^{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*

√*|a|** ^{p}*+

*|b|*

^{p}*+ ε*

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

*|b|*

^{p}*+ ε*

^{p}*≤*

^{√}

*|a|*

^{2}+

*|b|*

^{2}

*+ ε*

^{2}

=*⇒ (a + b)*^{2} *≤ a*^{2}*+ b*^{2}*+ ε*^{2} =*⇒ 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 ≤* *ε*

*√**p*

2^{p}*− 2*.
*2*

Using Lemma 3.1 and the expression of Ψ* _{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 > 0, Ψ***p**(x, ε) is continuously diﬀerentiable on IR*^{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}*∥(x**i**− l**i**, F*_{i}*(x), ε)∥*^{p}_{p}^{−1}*− 1,*
*(D** _{b}*)

_{ii}*(x, ε) =*

*sign(F*

*i*

*(x))|F*

*i*

*(x)|*

^{p}

^{−1}*∥(x**i**− l**i**, F*_{i}*(x), ε)∥*^{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}*∥(u**i**− x**i**, F*_{i}*(x), ε)∥*^{p}_{p}^{−1}*− 1,*
*(D** _{b}*)

_{ii}*(x, ε) =*

*−sign(F*

*i*

*(x))|F*

*i*

*(x)|*

^{p}

^{−1}*∥(u**i**− x**i**, F*_{i}*(x), ε)∥*^{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}_{p}^{−1}*− 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}_{p}^{−1}*− 1,*
*c*_{i}*(x, ε) =* *−sign(u*_{i}*− x**i*)*|u**i**− x**i**|*^{p}^{−1}

*∥(u**i**− x**i**, F*_{i}*(x), ε)∥*^{p}_{p}^{−1}*+ 1,*
*d*_{i}*(x, ε) =* *sign(F*_{i}*(x))|F**i**(x)|*^{p}^{−1}

*∥(u**i**− x**i**, F*_{i}*(x), ε)∥*^{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*

*∥Ψ**p**(x, ε*_{2})*− Ψ**p**(x, ε*_{1})*∥ ≤√*
*n*^{(}*√*^{p}

2 + 1^{)}*|ε*2*− ε*1*|,* *∀x ∈ IR*^{n}*.*
*Particularly, for any given ε > 0,*

*∥Ψ**p**(x, ε)− Φ**p**(x)∥ ≤√*
*n*^{(}*√*^{p}

2 + 1^{)}*ε,* *∀x ∈ IR*^{n}*.*

The Jacobian consistency property plays a crucial role in the analysis of local fast
convergence of the smoothing algorithm [12]. To show that the smoothing operator Ψ* _{p}*
satisﬁes the Jacobian consistency property, we need the following characterization of the

*generalized Jacobian ∂*

*C*Φ

*p*

*(x), which is direct by Lemma 2.2 (b).*

**Proposition 3.3 For any given x***∈ IR*^{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))* *̸= (0, 0), then*

*(D** _{a}*)

_{ii}*(x) =*

*sign(x*

*i*

*− l*

*i*)

*· |x*

*i*

*− l*

*i*

*|*

^{p}

^{−1}*∥(x**i**− l**i**, F**i**(x))∥*^{p}^{p}^{−1}*− 1,*
*(D** _{b}*)

_{ii}*(x)*=

*sign(F*

_{i}*(x))· |F*

*i*

*(x)|*

^{p}

^{−1}*∥(x**i**− l**i**, F*_{i}*(x))∥*^{p}^{p}^{−1}*− 1;*

*and otherwise*

*((D**a*)*ii**(x), (D**b*)*ii**(x))* *∈*^{{}*(ξ− 1, ζ − 1) ∈ IR*^{2} *| |ξ|*^{p}^{−1}* ^{p}* +

*|ζ|*

^{p}

^{−1}

^{p}*≤ 1*

^{}}

*.*

**(b) For i**∈ I*u*

*, if (u*

_{i}*− x*

*i*

*,−F*

*i*

*(x))̸= (0, 0), then*

*(D** _{a}*)

_{ii}*(x) =*

*sign(u*

_{i}*− x*

*i*)

*· |u*

*i*

*− x*

*i*

*|*

^{p−1}*∥(u**i**− x**i**,−F**i**(x))∥*^{p−1}^{p}*− 1,*
*(D** _{b}*)

_{ii}*(x)*=

*−sign(F*

*i*

*(x))· |F*

*i*

*(x)|*

^{p}

^{−1}*∥(u**i**− x**i**,−F**i**(x))∥*^{p}^{p}^{−1}*− 1;*

*and otherwise*

*((D** _{a}*)

_{ii}*(x), (D*

*)*

_{b}

_{ii}*(x))*

*∈*

^{{}

*(ξ− 1, ζ − 1) ∈ IR*

^{2}

*| |ξ|*

^{p}

^{−1}*+*

^{p}*|ζ|*

^{p}

^{−1}

^{p}*≤ 1*

^{}}

*.*

**(c) For i***∈ I**lu**, (D** _{a}*)

_{ii}*(x) = a*

_{i}*(x) + b*

_{i}*(x)c*

_{i}*(x) and (D*

*)*

_{b}

_{ii}*(x) = b*

_{i}*(x)d*

_{i}*(x) where, if*

*(x*

_{i}*− l*

*i*

*, ϕ*

_{p}*(u*

_{i}*− x*

*i*

*,−F*

*i*

*(x)))̸= (0, 0), then*

*a**i**(x) =* *sign(x*_{i}*− l**i*)*· |x**i**− l**i**|*^{p}^{−1}

*∥(x**i**− l**i**, ϕ**p**(u**i**− x**i**,−F**i**(x))∥*^{p}_{p}^{−1}*− 1,*

*b*_{i}*(x)* = *sign (ϕ*_{p}*(u*_{i}*− x**i**,−F**i**(x)))· |ϕ**p**(u*_{i}*− x**i**,−F**i**(x))|*^{p}^{−1}

*∥(x**i**− l**i**, ϕ*_{p}*(u*_{i}*− x**i**,−F**i**(x))∥*^{p}_{p}^{−1}*− 1,*

*and otherwise*

*(a*_{i}*(x), b*_{i}*(x))* *∈*^{{}*(ξ− 1, ζ − 1) ∈ IR*^{2} *| |ξ|*^{p−1}* ^{p}* +

*|ζ|*

^{p−1}

^{p}*≤ 1*

^{}};

*and if (u*

_{i}*− x*

*i*

*,−F*

*i*

*(x))*

*̸= (0, 0), then*

*c*_{i}*(x)* = *−sign(u**i* *− x**i*)*· |u**i**− x**i**|*^{p}^{−1}

*∥(u**i**− x**i**,−F**i**(x))∥*^{p}_{p}^{−1}*+ 1,*
*d**i**(x) =* *sign (F*_{i}*(x))· |F**i**(x)|*^{p}^{−1}

*∥(u**i**− x**i**,−F**i**(x))∥*^{p}_{p}^{−1}*+ 1,*
*and otherwise*

*(c*_{i}*(x), d*_{i}*(x))* *∈*^{{}*(ξ + 1, ζ + 1)∈ IR*^{2} *| |ξ|*^{p−1}* ^{p}* +

*|ζ|*

^{p−1}

^{p}*≤ 1*

^{}}

*.*

**(d) For i**∈ I*f*

*, (D*

*a*)

*ii*

*(x) = 0 and (D*

*b*)

*ii*

*(x) =−1.*

Now we are in a position to establish the Jacobian consistency of the operator Ψ* _{p}*.

**Proposition 3.4 Let Ψ**_{p}*be deﬁned by (10). Then, for any ﬁxed x∈ IR*^{n}*,*
lim*ε**↓0*dist(*∇**x*Ψ_{p}*(x, ε)*^{T}*, ∂** _{C}*Φ

_{p}*(x)) = 0.*

**Proof. For the sake of notation, for any given x**∈ IR* ^{n}*, we deﬁne the index sets:

*β*_{1}*(x) :=* *{i ∈ I**l* *| (x**i**− l**i**, F*_{i}*(x)) = (0, 0)}, ¯β*1*(x) := I*_{l}*\ β*1*(x),*

*β*_{2}*(x) :=* *{i ∈ I**u* *| (u**i**− x**i**, F*_{i}*(x)) = (0, 0)}, ¯β*2*(x) := I*_{u}*\ β*2*(x),* (16)
*β*_{3}*(x) :=* *{i ∈ I**lu* *| (x**i**− l**i**, ϕ*_{p}*(u*_{i}*− x**i**,−F**i**(x))) = (0, 0)}, ¯β*3*(x) := I*_{lu}*\ β*3*(x),*
*β*4*(x) :=* *{i ∈ ¯β*3*(x)| (u**i**− x**i**, F**i**(x)) = (0, 0)}, ¯β*4*(x) := ¯β*3*(x)\ β*4*(x).*

*We proceed the arguments by the cases i∈ I**l**∪ I**u**, i∈ I**lu* *and i∈ I**f*, respectively.

*Case 1: i∈ I**l**∪ I**u**. When i∈ β*1*(x)∪ β*2*(x), it is easy to see that*
*(D** _{a}*)

_{ii}*(x, ε) =−1 and (D*

*b*)

_{ii}*(x, ε) =*

*−1.*

By Proposition 3.2 (a1) and (a2),*∇**x*Ψ_{p,i}*(x, ε)** ^{T}* =

*−e*

^{T}

_{i}*− F*

_{i}

^{′}*(x) for all ε > 0. Since*(

*−1, −1) ∈*

^{{}

*(ξ− 1, ζ − 1) ∈ IR*

^{2}

*| |ξ|*

^{p}

^{−1}*+*

^{p}*|ζ|*

^{p}

^{−1}

^{p}*≤ 1*

^{}}

*,*(17) from Proposition 3.3 (a) and (b) we get

*∇*

*x*Ψ

_{p,i}*(x, ε)*

^{T}*∈ ∂*

*C*Φ

_{p,i}*(x). When i∈ ¯β*1

*(x)∪ ¯β*2

*(x),*

lim*ε**↓0* *(D** _{a}*)

_{ii}*(x, ε) = (D*

*)*

_{a}

_{ii}*(x) and lim*

*ε**↓0* *(D** _{b}*)

_{ii}*(x, ε) = (D*

*)*

_{b}

_{ii}*(x),*

which together with Proposition 3.2 (a1) and (a2) implies that

lim*ε**↓0* *∇**x*Ψ_{p,i}*(x, ε)*^{T}*= (D** _{a}*)

_{ii}*(x)e*

^{T}

_{i}*+ (D*

*)*

_{b}

_{ii}*(x)F*

_{i}

^{′}*(x)∈ ∂*

*C*Φ

_{p,i}*(x).*

*Since I**l**∪ I**u* *= β*1*(x)∪ β*2*(x)∪ ¯β*1*(x)∪ ¯β*2*(x), the last two subcases show that*

lim*ε**↓0* *∇**x*Ψ_{p,i}*(x, ε)*^{T}*∈ ∂**C*Φ_{p,i}*(x),* *∀ i ∈ I**l**∪ I**u**.* (18)

*Case 2: i∈ I**lu**. When i∈ β*3*(x), we have x**i**− l**i* *= 0, ϕ**p**(u**i**− x**i**,−F**i**(x)) = 0, u**i**− x**i* *> 0*
*and F*_{i}*(x) = 0. Hence, c*_{i}*(x) = 0 and d*_{i}*(x) = 1. From Proposition 3.3 (c), it follows that*

*∂** _{C}*Φ

_{p,i}*(x) ={a*

*i*

*(x)e*

^{T}

_{i}*+ b*

_{i}*(x)F*

_{i}

^{′}*(x)}*(19) with

*(a*_{i}*(x), b*_{i}*(x))∈*^{{}*(ξ− 1, ζ − 1) ∈ IR*^{2} *| |ξ|*^{p−1}* ^{p}* +

*|ζ|*

^{p−1}

^{p}*≤ 1*

^{}}

*.*

*On the other hand, since a*

*i*

*(x, ε) =*

*−1, d*

*i*

*(x, ε) = 1 and*

*b**i**(x, ε) =* *|ψ**p**(u*_{i}*− x**i**,−F**i**(x), ε)|*^{p}^{−1}

(*|ψ**p**(u*_{i}*− x**i**,−F**i**(x), ε)|*^{p}*+ ε** ^{p}*)

^{p}

^{−1}

^{p}*− 1,*

*c*

*i*

*(x, ε) = 1−*

*|u*

*i*

*− x*

*i*

*|*

^{p}

^{−1}(*|u**i**− x**i**|*^{p}*+ ε** ^{p}*)

^{(p}

^{−1)/p}*,*from Proposition 3.2 (a3) it follows that

*∇**x*Ψ*p,i**(x, ε)** ^{T}* = (

*−1 + b*

*i*

*(x, ε)c*

*i*

*(x, ε))e*

^{T}

_{i}*+ b*

*i*

*(x, ε)F*

_{i}

^{′}*(x).*(20) Taking

*ξ = 0 and ζ =* *|ψ**p**(u*_{i}*− x**i**,−F**i**(x), ε)|*^{p}* ^{−1}*
(

*|ψ*

*p*

*(u*

_{i}*− x*

*i*

*,−F*

*i*

*(x), ε)|*

^{p}*+ ε*

*)*

^{p}

^{p}

^{−1}

^{p}*,*
it is not hard to verify that *|ξ|*^{p−1}* ^{p}* +

*|ζ|*

^{p−1}

^{p}*≤ 1, and consequently*

*−e*^{T}*i* *+ b**i**(x, ε)F*_{i}^{′}*(x)∈ ∂**C*Φ*p,i**(x).*

Noting that
lim*ε↓0*

*∇**x*Ψ*p,i**(x, ε)*^{T}*−*^{(}*−e*^{T}*i* *+ b**i**(x, ε)F*_{i}^{′}*(x)*^{)
}
= lim

*ε↓0**∥b**i**(x, ε)c**i**(x, ε)e*^{T}_{i}*∥ = 0,*
it then follows that

lim*ε**↓0* dist^{(}*∇**x*Ψ_{p,i}*(x, ε)*^{T}*, ∂** _{C}*Φ

_{p,i}*(x)*

^{)}

*= 0,*

*i∈ β*3

*(x).*

*When i∈ ¯β*3*(x), we have lim*_{ε}_{↓0}*a*_{i}*(x, ε) = a*_{i}*(x) and lim*_{ε}_{↓0}*b*_{i}*(x, ε) = b*_{i}*(x). Also,*
*c*_{i}*(x, ε) = 1, d*_{i}*(x, ε) = 1* *for i∈ β*4*(x)*