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 Pan2
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]→ IRn with F = (F1, . . . , Fn)T, where l = (l1, . . . , ln)T and u = (u1, . . . , un)T with li ∈ IR ∪ {−∞} and ui ∈ IR ∪ {+∞} being given lower and upper bounds satisfying li < 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 = li =⇒ Fi(x∗)≥ 0, x∗i ∈ (li, ui) =⇒ Fi(x∗) = 0, x∗i = ui =⇒ Fi(x∗)≤ 0.
(1)
It is not hard to see that, when li =−∞ and ui = +∞ for all i = 1, 2, . . . , n, the MCP (1) is equivalent to solving the nonlinear system of equations
F (x) = 0; (2)
whereas when li = 0 and ui = +∞ for all i = 1, 2, . . . , n, it reduces to the nonlinear complementarity problems (NCP) which is to find a point x∈ IRn such that
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
ϕp(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 ϕp, they replace the 2-norm of (a, b) involved in the FB function by a more general p-norm. The function ϕp is still an NCP-function, that is, it satisfies the equivalence in (5). Moreover, it turns out that ϕp possesses all favorable properties of the FB function; see [6, 7, 8]. For example, ϕp is strongly semismooth and its square is a continuously differentiable NCP function. Particularly, numerical results in [9] 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 ϕp has similar numerical performance.
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, IRn denotes the space of n-dimensional real column vectors and ei 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 : IRn→ IRm 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
∂BG(x) :={V ∈ IRm×n | ∃{xk} ⊆ DG :{xk} → x and G′(xk)→ V},
where DG is 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 [30] 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 IRn×m whose i-th column is given by the gen- eralized gradient of the i-th component function Gi. In fact, by Proposition 2.6.2 of [10], ∂G(x)T ⊆ ∂CG(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 = (F1, . . . , Fn)T with Fi : IRn→ IR for i = 1, 2, . . . , n. Then, (a) the mapping F is called a P0-function if, for every x and y in IRn with x̸= y, there
is an index i∈ {1, 2, . . . , n} such that
xi ̸= yi and (xi− yi)(Fi(x)− Fi(y))≥ 0;
(b) the mapping F is called a P -function if, for every x and y in IRn with x ̸= y, there is an index i∈ {1, 2, . . . , n} such that
xi ̸= yi and (xi− yi)(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 IRn, there is an index i∈ {1, 2, . . . , n} such that
(xi− yi)(Fi(x)− Fi(y))≥ µ∥x − y∥2.
Definition 2.2 A matrix M ∈ IRn×n is called an
(a) P0-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 P0-function if and only if ∇F (x) is P0-matrix for all x ∈ IRn. For the P0-matrix, we also have the following important property.
Lemma 2.1 [17] A matrix M ∈ IRn×n is a P0-matrix if and only if for every nonzero vector x, there exists an index i such that xi ̸= 0 and xi(M x)i ≥ 0.
Next we recall some favorable properties of ϕp whose proofs can be found in [6, 7, 8].
Lemma 2.2 Let ϕp : IR× IR → IR be defined by (6). Then, the following results hold.
(a) ϕp is a strongly semismooth NCP-function.
(b) Given any point (a, b) ∈ IR2, each element in the generalized gradient ∂ϕp(a, b) has the representation (ξ− 1, ζ − 1) where, if (a, b) ̸= (0, 0),
(ξ, ζ) =
(sign(a)· |a|p−1
∥(a, b)∥pp−1 ,sign(b)· |b|p−1
∥(a, b)∥pp−1
)
,
and otherwise (ξ, ζ) is an arbitrary vector in IR2 satisfying |ξ|p−1p +|ζ|p−1p ≤ 1.
(c) The square of ϕp is a continuously differentiable NCP function.
(d) If {(ak, bk)} ⊆ IR2 satisfies (ak → −∞) or (bk → −∞) or (ak → ∞ and bk → ∞), then we have |ϕp(ak, bk)| → ∞ as k → ∞.
The following lemma establishes another property of ϕp, which plays a key role in the nonsmooth system reformulation of the MCP (1) with the generalized FB function.
Lemma 2.3 Let ϕp: IR× IR → IR be defined by (6). Then, the following limits hold.
(a) lim
li→−∞ϕp(xi− li, ϕp(ui− xi,−Fi(x))) =−ϕp(ui− xi,−Fi(x)).
(b) lim
ui→∞ϕp(xi− li, ϕp(ui− xi,−Fi(x))) = ϕp(xi− li, Fi(x)).
(c) lim
li→−∞ lim
ui→∞ϕp(xi− li, ϕp(ui− xi,−Fi(x))) =−Fi(x).
Proof. Let {ak} ⊆ IR be any sequence converging to +∞ as k → ∞ and b ∈ IR be any fixed real number. We will prove lim
k→∞ϕp(ak, b) = −b, and part (a) then follows by continuity arguments. Without loss of generality, assume that ak> 0 for each k. Then,
ϕp(ak, b) = ak(1 + (|b|/ak)p)1/p− ak− b
= ak
1 + 1 p
(|b|
ak
)p
+1− p 2p2
(|b|
ak
)2p
+· · · + (1− p) · · · (1 − pn + p)
n!pn
(|b|
ak
)np
+ o
((|b|
ak
)pn)]
− ak− 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 limt→0o(t)/t = 0. Since ak → +∞ as k → ∞, we have |b|np
(ak)np−1 → 0 for all n. This together with the last equation implies limk→∞ϕp(ak, b) =−b. This proves part (a). Part (b) and (c) are direct by part (a) and the continuity of ϕFB. 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≤ µ1 < µ2, the following functions f1(t) := (t + µ1)−p−1p − (t + µ2)−p−1p (t > 0) and
f2(t) := (t + µ2)p−1p − (t + µ1)p−1p (t≥ 0)
are decreasing on (0, +∞), and furthermore, f2(t)≤ f2(0) = µ(p2 −1)/p− µ(p1−1)/p.
3 The smoothing function and its properties
For convenience, in the rest of this paper, we adopt the following notations of index sets:
Il := {i ∈ {1, 2, . . . , n} | − ∞ < li < ui = +∞} , Iu := {i ∈ {1, 2, . . . , n} | − ∞ = li < ui < +∞} , Ilu := {i ∈ {1, 2, . . . , n} | − ∞ < li < ui < +∞} , If := {i ∈ {1, 2, . . . , n} | − ∞ = li < ui = +∞} .
(8)
With the generalized FB function, we define a operator Φp: IRn→IRncomponentwise as
Φp,i(x) :=
ϕp(xi− li, Fi(x)) if i∈ Il,
−ϕp(ui− xi,−Fi(x)) if i∈ Iu, ϕp(xi− li, ϕp(ui− xi,−Fi(x))) if i∈ Ilu,
−Fi(x) if i∈ If,
(9)
where the minus sign for i∈ Iu and i ∈ If 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∗ ∈ IRn is a solution of the MCP (1) if and only if x∗ solves the nonlinear system of equations Φp(x) = 0.
We want to point out that, unlike for the nonlinear complementarity problem, when writing the generalized FB function ϕp as ϕp(a, b) = (a + b)− ∥(a, b)∥p, the conclusion of Proposition 3.1 does not necessarily hold since, if Il = {1, 2, . . . , n}, then ¯x = l satisfies Φp(¯x) = 0, but F (¯x) ≥ 0 does not necessarily hold. Similar phenomenon also appears when replacing ϕp by the minimum NCP function.
Since ϕp is not differentiable at the origin, the system Φp(x) = 0 is nonsmooth. In this paper, we will find a solution of nonsmooth system Φp(x) = 0 by solving a sequence of smooth approximations Ψp(x, ε) = 0, where ε > 0 is a smoothing parameter and the operator Ψp : IRn× IR++ → IRn is defined componentwise as
Ψp,i(x, ε) :=
ψp(xi− li, Fi(x), ε) if i∈ Il,
−ψp(ui− xi,−Fi(x), ε) if i∈ Iu, ψp(xi− li, ψp(ui− xi,−Fi(x), ε), ε) if i∈ Ilu,
−Fi(x) if i∈ If,
(10)
with
ψp(a, b, ε) := p
√|a|p+|b|p+ εp− (a + b). (11)
In what follows, we concentrate on the favorable properties of the smoothing function ψp and the operator Ψp. First, let us state the favorable properties of ψp.
Lemma 3.1 Let ψp : IR3 → IR be defined by (11). Then, the following result holds.
(a) For any fixed ε > 0, ψp(a, b, ε) is continuously differentiable at all (a, b)∈ IR2 with
−2 < ∂ψp(a, b, ε)
∂a < 0, −2 < ∂ψp(a, b, ε)
∂b < 0. (12)
(b) For any fixed (a, b)∈ IR2, ψp(a, b, ε) is continuously differentiable, strictly increasing and convex with respect to ε > 0. Moreover, for any 0 < ε1 ≤ ε2,
0≤ ψp(a, b, ε2)− ψp(a, b, ε1)≤ (ε2− ε1). (13) In particular, |ψp(a, b, ε)− ϕp(a, b)| ≤ ε for all ε ≥ 0.
(c) For any fixed (a, b)∈ IR2, let ψp0(a, b) :=
(
limε↓0
∂ψp(a, b, ε)
∂a , lim
ε↓0
∂ψp(a, b, ε)
∂b
)
. Then,
h=(h1lim,h2)→(0,0)
ϕp(a + h1, b + h2)− ϕp(a, b)− ψp0(a + h1, b + h2)Th
∥h∥ = 0.
(d) For any given ε > 0, if p≥ 2, then ψp(a, b, ε) = 0 =⇒ a > 0, b > 0, 2ab ≤ ε2, and whenever p > 1, ψp(a, b, ε) = 0 =⇒ a > 0, b > 0, min{a, b} ≤ ε
√p
2p− 2 . Proof. (a) Using an elementary calculation, we immediately obtain that
∂ψp(a, b, ε)
∂a = sign(a)|a|p−1
(
p
√|a|p+|b|p+ εp )p−1
− 1,
∂ψp(a, b, ε)
∂b = sign(b)|b|p−1
(
p
√|a|p+|b|p+ εp
)p−1 − 1. (14)
For any fixed ε > 0, since ∂ψp(a, b, ε)
∂a and ∂ψp(a, b, ε)
∂b are continuous at all (a, b) ∈ IR2, it follows that ψp(a, b, ε) is continuously differentiable at all (a, b)∈ IR2. Noting that
sign(a)|a|p−1
(√p
|a|p+|b|p + εp )p−1
< 1 and
sign(b)|b|p−1
(√p
|a|p+|b|p+ εp )p−1
< 1, we readily get the inequality (12).
(b) For any ε > 0, an elementary calculation yields that
∂ψp(a, b, ε)
∂ε = εp−1
(
p
√|a|p+|b|p+ εp )p−1
> 0,
∂2ψp(a, b, ε)
∂ε2 = (p− 1)εp−2
(
p
√|a|p+|b|p+ εp)p−1
(
1− εp
|a|p+|b|p+ εp
)
≥ 0.
Therefore, for any fixed (a, b) ∈ IR2, ψp(a, b, ε) is continuously differentiable, strictly increasing and convex with respect to ε > 0. By the mean-value theorem, for any 0 < ε1 ≤ ε2, there exists some ε0 ∈ (ε1, ε2) such that
ψp(a, b, ε2)− ψp(a, b, ε1) = ∂ψp
∂ε (a, b, ε0)(ε2− ε1).
Since ∂ψ∂εp(a, b, ε0)≤ 1 by the proof of part (a), inequality (13) holds for all 0 < ε1 ≤ ε2. Letting ε1 ↓ 0, the desired result then follows.
(c) Using the formula (14), it is easy to calculate that
limε↓0
∂ψp(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);
limε↓0
∂ψp(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 ψp0(a, b) =(∂ϕp∂a(a,b),∂ϕp∂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
ϕp(h1, h2)− ϕp(0, 0)− ψp0(h1, h2)Th
= p
√|h1|p+|h2|p− |h1|p+|h2|p (p
√|h1|p +|h2|p)p−1
= p
√|h1|p+|h2|p−√p|h1|p+|h2|p
= 0.
(d) From the definition of ψp(a, b, ε), clearly, ψp(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+ εp > p
√|a|p+|b|p ≥ max{|a|, |b|} ≥ a + b,
i.e., ψp(a, b, ε) > 0. Hence, for any given ε > 0, ψp(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 ψp(a, b, ε) = 0 ⇐⇒ a + b = √p|a|p+|b|p+ εp ≤√|a|2+|b|2+ ε2
=⇒ (a + b)2 ≤ a2+ b2+ ε2 =⇒ 2ab ≤ ε2.
(ii) For p > 1, without loss of generality, we assume 0 < a ≤ b. For any fixed a ≥ 0, consider f (t) = (t + a)p − tp − ap − εp (t ≥ 0). It is easy to verify that the function f is strictly increasing on [0, +∞). Since ψp(a, b, ε) = 0, we have f (b) = 0 which says f (a) = (2p−2)ap−εp ≤ f(b) = 0. From this inequality, we get min{a, b} = a ≤ ε
√p
2p − 2. 2
Using Lemma 3.1 and the expression of Ψp, we readily obtain the following result.
Proposition 3.2 Let Ψp be defined by (10). Then, the following results hold.
(a) For any fixed ε > 0, Ψp(x, ε) is continuously differentiable on IRn with
∇xΨp(x, ε) = Da(x, ε) +∇F (x)Db(x, ε),
where Da(x, ε) and Db(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∈ Il,
(Da)ii(x, ε) = sign(xi− li)|xi− li|p−1
∥(xi− li, Fi(x), ε)∥pp−1 − 1, (Db)ii(x, ε) = sign(Fi(x))|Fi(x)|p−1
∥(xi− li, Fi(x), ε)∥pp−1 − 1.
(a2) For i∈ Iu,
(Da)ii(x, ε) = sign(ui− xi)|ui− xi|p−1
∥(ui− xi, Fi(x), ε)∥pp−1 − 1, (Db)ii(x, ε) = −sign(Fi(x))|Fi(x)|p−1
∥(ui− xi, Fi(x), ε)∥pp−1 − 1.
(a3) For i∈ Ilu,
(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
ai(x, ε) = sign(xi− li)|xi− li|p−1
∥(xi− li, ψp(ui− xi,−Fi(x), ε), ε)∥pp−1 − 1,
bi(x, ε) = sign(ψp(ui− xi,−Fi(x), ε))|ψp(ui− xi,−Fi(x), ε)|p−1
∥(xi− li, ψp(ui− xi,−Fi(x), ε), ε)∥pp−1 − 1, ci(x, ε) = −sign(ui − xi)|ui− xi|p−1
∥(ui− xi, Fi(x), ε)∥pp−1 + 1, di(x, ε) = sign(Fi(x))|Fi(x)|p−1
∥(ui− xi, Fi(x), ε)∥pp−1 + 1.
(a4) For i∈ If, (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 ∈ Il∪ Iu, and
−6 < (Da)ii(x, ε) < 0 and −4 < (Db)ii(x, ε) < 0 for i∈ Ilu.
(b) For any given ε1 > 0 and ε2 > 0, we have
∥Ψp(x, ε2)− Ψp(x, ε1)∥ ≤√ n(√p
2 + 1)|ε2− ε1|, ∀x ∈ IRn. Particularly, for any given ε > 0,
∥Ψp(x, ε)− Φp(x)∥ ≤√ n(√p
2 + 1)ε, ∀x ∈ IRn.
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 Ψp satisfies the Jacobian consistency property, we need the following characterization of the generalized Jacobian ∂CΦp(x), which is direct by Lemma 2.2 (b).
Proposition 3.3 For any given x ∈ IRn, ∂CΦp(x)T = {Da(x) +∇F (x)Db(x)}, where Da(x), Db(x) are n× n diagonal matrices whose diagonal elements are given as below:
(a) For i∈ Il, if (xi− li, Fi(x)) ̸= (0, 0), then
(Da)ii(x) = sign(xi− li)· |xi− li|p−1
∥(xi− li, Fi(x))∥pp−1 − 1, (Db)ii(x) = sign(Fi(x))· |Fi(x)|p−1
∥(xi− li, Fi(x))∥pp−1 − 1;
and otherwise
((Da)ii(x), (Db)ii(x)) ∈{(ξ− 1, ζ − 1) ∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}. (b) For i∈ Iu, if (ui− xi,−Fi(x))̸= (0, 0), then
(Da)ii(x) = sign(ui− xi)· |ui− xi|p−1
∥(ui− xi,−Fi(x))∥p−1p − 1, (Db)ii(x) = −sign(Fi(x))· |Fi(x)|p−1
∥(ui− xi,−Fi(x))∥pp−1 − 1;
and otherwise
((Da)ii(x), (Db)ii(x)) ∈{(ξ− 1, ζ − 1) ∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}.
(c) For i ∈ Ilu, (Da)ii(x) = ai(x) + bi(x)ci(x) and (Db)ii(x) = bi(x)di(x) where, if (xi− li, ϕp(ui− xi,−Fi(x)))̸= (0, 0), then
ai(x) = sign(xi− li)· |xi− li|p−1
∥(xi− li, ϕp(ui− xi,−Fi(x))∥pp−1 − 1,
bi(x) = sign (ϕp(ui− xi,−Fi(x)))· |ϕp(ui− xi,−Fi(x))|p−1
∥(xi− li, ϕp(ui − xi,−Fi(x))∥pp−1 − 1,
and otherwise
(ai(x), bi(x)) ∈{(ξ− 1, ζ − 1) ∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}; and if (ui− xi,−Fi(x)) ̸= (0, 0), then
ci(x) = −sign(ui − xi)· |ui− xi|p−1
∥(ui− xi,−Fi(x))∥pp−1 + 1, di(x) = sign (Fi(x))· |Fi(x)|p−1
∥(ui− xi,−Fi(x))∥pp−1 + 1, and otherwise
(ci(x), di(x)) ∈{(ξ + 1, ζ + 1)∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}. (d) For i∈ If, (Da)ii(x) = 0 and (Db)ii(x) =−1.
Now we are in a position to establish the Jacobian consistency of the operator Ψp.
Proposition 3.4 Let Ψp be defined by (10). Then, for any fixed x∈ IRn, limε↓0dist(∇xΨp(x, ε)T, ∂CΦp(x)) = 0.
Proof. For the sake of notation, for any given x∈ IRn, we define the index sets:
β1(x) := {i ∈ Il | (xi− li, Fi(x)) = (0, 0)}, ¯β1(x) := Il\ β1(x),
β2(x) := {i ∈ Iu | (ui− xi, Fi(x)) = (0, 0)}, ¯β2(x) := Iu\ β2(x), (16) β3(x) := {i ∈ Ilu | (xi− li, ϕp(ui− xi,−Fi(x))) = (0, 0)}, ¯β3(x) := Ilu\ β3(x), β4(x) := {i ∈ ¯β3(x)| (ui− xi, Fi(x)) = (0, 0)}, ¯β4(x) := ¯β3(x)\ β4(x).
We proceed the arguments by the cases i∈ Il∪ Iu, i∈ Ilu and i∈ If, respectively.
Case 1: i∈ Il∪ Iu. When i∈ β1(x)∪ β2(x), it is easy to see that (Da)ii(x, ε) =−1 and (Db)ii(x, ε) = −1.
By Proposition 3.2 (a1) and (a2),∇xΨp,i(x, ε)T =−eTi − Fi′(x) for all ε > 0. Since (−1, −1) ∈{(ξ− 1, ζ − 1) ∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}, (17) from Proposition 3.3 (a) and (b) we get∇xΨp,i(x, ε)T ∈ ∂CΦp,i(x). When i∈ ¯β1(x)∪ ¯β2(x),
limε↓0 (Da)ii(x, ε) = (Da)ii(x) and lim
ε↓0 (Db)ii(x, ε) = (Db)ii(x),
which together with Proposition 3.2 (a1) and (a2) implies that
limε↓0 ∇xΨp,i(x, ε)T = (Da)ii(x)eTi + (Db)ii(x)Fi′(x)∈ ∂CΦp,i(x).
Since Il∪ Iu = β1(x)∪ β2(x)∪ ¯β1(x)∪ ¯β2(x), the last two subcases show that
limε↓0 ∇xΨp,i(x, ε)T ∈ ∂CΦp,i(x), ∀ i ∈ Il∪ Iu. (18)
Case 2: i∈ Ilu. When i∈ β3(x), we have xi− li = 0, ϕp(ui− xi,−Fi(x)) = 0, ui− xi > 0 and Fi(x) = 0. Hence, ci(x) = 0 and di(x) = 1. From Proposition 3.3 (c), it follows that
∂CΦp,i(x) ={ai(x)eTi + bi(x)Fi′(x)} (19) with
(ai(x), bi(x))∈{(ξ− 1, ζ − 1) ∈ IR2 | |ξ|p−1p +|ζ|p−1p ≤ 1}. On the other hand, since ai(x, ε) = −1, di(x, ε) = 1 and
bi(x, ε) = |ψp(ui− xi,−Fi(x), ε)|p−1
(|ψp(ui− xi,−Fi(x), ε)|p+ εp)p−1p − 1, ci(x, ε) = 1− |ui− xi|p−1
(|ui− xi|p+ εp)(p−1)/p, from Proposition 3.2 (a3) it follows that
∇xΨp,i(x, ε)T = (−1 + bi(x, ε)ci(x, ε))eTi + bi(x, ε)Fi′(x). (20) Taking
ξ = 0 and ζ = |ψp(ui− xi,−Fi(x), ε)|p−1 (|ψp(ui− xi,−Fi(x), ε)|p+ εp)p−1p
, it is not hard to verify that |ξ|p−1p +|ζ|p−1p ≤ 1, and consequently
−eTi + bi(x, ε)Fi′(x)∈ ∂CΦp,i(x).
Noting that limε↓0
∇xΨp,i(x, ε)T −(−eTi + bi(x, ε)Fi′(x)) = lim
ε↓0∥bi(x, ε)ci(x, ε)eTi ∥ = 0, it then follows that
limε↓0 dist(∇xΨp,i(x, ε)T, ∂CΦp,i(x))= 0, i∈ β3(x).
When i∈ ¯β3(x), we have limε↓0ai(x, ε) = ai(x) and limε↓0bi(x, ε) = bi(x). Also, ci(x, ε) = 1, di(x, ε) = 1 for i∈ β4(x)
