The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

Abstract

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.

1. Introduction

The second-order cone complementarity problem (SOCCP) [1] is to find (𝑥,𝑦,𝜁)∈𝑅^𝑛×𝑅^𝑛×𝑅^𝑙 such that 𝑥∈𝐾,𝑦∈𝐾,𝑥^𝑇𝑦=0,𝐹𝑥,𝑦,𝜁=0, (1) where 𝐹∶𝑅^𝑛×𝑅^𝑛×𝑅^𝑙→𝑅^𝑛×𝑅^𝑙 is a continuously differentiable mapping, 𝐾⊂𝑅^𝑛 is the Cartesian product of second-order cones (SOC), that is, 𝐾=𝐾^𝑛1×𝐾^𝑛2×⋯×𝐾^𝑛𝑚 with 𝑛=𝑛₁+𝑛₂+⋯+𝑛_𝑚 and the 𝑛_𝑖-dimensional SOC defined by 𝐾^𝑛𝑖=𝑥^𝑖=𝑥^𝑖₁;𝑥^𝑖₂∈𝑅×𝑅^𝑛𝑖−1∶𝑥^𝑖₁−𝑥^𝑖₂≥0. (2)

Without loss of generality, we may assume that 𝑚=1 and 𝐾=𝐾^𝑛 in the following analysis, since our analysis can be easily extended to the general case.

The SOCCP contains a wide class of problems, such as nonlinear complementarity problems [2], second-order cone programming [1, 3, 4], and has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [5, 6].

Recently, great attention has been paid to smoothing methods, partially due to their superior theoretical and numerical performances [7–10]. Smoothing methods usually reformulate the SOCCP as a system of equations by using smoothing functions of SOC complementarity functions [10, 11]. The smoothing parameter involved in smoothing functions may be treated as a variable [9] or a parameter with an appropriate parameter control [7]. In the latter case, the Jacobian consistency plays a key role for achieving a rapid convergence of Newton the methods or the Newton-like methods. Hayashi et al. [7] propose a combined smoothing and regularized method for monotone SOCCP and show its global and quadratic convergence based on the Jacobian consistency of the smoothing natural residual function. Ogasawara and Narushima [12] show the Jacobian consistency of a smoothed Fischer-Burmeister (FB) function. Chen et al. [13] present a smoothing function of a generalized FB function in the context of nonlinear complementarity programming and study some of its favorable properties, including the Jacobian consistency property. Based on the results, they [13] propose a smoothing algorithm for the mixed complementarity problem, which is shown to possess global convergence and local superlinear (or quadratic) convergence.

In this paper, we are concerned with the one-parametric class of SOC complementarity functions 𝜙_𝜏∶𝑅^𝑛→𝑅^𝑛 defined by [14] 𝜙_𝜏−𝑥,𝑦∶=𝑥+𝑦𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦, (3) where 𝜏∈(0,4) is an arbitrary but fixed parameter. When 𝜏=2,  𝜙_𝜏 reduces to the vector-valued Fischer-Burmeister function given by 𝜙_FB𝑥,𝑦∶=𝑥+𝑦−𝑥²+𝑦², (4) and as 𝜏→0, it becomes a multiple of the following vector-valued residual function: 𝜙_NR𝑥,𝑦∶=𝑥−Π_𝐾^𝑛𝑥−𝑦, (5) where Π_𝐾^𝑛(⋅) denotes the metric projection on the second-order cone 𝐾^𝑛. Thus, the one-parametric class of vector-valued functions (3) cover two popular second-order cone complementarity functions.

In this paper, we aim to show the Jacobian consistency of smoothing functions of the one-parametric class of SOC complementarity functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of the smoothing functions, which will help to adjust a parameter appropriately in smoothing methods.

The organization of this paper is as follows. In Section 2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and subdifferentials. In Section 3, we derive the computable formula for the Jacobian of the one-parametric class of smoothing functions in the SOCCP. In Section 4, we prove the Jacobian consistency of the one-parametric class of smoothing functions and estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions. In Section 5, we study the directional derivative and 𝐵-subdifferential of the one-parametric class of SOC complementarity functions and then present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions. Finally, we close this paper with some conclusions in Section 6.

In what follows, we denote the nonnegative orthant of 𝑅 by 𝑅₊. We use the symbol ‖⋅‖ to denote the Euclidean norm defined by ‖𝑥‖∶=𝑥^𝑇𝑥 for a vector 𝑥 or the corresponding induced matrix norm. For simplicity, we often use 𝑥=(𝑥₁;𝑥₂) for the column vector 𝑥=(𝑥₁,𝑥^𝑇₂)^𝑇. For the SOC 𝐾^𝑛,  int𝐾^𝑛 and bd𝐾^𝑛 mean the topological interior and the boundary of 𝐾^𝑛, respectively. For a given set 𝑆⊂𝑅^𝑚×𝑛,  conv𝑆 denotes the convex hull of 𝑆 in 𝑅^𝑚×𝑛, and dist(𝑋,𝑆) denotes inf{‖𝑋−𝑌‖∶𝑌∈𝑆} for a matrix 𝑋∈𝑅^𝑚×𝑛.

2. Preliminaries

In this section, we recall some concepts and results, which include the Euclidean Jordan algebra [3, 15] associated with the SOC 𝐾^𝑛 and subdifferentials [16].

First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions. The Euclidean Jordan algebra for the SOC 𝐾^𝑛 is the algebra defined by 𝑥∘𝑦=𝑥^𝑇𝑦;𝑥₁𝑦₂+𝑦₁𝑥₂,∀𝑥,𝑦∈𝑅^𝑛, (6) with 𝑒=(1,0,…,0)∈𝑅^𝑛 being its unit element. Given an element 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1, we define 𝐿(𝑥)=𝑥₁𝑥^𝑇₂𝑥₂𝑥₁𝐼, (7) where 𝐼 represents the (𝑛−1)×(𝑛−1) identity matrix. It is easy to verify that 𝑥∘𝑦=𝐿(𝑥)𝑦 for any 𝑦∈𝑅^𝑛. Moreover, 𝐿(𝑥) is symmetric positive definite (and hence invertible) if and only if 𝑥∈int𝐾^𝑛.

Now, we give the spectral factorization of vectors in 𝑅^𝑛 associated with the SOC 𝐾^𝑛. Let 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1. Then 𝑥 can be decomposed as 𝑥=𝜆₁𝑢⁽¹⁾+𝜆₂𝑢⁽²⁾, (8) where 𝜆₁,  𝜆₂, and 𝑢⁽¹⁾,  𝑢⁽²⁾ are the spectral values and the associated spectral vectors of 𝑥 given by 𝜆_𝑖=𝑥₁+(−1)^𝑖𝑥₂,𝑢^(𝑖)=121,(−1)^𝑖𝑥₂𝑥₂if𝑥₂1≠0,21,(−1)^𝑖𝜔otherwise, (9) for 𝑖=1,2, with any 𝜔∈𝑅^𝑛−1 such that ‖𝜔‖=1. By the spectral factorization, a scalar function can be extended to a function for the SOC. For any 𝑥∈𝑅^𝑛, we define 𝑥²=𝜆²₁𝑢⁽¹⁾+𝜆²₂𝑢⁽²⁾. (10) Since both eigenvalues of any 𝑥∈𝐾^𝑛 are nonnegative, we define √𝑥=𝜆₁𝑢⁽¹⁾+𝜆₂𝑢⁽²⁾. (11)

For any 𝑥=(𝑥₁;𝑥₂)∈𝑅×𝑅^𝑛−1, we define [12] 𝑥^′=(𝑥₁;−𝑥₂). Obviously, 𝑥^′′=𝑥,  (𝑥+𝑦)^′=𝑥^′+𝑦^′, and (𝑐𝑥)^′=𝑐𝑥^′ for any 𝑐∈𝑅. Moreover, 𝑥∘𝑥^′=𝑥²₁−‖𝑥₂‖²=0 if 𝑥∈bd𝐾^𝑛.

Let 𝐺∶𝑅^𝑚→𝑅^𝑛 be a locally Lipschitzian function. Then, 𝐺 is differentiable almost everywhere by Rademacher’s theorem [17]. The Bouligand- (B-) subdifferential and the Clarke subdifferential of 𝐺 at 𝑧 are defined by 𝜕𝐺_𝐵(𝑧)𝑧∶=lim_→𝑧∇𝐺(𝑧)∶𝑧∈𝐷_𝐺,𝜕𝐺(𝑧)=conv𝜕𝐺_𝐵(𝑧), (12) respectively, where 𝐷_𝐺 denotes the set of points at which 𝐺 is differentiable. It is obvious that 𝜕𝐺(𝑧)={∇𝐺(𝑧)} if 𝐺 is continuously differentiable at 𝑧.

By using the concepts of subdifferentials, we give the definition of the Jacobian consistency, which was first introduced by Chen et al. [16], which is a concept relating the generalized Jacobian of a nonsmooth function with the Jacobian of a smoothing function [7].

Definition 1 (see [16]). Let 𝐺∶𝑅^𝑚→𝑅^𝑛 be a locally Lipschitzian function. Let 𝐺_𝜀∶𝑅^𝑚→𝑅^𝑛 be a continuously differentiable function for any 𝜀>0 such that lim_𝜀↓0𝐺_𝜀(𝑧)=𝐺(𝑧) for any 𝑧∈𝑅^𝑚. We say that 𝐺_𝜀 satisfies the Jacobian consistency property if for any 𝑧∈𝑅^𝑚,  lim_𝜀↓0dist(∇𝐺_𝜀(𝑧),𝜕𝐺(𝑧))=0.

It should be noted that the “inf” appearing in the definition of dist(∇𝐺_𝜀(𝑧),𝜕𝐺(𝑧)) can be replaced by “min,” since the set 𝜕𝐺(𝑧) is compact at all 𝑧∈𝑅^𝑚 [17].

3. Smoothing Function

In this section, we propose a smoothing function of the one-parametric class of SOC complementarity functions and derive the computable formula for its Jacobian.

Since the one-parametric class of SOC complementarity functions 𝜙_𝜏 defined by (3) is nonsmooth, we consider the function 𝜙_𝜏,𝜀 defined by 𝜙_𝜏,𝜀−𝑥,𝑦∶=𝑥+𝑦𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦+2𝜀²𝑒, (13) where the smoothing parameter 𝜀∈𝑅.

Definition 2 (see [7]). For a nondifferentiable function ℎ∶𝑅^𝑚→𝑅^𝑛, one considers a function ℎ_𝜀∶𝑅^𝑚→𝑅^𝑛 with a parameter 𝜀>0 that has the following properties:(i)ℎ_𝜀 is differentiable for any 𝜀>0;(ii)lim_𝜀↓0ℎ_𝜀(𝑥)=ℎ(𝑥) for any 𝑥∈𝑅^𝑚.
Such a function ℎ_𝜀 is called a smoothing function of ℎ.

In the following, we will show that the function 𝜙_𝜏,𝜀  given by (13) is a smoothing function of 𝜙_𝜏. Thus, we can solve a family of smoothing subproblems 𝜙_𝜏,𝜀(𝑥,𝑦)=0 for 𝜀>0 and obtain a solution of 𝜙_𝜏(𝑥,𝑦)=0 by letting 𝜀↓0.

For convenience, we give some notations. For any 𝑥=(𝑥₁;𝑥₂),  𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1, and any 𝜀∈𝑅, we define the mapping 𝑧^𝜀∶𝑅^2𝑛→𝑅×𝑅^𝑛−1 by 𝑧^𝜀=𝑧^𝜀₁;𝑧^𝜀₂=𝑧^𝜀𝑥,𝑦∶=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦+2𝜀²𝑒, (14) and drop the subscript for simplicity for 𝜀=0, and thus, 𝑧=𝑧₁;𝑧₂=𝑧𝑥,𝑦∶=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦. (15) By direct calculations, we obtain 𝑧^𝜀₁=‖𝑥‖²+𝑦²+(𝜏−2)𝑥^𝑇𝑦+2𝜀²=2𝜀²+𝑧₁,𝑧^𝜀₂=2𝑥₁𝑥₂+2𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂+𝑦₁𝑥₂=𝑧₂, (16) and therefore 𝑧^𝜀=(𝑧^𝜀₁;𝑧₂). Then, the spectral factorization of 𝑧^𝜀 is 𝑧^𝜀=𝜆₁𝑧^𝜀𝑢₁(𝑧)+𝜆₂𝑧^𝜀𝑢₂(𝑧), (17) where 𝜆₁(𝑧^𝜀),  𝜆₂(𝑧^𝜀), and 𝑢₁(𝑧),  𝑢₂(𝑧) are the spectral values and the associated spectral vectors of 𝑧^𝜀 given by 𝜆_𝑖𝑧^𝜀=2𝜀²+‖𝑥‖²+𝑦²+(𝜏−2)𝑥^𝑇𝑦+(−1)^𝑖2𝑥₁𝑥₂+2𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂+𝑦₁𝑥₂,𝑢_𝑖(𝑧)=121;(−1)^𝑖𝑧₂, (18)(19) for 𝑖=1,2. Here, 𝑧₂𝑧∶=²𝑧₂=2𝑥₁𝑥₂+2𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂+𝑦₁𝑥₂2𝑥₁𝑥₂+2𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂+𝑦₁𝑥₂, (20) if 𝑧₂≠0, and otherwise, 𝑧₂ is any vector in 𝑅^𝑛−1 such that ‖𝑧₂‖=1.

For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛, it is not difficult to verify that 𝑧^𝜀=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦+2𝜀²𝑒=𝑥+𝜏−22𝑦²+𝜏(4−𝜏)4𝑦²+2𝜀²𝑒=𝑦+𝜏−22𝑥²+𝜏(4−𝜏)4𝑥²+2𝜀²𝑒∈int𝐾^𝑛, (21) for any 𝜀≠0, and 𝑧=𝑥²+𝑦²+(=𝜏−2)𝑥∘𝑦𝑥+𝜏−22𝑦²+𝜏(4−𝜏)4𝑦²=𝑦+𝜏−22𝑥²+𝜏(4−𝜏)4𝑥²∈𝐾^𝑛. (22) Therefore, for any 𝑥=(𝑥₁;𝑥₂),𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1, and any 𝜀∈𝑅, we can also define 𝑤^𝜀=𝑤^𝜀₁;𝑤^𝜀₂=𝑤^𝜀𝑥,𝑦∶=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦+2𝜀²𝑒, (23) and for 𝜀=0, 𝑤=𝑤₁;𝑤₂=𝑤𝑥,𝑦∶=𝑥²+𝑦²+(𝜏−2)𝑥∘𝑦. (24) The spectral factorization of 𝑤^𝜀 and 𝑤 is given by, respectively, 𝑤^𝜀=𝜆₁(𝑧^𝜀)𝑢₁(𝑧)+𝜆₂(𝑧^𝜀)𝑢₂(𝑧),𝑤=𝜆₁(𝑧)𝑢₁(𝑧)+𝜆₂(𝑧)𝑢₂(𝑧). (25) By (22), we can partition 𝑅^2𝑛 as 𝑅^2𝑛=𝑆₁∪𝑆₂∪{(0,0)}, where 𝑆₁∶=𝑥,𝑦∈𝑅^2𝑛∶𝑧∈int𝐾^𝑛=𝑥,𝑦∈𝑅^2𝑛∶𝜆₂(𝑧)≥𝜆₁(𝑧),𝑆>0₂∶=𝑥,𝑦∈𝑅^2𝑛∶𝑧∈bd𝐾^𝑛⧵{0=}𝑥,𝑦∈𝑅^2𝑛∶2𝑧₁=𝜆₂(𝑧)>𝜆₁(𝑧)=0. (26)

Theorem 3. For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛 and 𝜀≠0, let 𝜙_𝜏 and 𝜙_𝜏,𝜀 be, respectively, defined by (3) and (13). Then, the following results hold.(i) The function 𝜙_𝜏,𝜀 is continuously differentiable everywhere, and its Jacobian is given by ∇𝜙_𝜏,𝜀=𝑥,𝑦𝐼−𝐿𝑥+𝜏−22𝑦𝐿⁻¹𝑤^𝜀𝐼−𝐿𝑦+𝜏−22𝑥𝐿⁻¹𝑤^𝜀, (27) where 𝐿⁻¹(𝑤^𝜀)=(1/𝑧^𝜀₁)𝐼 if 𝑧₂=0, and otherwise, 𝐿⁻¹𝑤^𝜀=𝐿₁𝑧^𝜀+𝐿₂𝑧^𝜀=𝑏_𝜀𝑐_𝜀𝑧^𝑇₂𝑐_𝜀𝑧₂𝑎_𝜀𝐼+𝑏_𝜀−𝑎_𝜀𝑧₂𝑧^𝑇₂, (28) with 𝐿₁𝑧^𝜀=12𝜆₁(𝑧^𝜀)1−𝑧^𝑇₂−𝑧₂𝑧₂𝑧^𝑇₂,𝐿₂𝑧^𝜀=12𝜆₂(𝑧^𝜀)1𝑧^𝑇₂𝑧₂𝑧₂𝑧^𝑇₂+𝑎_𝜀00^𝑇0𝐼−𝑧₂𝑧^𝑇₂,𝑎_𝜀=2𝜆₁(𝑧^𝜀)+𝜆₂(𝑧^𝜀),𝑏_𝜀=121𝜆₁(𝑧^𝜀)+1𝜆₂(𝑧^𝜀),𝑐_𝜀=121𝜆₂(𝑧^𝜀)−1𝜆₁(𝑧^𝜀). (29)(30)(ii) For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛,  lim_𝜀↓0𝜙_𝜏,𝜀(𝑥,𝑦)=𝜙_𝜏(𝑥,𝑦). Thus, 𝜙_𝜏,𝜀 is a smoothing function of 𝜙_𝜏.

Proof. (i) For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛 and any 𝜀≠0, it follows from Corollary  5.4 in [1], the chain rule for differentiation, and (21) that formula (27) holds. Formula (28) is due to Proposition  5.2 and its proof in [1].
(ii) Fix any 𝑥=(𝑥₁;𝑥₂),  𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1. For any 𝜀>0, it follows from the spectral factorization of 𝑧 and 𝑧^𝜀 that 𝜙_𝜏−𝑥,𝑦=𝑥+𝑦𝜆₁(𝑧)𝑢₁(𝑧)+𝜆₂(𝑧)𝑢₂(𝑧),𝜙_𝜏,𝜀−𝑥,𝑦=𝑥+𝑦𝜆₁(𝑧^𝜀)𝑢₁(𝑧)+𝜆₂(𝑧^𝜀)𝑢₂(𝑧), (31) where 𝜆_𝑖(𝑧)=‖𝑥‖²+𝑦²+(𝜏−2)𝑥^𝑇𝑦+(−1)^𝑖2𝑥₁𝑥₂+2𝑦₁𝑦₂+(𝜏−2)𝑥₁𝑦₂+𝑦₁𝑥₂, (32) and 𝜆_𝑖(𝑧^𝜀) and 𝑢_𝑖(𝑧) are, respectively, given by (18) and (19) for 𝑖=1,2. It is obvious that 𝜆_𝑖(𝑧^𝜀)=𝜆_𝑖(𝑧)+2𝜀² for 𝑖=1,2. Then, lim_𝜀↓0𝜆₁(𝑧^𝜀)𝑢₁(𝑧)+𝜆₂(𝑧^𝜀)𝑢₂(𝑧)=lim_𝜀↓0𝜆₁(𝑧)+2𝜀²𝑢₁(𝑧)+𝜆₂(𝑧)+2𝜀²𝑢₂(𝑧)=𝜆₁(𝑧)𝑢₁(𝑧)+𝜆₂(𝑧)𝑢₂(𝑧), (33) and hence, lim_𝜀↓0𝜙_𝜏,𝜀(𝑥,𝑦)=𝜙_𝜏(𝑥,𝑦). Therefore, it follows from (i) and Definition 2 that 𝜙_𝜏,𝜀 is a smoothing function of 𝜙_𝜏.

Next, we give some properties of 𝜙_𝜏 [14], which will be used in the subsequent analysis.

Lemma 4. For any 𝑥=(𝑥₁;𝑥₂),  𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1, let 𝑧=(𝑧₁;𝑧₂)  =  𝑥²+𝑦²+  (𝜏−2)(𝑥∘𝑦)∈𝑏𝑑𝐾^𝑛. Then one has 𝑥+𝜏−22𝑦∘𝑥+𝜏−22𝑦^′=0,𝑦∘𝑦^′=0,𝑥+𝜏−22𝑦∘𝑧^′=0,𝑦∘𝑧^′=0,𝑥+𝜏−22𝑦∘𝑦^′=0,𝑦+𝜏−22𝑥∘𝑥^′=0,𝑦+𝜏−22𝑥∘𝑦+𝜏−22𝑥^′=0,𝑥∘𝑥^′=0,𝑦+𝜏−22𝑥∘𝑧^′=0,𝑥∘𝑧^′=0,𝑥+𝜏−22𝑦∘𝑦+𝜏−22𝑥^′=0,𝑥∘𝑦^′=0. (34)(35)(36)(37)(38)(39) Moreover, the following equivalence holds: 𝑧₁=0⟺𝑧₂=0⟺𝑧=0⟺𝑥₁=𝑦₁=0⟺𝑥₂=𝑦₂⟺=().=0𝑥,𝑦0,0 (40)

Proof. From Lemma 3.3 and its proof in [14], it is not difficult to see that relations (34)–(39) hold. The equivalence is also true, since 𝑧=𝑥²+𝑦²+(=𝜏−2)𝑥∘𝑦𝑥+𝜏−22𝑦²+𝜏(4−𝜏)4𝑦²=𝑦+𝜏−22𝑥²+𝜏(4−𝜏)4𝑥²∈bd𝐾^𝑛. (41) This completes the proof.

4. The Jacobian Consistency

In this section, we show the Jacobian consistency of the smoothing function 𝜙_𝜏,𝜀, which will play an important role for establishing the rapid convergence of smoothing methods. Moreover, we estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions, which will help to adjust a parameter appropriately in smoothing methods.

It has been shown in Proposition 3.1 in [14] that the function 𝜙_𝜏 with any 𝜏∈(0,4) satisfies 𝜙_𝜏𝑥,𝑦=0⟺𝑥∈𝐾^𝑛,𝑦∈𝐾^𝑛,⟨𝑥,𝑦⟩=0. (42) Let (𝑥,𝑦,𝜁)∈𝑅^𝑛×𝑅^𝑛×𝑅^𝑙 and define Φ_𝜏,𝜀𝐹𝜙𝑥,𝑦,𝜁∶=𝑥,𝑦,𝜁_𝜏,𝜀,Φ𝑥,𝑦_𝜏𝐹𝜙𝑥,𝑦,𝜁∶=𝑥,𝑦,𝜁_𝜏.𝑥,𝑦 (43) It is easy to see that Φ_𝜏,𝜀(𝑥,𝑦,𝜁)=0 is the perturbation of the system of equations Φ_𝜏(𝑥,𝑦,𝜁)=0. On account of (1), (42), and (43), we have Φ_𝜏𝑥,𝑦,𝜁=0⟺𝑥,𝑦,𝜁solves(1). (44) Since Φ_𝜏(𝑥,𝑦,𝜁) is typically nonsmooth, we can solve approximately the smooth system Φ_𝜏,𝜀(𝑥,𝑦,𝜁)=0 by using Newton’s method at each iteration, and then obtain a solution of Φ_𝜏(𝑥,𝑦,𝜁)=0 by reducing the parameter 𝜀 to zero.

First, we show that the function Φ_𝜏,𝜀(𝑥,𝑦,𝜁) satisfies the Jacobian consistency.

Lemma 5. For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛, one has 𝐽⁰_𝜙𝜏𝑥,𝑦∶=lim_𝜀→0∇𝜙_𝜏,𝜀=𝑥,𝑦𝐼−𝐿𝑥+𝜏−22𝑦𝐽𝐼−𝐿𝑦+𝜏−22𝑥𝐽, (45) where 𝐿𝐽∶=⁻¹(𝑤)𝑖𝑓𝑥,𝑦∈𝑆₁,122𝑧₁1𝑧^𝑇₂𝑧₂4𝐼−3𝑧₂𝑧^𝑇₂𝑖𝑓𝑥,𝑦∈𝑆₂,𝑂𝑖𝑓𝑥,𝑦=(0,0). (46)

Proof. By (27) and the symmetry of 𝑥 and 𝑦, it suffices to show that lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹𝑤^𝜀=𝐿𝑥+𝜏−22𝑦𝐽. (47)Case  (i). If (𝑥,𝑦)∈𝑆₁, we have from (18) that lim_𝜀→0𝜆_𝑖𝑧^𝜀=lim_𝜀→0𝜆_𝑖(𝑧)+2𝜀²=𝜆_𝑖(𝑧)>0, (48) for 𝑖=1,2, and then, lim_𝜀→0𝑤^𝜀=lim_𝜀→0𝜆₁(𝑧^𝜀)𝑢₁(𝑧)+𝜆₂(𝑧^𝜀)𝑢₂(𝑧=)𝜆₁(𝑧)𝑢₁(𝑧)+𝜆₂(𝑧)𝑢₂(𝑧)=𝑤∈int𝐾^𝑛. (49) Therefore, lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹𝑤^𝜀=𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤). (50)Case  (ii). If (𝑥,𝑦)∈𝑆₂, we obtain (𝑥,𝑦)≠(0,0), and (41) holds, and thus, 𝑧₂=𝑧₁=𝑥+𝜏−22𝑦²+𝜏(4−𝜏)4𝑦²>0,2𝑧₁=𝜆₂(𝑧)>𝜆₁(𝑧)=0. (51) Then we have from (18) that 𝜆₁𝑧^𝜀=𝜆₁(𝑧)+2𝜀²=2𝜀²𝜆>0,₂𝑧^𝜀=𝜆₂(𝑧)+2𝜀²=2𝑧₁+2𝜀²>0. (52) For any 𝜀≠0, it follows from (28) that 𝐿⁻¹(𝑤^𝜀)=𝐿₁(𝑧^𝜀)+𝐿₂(𝑧^𝜀). We first show that 𝐿(𝑥+((𝜏−2)/2)𝑦)𝐿₁(𝑧^𝜀)=𝑂 for any 𝜀≠0. Let 𝜔∶=1;𝑧₂=1𝑧₂𝑧₁;𝑧₂=𝑧𝑧₁. (53) By (35), we have 𝑥+𝜏−22𝑦∘𝜔^′=𝑥+𝜏−22𝑦∘𝑧^′𝑧₁=0, (54) and therefore, 𝐿𝑥+𝜏−22𝑦𝐿₁𝑧^𝜀=12𝜆₁(𝑧^𝜀)𝐿𝑥+𝜏−22𝑦𝜔^′𝜔^′𝑇=12√2|𝜀|𝑥+𝜏−22𝑦∘𝜔^′𝜔^′𝑇=𝑂. (55) We next show that lim_𝜀→0𝐿₂(𝑧^𝜀)=𝐽. In fact, we obtain from (52) lim_𝜀→0𝐿₂𝑧^𝜀=lim_𝜀→0122𝑧₁+2𝜀²1𝑧^𝑇₂𝑧₂𝑧₂𝑧^𝑇₂+lim_𝜀→02√2𝜀²+2𝑧₁+2𝜀²00^𝑇0𝐼−𝑧₂𝑧^𝑇₂,=122𝑧₁1𝑧^𝑇₂𝑧₂𝑧₂𝑧^𝑇₂+22𝑧₁00^𝑇0𝐼−𝑧₂𝑧^𝑇₂,=122𝑧₁1𝑧^𝑇₂𝑧₂4𝐼−3𝑧₂𝑧^𝑇₂=𝐽. (56) Combining (55) and (56) yields lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹𝑤^𝜀=lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿₂𝑧^𝜀=𝐿𝑥+𝜏−22𝑦𝐽. (57)Case  (iii). If (𝑥,𝑦)=(0,0), then 𝑤^𝜀=√𝑧^𝜀=√2|𝜀|𝑒∈int𝐾^𝑛, and lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹𝑤^𝜀=lim_𝜀→01𝑂⋅√2|𝜀|𝑒=𝑂=𝐿𝑥+𝜏−22𝑦𝐽. (58) This completes the proof.

Lemma 6. For any (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛, one has 𝐼−𝑈_𝑥𝐼−𝑈_𝑦∈𝜕_𝐵𝜙_𝜏𝑥,𝑦, (59) where 𝑈_𝑥=±𝑍+𝐿𝑥+𝜏−22𝑦𝑈𝐽,_𝑦=±𝜏−22𝑍+𝐿𝑦+𝜏−22𝑥𝐽,𝑍=𝑂𝑖𝑓𝑥,𝑦∈𝑆₁,121−𝑧^𝑇₂−𝑧₂𝑧₂𝑧^𝑇₂𝑖𝑓𝑥,𝑦∈𝑆₂,=(),𝐿𝐼𝑖𝑓𝑥,𝑦0,0𝐽=⁻¹(𝑤)𝑖𝑓𝑥,𝑦∈𝑆₁,122𝑧₁1𝑧^𝑇₂𝑧₂4𝐼−3𝑧₂𝑧^𝑇₂𝑖𝑓𝑥,𝑦∈𝑆₂,𝑂𝑖𝑓𝑥,𝑦=(0,0). (60)(61)

Proof. It follows from Proposition  5.2 [1] and the chain rule for differentiation that the one-parametric class of SOC complementarity functions 𝜙_𝜏 is continuously differentiable at any (𝑥,𝑦)∈𝑆₁ with ∇𝜙_𝜏=𝑥,𝑦𝐼−𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)𝐼−𝐿𝑦+𝜏−22𝑥𝐿⁻¹(𝑤)∈𝜕_𝐵𝜙_𝜏𝑥,𝑦. (62) Therefore, it suffices to consider the two cases: (𝑥,𝑦)∈𝑆₂ and (𝑥,𝑦)=(0,0).
For any (𝑥,𝑦)∈𝑆₂ or (𝑥,𝑦)=(0,0), let (𝑥,𝑦)=(𝑥+𝜀𝑒,𝑦) with sufficiently small 𝜀≠0, and define 𝑧=𝑧₁;𝑧₂∶=𝑥²+𝑦²+(,𝑤𝜏−2)𝑥∘𝑦𝑤=₁;𝑤₂∶=√𝑧,𝑧₂𝑧∶=²𝑧₂,𝜆_𝑖=𝜆_𝑖(𝑧)∶=𝑧₁+(−1)^𝑖𝑧₂,𝑖=1,2. (63) Direct calculations yield 𝑧=(𝑥+𝜀𝑒)²+𝑦²+(𝜏−2)(𝑥+𝜀𝑒)∘𝑦=𝑥²+𝑦²+()()𝜏−2𝑥∘𝑦+2𝜀𝑥+𝜀𝜏−2𝑦+𝜀²𝑒=𝑧+2𝜀𝑥+𝜀(𝜏−2)𝑦+𝜀²𝑒. (64) Therefore, we obtain 𝑧₁=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁+𝜀²,𝑧₂=𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂,𝜆_𝑖=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁+𝜀²+(−1)^𝑖×𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂,𝑖=1,2. (65)(66) It is obvious that as 𝜀→0, we have (𝑥,𝑦)→(𝑥,𝑦),  𝑧→𝑧,𝑤→𝑤, and 𝜆_𝑖→𝜆_𝑖(𝑧) for 𝑖=1,2.
By the definition of 𝐵-subdifferential and (3), it suffices to show that lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)=𝑈_𝑥,lim_𝜀→0𝐿𝑦+𝜏−22𝑥𝐿⁻¹(𝑤)=𝑈_𝑦, (67) if 𝜙_𝜏 is differentiable at (𝑥,𝑦).
Case  (i). If (𝑥,𝑦)=(0,0), it is easy to see that 𝑧=𝜀²𝑒∈int𝐾^𝑛,  𝑤=|𝜀|𝑒, and 𝜙_𝜏 is differentiable at (𝑥,𝑦). Then, we have lim_𝜀→±0𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)=lim_𝜀→±01𝜀𝐼⋅|𝜀|𝐼=lim_𝜀→±0sgn(𝜀)𝐼=±𝐼=𝑈_𝑥,lim_𝜀→±0𝐿𝑦+𝜏−22𝑥𝐿⁻¹(𝑤)=lim_𝜀→±0𝜏−221𝜀𝐼⋅|𝜀|𝐼=lim_𝜀→±0𝜏−22sgn(𝜀)𝐼=±𝜏−22𝐼=𝑈_𝑦. (68)Case  (ii). If (𝑥,𝑦)∈𝑆₂, we obtain 𝑧∈bd𝐾^𝑛⧵{0}, and thus, from (38) and (39), 𝑧₂²=𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂²=𝑧₂²+4𝜀²𝑥₂²+𝜀²(𝜏−2)²𝑦₂²+4𝜀𝑧^𝑇₂𝑥₂+2𝜀(𝜏−2)𝑧^𝑇₂𝑦₂+4𝜀²(𝜏−2)𝑥^𝑇₂𝑦₂=𝑧²₁+4𝜀²𝑥²₁+𝜀²(𝜏−2)²𝑦²₁+4𝜀𝑧₁𝑥₁+2𝜀(𝜏−2)𝑧₁𝑦₁+4𝜀²(𝜏−2)𝑥₁𝑦₁=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁². (69) Since 𝑧₁>0 and 𝜀 is sufficiently small, we obtain 𝑧₂=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁>0. (70) By (66) and (70), we have 𝜆₁=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁+𝜀²−𝑧₂=𝜀²𝜆>0,₂=𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁+𝜀²+𝑧₂=2𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁+𝜀²>0. (71)(72) Relations (71) and (72) imply that 𝑧∈int𝐾^𝑛, and thus, 𝜙_𝜏 is differentiable at (𝑥,𝑦).
Now we will show that lim_𝜀→0𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)=𝑈_𝑥. (73) By Proposition 5.2 in [1], we have 𝐿⁻¹(𝑤)=𝐿₁(𝑧)+𝐿₂(𝑧), where 𝐿₁(𝑧) and 𝐿₂(𝑧) are given by (29) with 𝑧 and 𝑧₂ replacing 𝑧^𝜀 and 𝑧₂, respectively. By (70), 𝜔∶=1;𝑧₂=1𝑧₂𝑧₂;𝑧₂=1𝑧₂𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂;𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂=1𝑧₂𝑧₁+2𝜀𝑥₁+𝜀(𝜏−2)𝑦₁;𝑧₂+2𝜀𝑥₂+𝜀(𝜏−2)𝑦₂=1𝑧₂𝑧+2𝜀𝑥+𝜀(𝜏−2)𝑦. (74) According to (35), (39), (71), and (74), we have 𝑥+𝜏−22𝑦∘𝜔^′=1𝑧₂𝑥+𝜏−22𝑦∘𝑧+2𝜀𝑥+𝜀(𝜏−2)𝑦^′=1𝑧₂𝑥+𝜏−22𝑦∘𝑧^′+2𝜀𝑥+𝜏−22𝑦∘𝑥+𝜏−22𝑦^′=0, (75) and thus, 𝐿𝑥+𝜏−22𝑦𝐿₁(𝑧)=12𝜆₁𝑥+𝜀𝑒+𝜏−22𝑦∘𝜔^′𝜔^′𝑇=12|𝜀|𝑥+𝜏−22𝑦∘𝜔^′𝜔^′𝑇+𝜀𝜔^′𝜔^′𝑇=sgn(𝜀)2𝜔^′𝜔^′𝑇=sgn(𝜀)21−𝑧^𝑇₂−𝑧₂𝑧₂𝑧^𝑇₂. (76) It follows from (70), (71), (72), and (74) that 𝜆₁→𝜆₁(𝑧)=0,  𝜆₂→𝜆₂(𝑧)=2𝑧₁, and 𝑧₂→𝑧₂ as 𝜀→0. Then, by following the proof of Case (ii) in Lemma 5, we obtain lim_𝜀→0𝐿₂(𝑧)=122𝑧₁1𝑧^𝑇₂𝑧₂4𝐼−3𝑧₂𝑧^𝑇₂=𝐽. (77) Hence, we have from (76) and (77) that lim_𝜀→±0𝐿𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)=lim_𝜀→±0𝐿𝑥+𝜏−22𝑦𝐿₁(𝑧)+lim_𝜀→±0𝐿𝑥+𝜏−22𝑦𝐿₂(𝑧)=lim_𝜀→±0sgn(𝜀)2𝜔^′𝜔^′𝑇+lim_𝜀→±0𝐿𝑥+𝜏−22𝑦𝐽=±𝑍+𝐿𝑥+𝜏−22𝑦𝐽=𝑈_𝑥. (78)
Next, we will show that lim_𝜀→0𝐿𝑦+𝜏−22𝑥𝐿⁻¹(𝑤)=𝑈_𝑦. (79) By (38), (39), and (74), we have 𝑦+𝜏−22𝑥∘𝜔^′=1𝑧₂𝑦+𝜏−22𝑥∘𝑧+2𝜀𝑥+𝜀(𝜏−2)𝑦^′=1𝑧₂𝑦+𝜏−22𝑥∘𝑧^′+2𝜀𝑦+𝜏−22𝑥∘𝑥+𝜏−22𝑦^′=0, (80) and then, 𝐿𝑦+𝜏−22𝑥𝐿₁(𝑧)=12𝜆₁𝑦+𝜏−22𝑥+𝜏−22𝜀𝑒∘𝜔^′𝜔^′𝑇=12|𝜀|𝑦+𝜏−22𝑥∘𝜔^′𝜔^′𝑇+𝜏−22𝜀𝜔^′𝜔^′𝑇=𝜏−24sgn(𝜀)𝜔^′𝜔^′𝑇=𝜏−24(𝜀)𝑧sgn1−^𝑇₂−𝑧₂𝑧₂𝑧^𝑇₂. (81) Hence, we have from (77) and (81) that lim_𝜀→±0𝐿𝑦+𝜏−22𝑥𝐿⁻¹(𝑤)=lim_𝜀→±0𝐿𝑦+𝜏−22𝑥𝐿₁(𝑧)+lim_𝜀→±0𝐿𝑦+𝜏−22𝑥𝐿₂(𝑧)=lim_𝜀→±0𝜏−24sgn(𝜀)𝜔^′𝜔^′𝑇+lim_𝜀→±0𝐿𝑦+𝜏−22𝑥𝐽=±𝜏−22𝑍+𝐿𝑦+𝜏−22𝑥𝐽=𝑈_𝑦. (82) This completes the proof.

Theorem 7. The function Φ_𝜏,𝜀 defined by (43) with 𝜀>0 satisfies the Jacobian consistency.

Proof. By (43), it suffices to show the Jacobian consistency of the function 𝜙_𝜏,𝜀 with 𝜀>0. Let 𝑉^𝑖∶=𝐼−𝑈^𝑖_𝑥𝐼−𝑈^𝑖_𝑦, (83) where 𝑈^𝑖_𝑥=(−1)^𝑖𝑍+𝐿𝑥+𝜏−22𝑦𝑈𝐽,_𝑖𝑦=(−1)^𝑖𝜏−22𝑍+𝐿𝑦+𝜏−22𝑥𝐽, (84) for 𝑖=1,2, and 𝑍 and 𝐽 are given by (61). Define 1𝑉∶=2𝑉¹+𝑉²=𝐼−𝐿𝑥+𝜏−22𝑦𝐽𝐼−𝐿𝑦+𝜏−22𝑥𝐽. (85) On the one hand, we obtain from Lemma 5 that 𝑉=𝐽⁰_𝜙𝜏(𝑥,𝑦)=lim_𝜀→0∇𝜙_𝜏,𝜀(𝑥,𝑦). On the other hand, we have by Lemma 6 that 𝑉¹,𝑉²∈𝜕_𝐵𝜙_𝜏(𝑥,𝑦), and therefore, 𝑉=(1/2)(𝑉¹+𝑉²)∈𝜕𝜙_𝜏(𝑥,𝑦). This together with Theorem 3 and Definition 1 implies the Jacobian consistency of 𝜙_𝜏,𝜀 with 𝜀>0.

Now, we are in a position to estimate an upper bound of the parameter 𝜀>0 for the predicted accuracy of the distance between the gradient of Φ_𝜏,𝜀 and the subgradient of Φ_𝜏.

Theorem 8. Let 𝜏∈(0,4),  𝛿>0 be given, and (𝑥,𝑦,𝜁)∈𝑅^2𝑛+𝑙 be any point. Let 𝜃_𝜏(𝑥,𝑦) be any function such that 𝜃_𝜏≥𝐿𝑥,𝑦𝑥+𝜏−22𝑦𝐽𝐿𝑦+𝜏−22𝑥𝐽, (86) and let 𝜀∶𝑅^2𝑛×𝑅₊→𝑅₊∪{+∞} be defined by 𝜀𝜆𝑥,𝑦,𝛿∶=¹(𝑧)𝛿2𝜃_𝜏𝑥,𝑦²−𝜆₁(𝑧)𝛿²𝑖𝑓𝑥,𝑦∈𝑆₁𝜃,𝛿<^𝜏𝑥,𝑦𝜆₁(𝑧),𝑧₁𝛿2𝜃_𝜏𝑥,𝑦2𝜃_𝜏𝑥,𝑦−𝛿2𝑧₁𝑖𝑓𝑥,𝑦∈𝑆₂,𝛿<2𝜃_𝜏𝑥,𝑦2𝑧₁,+∞𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (87) Then, for any 𝜀∈𝑅 such that 0<|𝜀|≤𝜀(𝑥,𝑦,𝛿), we have dist∇Φ_𝜏,𝜀𝑥,𝑦,𝜁,𝜕Φ_𝜏𝑥,𝑦,𝜁<𝛿. (88)

Proof. Since it follows from the proof of Theorem 7 that 𝐽⁰_𝜙𝜏(𝑥,𝑦)=𝑉∈𝜕𝜙_𝜏(𝑥,𝑦) for any (𝑥,𝑦,𝜁)∈𝑅^2𝑛+𝑙, we obtain 𝐽⁰_Φ𝜏𝐽𝑥,𝑦,𝜁∶=^0𝜙𝜏𝑥,𝑦∇_𝑥,𝑦𝐹𝑥,𝑦,𝜁𝑂∇_𝜁𝐹𝑥,𝑦,𝜁∈𝜕Φ_𝜏𝑥,𝑦,𝜁. (89) Therefore, we have from (27) and (85) that dist∇Φ_𝜏,𝜀𝑥,𝑦,𝜁,𝜕Φ_𝜏𝑥,𝑦,𝜁=min∇Φ_𝜏,𝜀𝑥,𝑦,𝜁−𝑊∶𝑊∈𝜕Φ_𝜏≤𝑥,𝑦,𝜁∇Φ_𝜏,𝜀𝑥,𝑦,𝜁−𝐽⁰_Φ𝜏=𝑥,𝑦,𝜁∇𝜙_𝜏,𝜀𝑥,𝑦,𝜁−𝐽⁰_𝜙𝜏=𝐿𝑥,𝑦,𝜁𝑥+𝜏−22𝑦𝐿⁻¹(𝑤)𝐿−𝐽𝑦+𝜏−22𝑥𝐿⁻¹(𝑤).−𝐽 (90) Then, by following the proof of Proposition 4.1 in [12], we obtain dist∇Φ_𝜏,𝜀𝑥,𝑦,𝜁,𝜕Φ_𝜏≤ℎ𝑥,𝑦,𝜁_𝜏,0𝑥,𝑦−ℎ_𝜏,𝜀×𝐿𝑥,𝑦𝑥+𝜏−22𝑦𝐽𝐿𝑦+𝜏−22𝑥𝐽, (91) where ℎ_𝜏,𝜀∶𝑅^2𝑛→𝑅₊ is defined as ℎ_𝜏,𝜀1𝑥,𝑦∶=𝜆₁(𝑧)+2𝜀²if𝑥,𝑦∈𝑆₁,√2𝑧₁+𝜀²+|𝜀|if𝑥,𝑦∈𝑆₂,0if𝑥,𝑦=(0,0). (92) Hence, similar to the proof of Theorem 4.1 in [12], we have the desired result.

5. An Alternative Proof

In this section, we study the directional derivative and 𝐵-subdifferential of the one-parametric class of SOC complementarity functions 𝜙_𝜏. Based on these results, we present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions 𝜙_𝜏,𝜀.

By Corollary 3.3 in [18], it is not difficult to see that the function 𝜙_𝜏 given as (3) is directionally differentiable everywhere. However, as far as we know, the expression of its directional derivative is not given in the available literature. In this section, we derive its expression and prove that the 𝐵-subdifferential of 𝜙_𝜏(𝑥,𝑦) at a general point coincides with that of its directional derivative function at the origin.

In light of the 𝐵-subdifferential of 𝜙_FB(𝑥,𝑦) [4, 10, 19], we obtain the following four results, which can be shown by following the proofs of Proposition 9, Lemma 11, Lemma 12, and Proposition 13 in [4], respectively.

Proposition 9. For any given 𝑥=(𝑥₁;𝑥₂),  𝑦=(𝑦₁;𝑦₂)∈𝑅×𝑅^𝑛−1, the directional derivative 𝜙^′_𝜏((𝑥,𝑦);(𝑔,ℎ)) of 𝜙_𝜏 at (𝑥,𝑦) has the following form.(i) If (𝑥,𝑦)=(0,0), then 𝜙^′_𝜏((𝑥,𝑦);(𝑔,ℎ))=𝜙_𝜏(𝑔,ℎ).(ii) If (𝑥,𝑦)∈𝑆₁, then 𝜙^′_𝜏𝑥,𝑦;𝑔,ℎ=𝐼−𝐿⁻¹(𝑤)𝐿𝑥+𝜏−22𝑦𝑔+𝐼−𝐿⁻¹(𝑤)𝐿𝑦+𝜏−22𝑥ℎ. (93)(iii) If (𝑥,𝑦)∈𝑆₂, then 𝜙^′_𝜏;𝜑𝑥,𝑦𝑔,ℎ=𝑔+ℎ−𝑔,ℎ21−𝑧₂+𝑥₂+((𝜏−2)/2)𝑦₂^𝑇𝑔₂+𝑦₂+((𝜏−2)/2)𝑥₂^𝑇ℎ₂2𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁0𝑧₂−12𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×𝑥+𝜏−22𝑦^𝑇𝑔+𝑦+𝜏−22𝑥^𝑇ℎ2𝑥₁+𝜏−22𝑦₁𝑔₂+𝑔₁𝑥₂+𝜏−22𝑦₂+2𝑦₁+𝜏−22𝑥₁ℎ₂+ℎ₁𝑦₂+𝜏−22𝑥₂, (94)where 𝑔=(𝑔₁;𝑔₂),  ℎ=(ℎ₁;ℎ₂)∈𝑅×𝑅^𝑛−1,  𝑧₂ is given as (20), and the function 𝜑∶𝑅^𝑛×𝑅^𝑛→𝑅 is defined by 𝜑𝑔,ℎ=𝜃₁𝑔,ℎ²+𝜃₂𝑔,ℎ²𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁, (95) with 𝜃₁𝑔,ℎ=𝑥₁+𝜏−22𝑦₁𝑔₁−𝑥₂+𝜏−22𝑦₂^𝑇𝑔₂+𝑦₁+𝜏−22𝑥₁ℎ₁−𝑦₂+𝜏−22𝑥₂^𝑇ℎ₂,𝜃₂𝑔,ℎ=𝑥₁+𝜏−22𝑦₁ℎ₂−ℎ₁𝑥₂+𝜏−22𝑦₂+𝑔₁𝑦₂+𝜏−22𝑥₂−𝑦₁+𝜏−22𝑥₁𝑔₂. (96)

Lemma 10. For any given (𝑥,𝑦)∈𝑆₂, let 𝜑 be defined by (95). Then, the B-subdifferential of the function 𝜑 at (0,0) is 𝜕_𝐵𝜑()=𝜁0,0_𝑇2𝑦₂+((𝜏−2)/2)𝑥₂+𝜁₁𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁;−𝑦₁+((𝜏−2)/2)𝑥₁𝜁₂−𝜁₁𝑥₂+((𝜏−2)/2)𝑦₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁−𝜁^𝑇₂𝑥₂+((𝜏−2)/2)𝑦₂+𝜁₁𝑦₁+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁;𝑥₁+((𝜏−2)/2)𝑦₁𝜁₂−𝜁₁𝑦₂+((𝜏−2)/2)𝑥₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁𝜁=𝜁₁;𝜁₂∈𝑅×𝑅^𝑛−1𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔𝜁²₁+𝜁₂²=1. (97)

Lemma 11. For any given (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛, let 𝜓_𝜏(⋅,⋅)≡𝜙^′_𝜏((𝑥,𝑦);(⋅,⋅)). Then, 𝜕_𝐵𝜙_𝜏𝑥,𝑦=𝜕_𝐵𝜓_𝜏(0,0). (98)

Proposition 12. For any given (𝑥,𝑦)∈𝑆₂, we have 𝜕_𝐵𝜙_𝜏=𝑥,𝑦𝐼−𝑋−𝑝𝑢₁(𝑧)^𝑇𝐼−𝑌−𝑞𝑢₁(𝑧)^𝑇𝑝=𝑝₁;𝑝₂,𝑞=𝑞₁;𝑞₂𝜁𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔𝑝=^𝑇2𝑦₂+((𝜏−2)/2)𝑥₂+𝜁₁𝑥₁+((𝜏−2)/2)𝑦₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁;−𝑦₁+((𝜏−2)/2)𝑥₁𝜁₂−𝜁₁𝑥₂+((𝜏−2)/2)𝑦₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁,𝑞=−𝜁^𝑇₂𝑥₂+((𝜏−2)/2)𝑦₂+𝜁₁𝑦₁+((𝜏−2)/2)𝑥₁𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁;𝑥₁+𝜏−22𝑦₁𝜁₂−𝜁₁𝑦₂+𝜏−22𝑥₂𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝜁=𝜁₁;𝜁₂𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜁²₁+𝜁₂²=1, (99) where 𝑢₁(𝑧) and 𝑧₂ are defined by (19) and (20), respectively, and 1𝑋=2𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×𝑥₁+𝜏−22𝑦₁𝑥^𝑇₂+𝜏−22𝑦^𝑇₂𝑥₂+𝜏−22𝑦₂2𝑥₁+𝜏−22𝑦₁𝐼−𝑥₂+𝜏−22𝑦₂𝑧^𝑇₂,1𝑌=2𝑥²₁+𝑦²₁+(𝜏−2)𝑥₁𝑦₁×𝑦₁+𝜏−22𝑥₁𝑦^𝑇₂+𝜏−22𝑥^𝑇₂𝑦₂+𝜏−22𝑥₂2𝑦₁+𝜏−22𝑥₁𝐼−𝑦₂+𝜏−22𝑥₂𝑧^𝑇₂. (100)

Theorem 13 13. The function 𝜙_𝜏,𝜀 defined by (13) with 𝜀>0 satisfies the Jacobian consistency.

Proof. By the proof of Theorem 7, it suffices to show that 𝑉∈𝜕𝜙_𝜏(𝑥,𝑦), where 𝑉 is defined by (85). From Proposition 9 and Lemma 11, it is obvious that the result holds if (𝑥,𝑦)∈𝑆₁ or (𝑥,𝑦)=(0,0). Now, we prove 𝑉∈𝜕𝜙_𝜏(𝑥,𝑦) if (𝑥,𝑦)∈𝑆₂. From Proposition 12, it is not difficult to see that 𝐼−𝑋𝐼−𝑌∈𝜕𝜙_𝜏𝑥,𝑦, (101) for any given (𝑥,𝑦)∈𝑆₂. Direct calculations together with Lemma 4 yield 𝑋=𝐿𝑥+𝜏−22𝑦𝐽,𝑌=𝐿𝑦+𝜏−22𝑥𝐽, (102) if (𝑥,𝑦)∈𝑆₂. Therefore, it follows from (101) that for any given (𝑥,𝑦)∈𝑆₂, 𝑉≡𝐼−𝑋𝐼−𝑌∈𝜕𝜙_𝜏𝑥,𝑦, (103) which completes the proof.

6. Conclusions

In this paper, we show the Jacobian consistency of the one-parametric class of smoothing functions 𝜙_𝜏,𝜀, which will play an important role for establishing the rapid convergence of smoothing methods. Moreover, we estimate the distance between the gradient of the smoothing functions 𝜙_𝜏,𝜀 and the subgradient of the one-parametric class of the SOC complementarity functions 𝜙_𝜏, which will help to adjust a parameter appropriately in smoothing methods. The related topics for further study are about smoothing methods for solving the SOCCP via the Jacobian consistency of the one-parametric class of smoothing functions.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 71171150), China Postdoctoral Science Foundation (no. 2012M511651), and the Excellent Youth Project of Hubei Provincial Department of Education (no. Q20122709), China. The authors are grateful to the editor and the anonymous referees for their valuable comments on this paper. In particular, the authors thank one of the referees and Professor Shaohua Pan for their helpful suggestions on the alternative proof of the Jacobian consistency, which have greatly improved this paper.