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

and Applied Analysis 3 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 : Rm → Rn be a locally Lipschitzian function. Let G ε : Rm → Rn be a continuously differentiable function for any ε > 0 such that lim ε↓0 G ε (z) = G(z) for any z ∈ R. We say that G ε satisfies the Jacobian consistency property if for any z ∈ R, lim ε↓0 dist(∇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 ∈ Rm [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 + y − √x + y + (τ − 2) (x ∘ y) + 2εe, (13) where the smoothing parameter ε ∈ R. Definition 2 (see [7]). For a nondifferentiable function h : R m → R , 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↓0 h ε (x) = h(x) for any x ∈ R. 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 = (x 1 ; x 2 ), y = (y 1 ; y 2 ) ∈ R × R, and any ε ∈ R, we define the mapping zε : R2n → R × Rn−1 by

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][8][9][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 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 -subdifferential of the oneparametric 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  by  + .We use the symbol ‖ ⋅ ‖ to denote the Euclidean norm defined by ‖‖ := √    for a vector  or the corresponding induced matrix norm.For simplicity, we often use  = ( 1 ;  2 ) for the column vector  = ( 1 ,   2 )  .For the SOC   , int   and bd   mean the topological interior and the boundary of   , respectively.For a given set  ⊂  × , conv  denotes the convex hull of  in  × , and dist(, ) denotes inf{‖ − ‖ :  ∈ } for a matrix  ∈  × .

Preliminaries
In this section, we recall some concepts and results, which include the Euclidean Jordan algebra [3,15] associated with the SOC   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   is the algebra defined by with  = (1, 0, . . ., 0) ∈   being its unit element.Given an element  = ( 1 ;  2 ) ∈  ×  −1 , we define where  represents the ( − 1) × ( − 1) identity matrix.It is easy to verify that  ∘  = () for any  ∈   .Moreover, () is symmetric positive definite (and hence invertible) if and only if  ∈ int   .Now, we give the spectral factorization of vectors in   associated with the SOC   .Let  = ( 1 ;  2 ) ∈  ×  −1 .Then  can be decomposed as where  1 ,  2 , and  (1) ,  (2) are the spectral values and the associated spectral vectors of  given by for  = 1, 2, with any  ∈  −1 such that ‖‖ = 1.By the spectral factorization, a scalar function can be extended to a function for the SOC.For any  ∈   , we define Since both eigenvalues of any  ∈   are nonnegative, we define For any  = ( 1 ;  2 ) ∈  ×  −1 , we define [12]   = ( 1 ; − 2 ).Obviously,   = , ( + )  =   +   , and ()  =   for any  ∈ .Moreover, Let  :   →   be a locally Lipschitzian function.Then,  is differentiable almost everywhere by Rademacher's theorem [17].The Bouligand-(B-) subdifferential and the Clarke subdifferential of  at  are defined by Abstract and Applied Analysis 3 respectively, where   denotes the set of points at which  is differentiable.It is obvious that () = {∇()} if  is continuously differentiable at .
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].
It should be noted that the "inf" appearing in the definition of dist(∇  (), ()) can be replaced by "min, " since the set () is compact at all  ∈   [17].

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   defined by ( 3) is nonsmooth, we consider the function  , defined by where the smoothing parameter  ∈ .
Definition 2 (see [7]).For a nondifferentiable function ℎ :   →   , one considers a function ℎ  :   →   with a parameter  > 0 that has the following properties: In the following, we will show that the function  , given by ( 13) is a smoothing function of   .Thus, we can solve a family of smoothing subproblems  , (, ) = 0 for  > 0 and obtain a solution of   (, ) = 0 by letting  ↓ 0.
Next, we give some properties of   [14], which will be used in the subsequent analysis.

Lemma 4. For any 𝑥 = (𝑥
Moreover, the following equivalence holds: 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 This completes the proof.
By the definition of -subdifferential and (3), it suffices to show that lim if   is differentiable at ( x, ).

An Alternative Proof
In this section, we study the directional derivative and subdifferential of the one-parametric class of SOC complementarity functions   .Based on these results, we present an alternative way to prove the Jacobian consistency of the oneparametric class of smoothing functions  , .By Corollary 3.3 in [18], it is not difficult to see that the function   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 subdifferential of   (, ) at a general point coincides with that of its directional derivative function at the origin.
In light of the -subdifferential of  FB (, ) [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.