Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented.+en, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. +ese results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.


Introduction
e weighted second-order cone complementarity problem (WSOCCP) is, for a given weight vector w ∈ K and a continuously differentiable function F: R n × R n × R m ⟶ R n+m , to find vectors (x, s, y) ∈ R n × R n × R m such that where ∘ represents the Jordan product and K is the Cartesian product of second-order cone, that is, K � K n 1 × K n 2 × · · · × K n r with r i�1 n i � n, i � 1, . . . , r. e set K n i (i � 1, . . . , r) is the second-order cone (SOC) of dimension n i defined by � � � � � � � � ≥ 0 , (2) and the interior of the SOC K n i is the set Here ‖·‖ is the Euclidean norm, and intK � intK n 1 × intK n 2 × · · · × intK n r .
Obviously, if w � 0, WSOCCP (1) reduces to second-order cone complementarity problem (SOCCP). In this article, we may assume that r � 1 and K � K n in the following analysis, since it can easily be extended to the general case.
In order to reformulate several equilibrium problems in economics and study highly efficient algorithms to solve these problems, Potra [1] introduced the notion of a weighted complementarity problem (WCP). He showed that the Fisher market equilibrium problem can be modeled as a monotone linear WCP. Moreover, the linear programming and weighted centering (LPWC) problem, which was introduced by Anstreicher [2], can also be formulated as a monotone linear WCP. And Potra [1] analyzed two interiorpoint methods for solving the monotone linear WCP over the nonnegative orthant. Since then, many scholars are dedicated to investigating the theories and solution methods of WCP. Tang [3] gave a new nonmonotone smoothing-type algorithm to solve the linear WCP. Chi et al. [4] studied the existence and uniqueness of the solution for a class of WCPs.
As is well known, smoothing methods have superior theoretical and numerical performances. For solving the SOCCP by smoothing methods, we usually reformulate the SOCCP as a system of equations based on parametric smoothing functions of SOC complementarity functions [5,6]. e smoothing parameter involved in smoothing functions may be treated as a variable [7] or a parameter with an appropriate parameter control [8]. In the latter case, the Jacobian consistency is important to achieve a rapid convergence of Newton methods or Newton-like methods. Hayashi et al. [8] proposed a combined smoothing and regularized method for monotone SOCCP, and based on the Jacobian consistency of the smoothing natural residual function, they proved that the method has global and quadratic convergence. Krejić and Rapajić [9] gave a nonmonotone Jacobian smoothing inexact Newton method for nonlinear complementarity problem and proved the global and local superlinear convergence of the method. Chen et al. [10] presented a modified Jacobian smoothing method for the nonsmooth complementarity problem and established the global and fast local convergence for the method.
In this paper, we consider the function φ: with a given vector w ∈ K n . If w � 0, φ (5) reduces to the SOC complementarity function [6] with τ � 3: Since φ is nonsmooth, we define the following smoothing function φ μ : where μ ∈ R is a smoothing parameter. e main contribution of this paper is to show the Jacobian consistency of the smoothing function (7) and estimate the distance between the subgradient of the weighted SOC complementarity function (5) and the gradient of its smoothing function (7). ese properties will be critical to solve weighted SOC complementarity problems by smoothing methods. e paper is organized as follows. In Section 2, we review some concepts and properties. In Section 3, we derive the computable formula for the Jacobian of the smoothing function in WSOCCP. In Section 4, we show the Jacobian consistency of the smoothing function and estimate the distance between the gradient of smoothing function and the subgradient of the weighted SOC complementarity function. Some conclusions are reported in Section 5. roughout this paper, R + denotes the set of nonnegative numbers. R n and R m×n denote the space of n-dimensional real column vectors and the space of matrices, respectively. We use ‖ · ‖ to denote the Euclidean norm and define ‖x‖ ≔ for a vector x or the corresponding induced matrix norm. For simplicity, we often use x � (x 0 ; x 1 ) instead of the column vector x � (x 0 , x T 1 ) T . intK n and bdK n mean the topological interior and the boundary of the SOC K n , respectively. For a given set S ⊂ R m×n , convS denotes the convex hull of S in R m×n , and for any matrix X ∈ R m×n , dist(X, S) denotes inf ‖X − Y‖: { Y ∈ S}.

Preliminaries
In this section, we briefly recall some definitions and results about the Euclidean Jordan algebra [11] associated with the SOC K n and subdifferentials [12].
For any x, s ∈ R n , their Jordan product is defined as x ∘ s � (x T s; x 0 s 1 + s 0 x 1 ), and e � (1, 0, . . . , 0) ∈ R n is unit element of this algebra. Given an element x � (x 0 ; x 1 ) ∈ R × R n− 1 , we define the symmetric matrix where I represents the (n − 1) × (n − 1) identity matrix. It is easy to verify that x ∘ s � L(x)s for any s ∈ R n . Moreover, L(x) is positive definite (and hence invertible) if and only if x ∈ intK n .
For each x � (x 0 ; x 1 ) ∈ R × R n− 1 , let λ 1 , λ 2 and u (1) , u (2) be the spectral values and the associated spectral vectors of x, given by for i � 1, 2, with any x 1 ∈ R n− 1 such that ‖x 1 ‖ � 1. en, x admits a spectral factorization associated with SOC K n in the form of For is a locally Lipschitzian function; then, from Rademacher's theorem [14], G is differentiable almost everywhere.
e Bouligand (B-) subdifferential and the Clarke subdifferential of G at z are defined by where D G denotes the set of points at which G is differ- Definition 1 (see [12]). Let G: R m ⟶ R n be a locally Lipschitzian function and G μ : R m ⟶ R n be a continuously differentiable function for any μ > 0, and for any z ∈ R m , we have lim μ⟶0 G μ (z) � G(z). en, G μ satisfies the Jacobian consistency property if for any z ∈ R m , lim μ⟶0 dist (G μ ′ (z), zG(z)) � 0.

Smoothing Function
In this section, we study the properties of the smoothing function (7).
Definition 2 (see [8]). For a nondifferentiable function f: R m ⟶ R n , we consider a function f μ : R m ⟶ R n with a parameter μ > 0 that has the following properties: Such a function f μ is called a smoothing function of f.

Lemma 1.
For any w ∈ K n and μ ∈ R, one has and hence at is, φ μ (x, s, w) � 0. Conversely, suppose that φ μ (x, s, w) � 0; then, it follows from (7) that Upon squaring both sides of it, we obtain Let which implies ω ∈ K n , erefore, Further, it follows from Proposition 3.4 [15] that For simplicity, we use υ to denote υ μ when μ � 0, that is, By direct calculations, we have . From the definition of spectral factorization, υ μ can be decomposed as where λ 1 (υ μ ), λ 2 (υ μ ), and u 1 (υ), u 2 (υ) are the spectral values and the associated spectral vectors of υ μ given by and for i � 1, 2, where if υ 1 ≠ 0; otherwise, υ 1 is any vector in R n− 1 such that ‖υ 1 ‖ � 1. For any given w � (w 0 ; w 1 ) ∈ K n and any (x, s) ∈ R n × R n , it can be verified that for any μ > 0, and

us, by (i)
and Definition 2, φ μ is a smoothing function of φ. (iii) By following the proof of Proposition 5.1 [15], we obtain the desired result. □ Next, we study some properties of φ, which will be used in the subsequent analysis.

Jacobian Consistency
In this section, we will show the Jacobian consistency property and estimate the distance between the gradient of the smoothing function (7) and the subgradient of the WSOCCP complementarity function (5). For any μ ∈ R, w ∈ K n , let z ≔ (x, s, y) ∈ R n × R n × R m . Based on smoothing function (7), we define Φ μ : From (1) and (56) and Lemma 1, Since the function Φ(z) is typically nonsmooth, Newton's method cannot be applied to the system Φ(z) � 0 directly. us, we can approximately solve the smooth system Φ μ (z) � 0 at each iteration and make ‖Φ μ (z)‖ decrease gradually by reducing μ to zero. First, we show that the function Φ μ (z) satisfies the Jacobian consistency.
Proof. By Proposition 5.2 [15] and the chain rule for differentiation, the complementarity function φ is continuously differentiable at any (x, s) ∈ I with us, it suffices to consider the two cases: (x, s) ∈ B and (x, s) ∈ O.