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

Authors: Wenli Liu; Xiaoni Chi; Qili Yang; Ranran Cui
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, 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. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.


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## Body

## 1. Introduction

The weighted second-order cone complementarity problem (WSOCCP) is, for a given weight vectorw∈K and a continuously differentiable function F:ℝn×ℝn×ℝm⟶ℝn+m, to find vectors x,s,y∈ℝn×ℝn×ℝm such that(1)x∘s=w,Fx,s,y=0,x∈K,s∈K,where ∘ represents the Jordan product and K is the Cartesian product of second-order cone, that is, K=Kn1×Kn2×⋯×Knr with ∑i=1rni=n,i=1,…,r. The set Knii=1,…,r is the second-order cone (SOC) of dimension ni defined by(2)Kni≔xi=xi0,xi1∈ℝ×ℝni−1:xi0−xi1≥0,and the interior of the SOC Kni is the set(3)intKni=xi=xi0,xi1∈ℝ×ℝni−1:xi0−xi1>0.Here⋅ is the Euclidean norm, and(4)intK=intKn1×intKn2×⋯×intKnr.Obviously, ifw=0, WSOCCP (1) reduces to second-order cone complementarity problem (SOCCP). In this article, we may assume that r=1 and K=Kn 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 interior-point 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]. The 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 functionφ:ℝn×ℝn⟶ℝn for WSOCCP(5)φx,s,w≔x+s−x2+s2+x∘s+w,with a given vector w∈Kn. If w=0, φ (5) reduces to the SOC complementarity function [6] with τ=3:(6)φx,s,0≔x+s−x2+s2+x∘s.Sinceφ is nonsmooth, we define the following smoothing function φμ:(7)φμx,s,w≔x+s−x2+s2+x∘s+w+μ2e,where μ∈ℝ is a smoothing parameter.The 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). These properties will be critical to solve weighted SOC complementarity problems by smoothing methods.The paper is organized as follows. In Section2, 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.Throughout this paper,ℝ+ denotes the set of nonnegative numbers. ℝn and ℝm×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≔xTx for a vector x or the corresponding induced matrix norm. For simplicity, we often use x=x0;x1 instead of the column vector x=x0,x1TT. intKn and bdKn mean the topological interior and the boundary of the SOC Kn, respectively. For a given set S⊂ℝm×n, convS denotes the convex hull of S in ℝm×n, and for any matrix X∈ℝm×n, distX,S denotes infX−Y:Y∈S.


## 2. Preliminaries

In this section, we briefly recall some definitions and results about the Euclidean Jordan algebra [11] associated with the SOC Kn and subdifferentials [12].For anyx,s∈ℝn, their Jordan product is defined as x∘s=xTs;x0s1+s0x1, and e=1,0,…,0∈ℝn is unit element of this algebra. Given an element x=x0;x1∈ℝ×ℝn−1, we define the symmetric matrix(8)Lx=x0x1Tx1x0I,where I represents the n−1×n−1 identity matrix. It is easy to verify that x∘s=Lxs for any s∈ℝn. Moreover, Lx is positive definite (and hence invertible) if and only if x∈intKn.For eachx=x0;x1∈ℝ×ℝn−1, let λ1,λ2 and u1,u2 be the spectral values and the associated spectral vectors of x, given by(9)λi=x0+−1ix1,ui=121;−1ix1x1,ifx1≠0,121;−1ix¯1,otherwise,for i=1,2, with any x¯1∈ℝn−1 such that x¯1=1. Then, x admits a spectral factorization associated with SOC Kn in the form of(10)x=λ1u1+λ2u2.For anyx=x0;x1∈ℝ×ℝn−1, let x′=x0;−x1[13]. Then, x″=x, x+s′=x′+s′, and cx′=cx′ for any c∈ℝ. Moreover, x°x′=x02−x12=0 if x∈bdKn.Suppose thatG:ℝm⟶ℝn is a locally Lipschitzian function; then, from Rademacher’s theorem [14], G is differentiable almost everywhere. The Bouligand (B-) subdifferential and the Clarke subdifferential of G at z are defined by(11)∂BGz≔limz^⟶zG′z^:z^∈DGand∂Gz=conv∂BGz,where DG denotes the set of points at which G is differentiable. Obviously, ∂Gz=G′z if G is continuously differentiable at z.Definition 1 (see [12]).
LetG:ℝm⟶ℝn be a locally Lipschitzian function and Gμ:ℝm⟶ℝn be a continuously differentiable function for any μ>0, and for any z∈ℝm, we have limμ⟶0Gμz=Gz. Then, Gμ satisfies the Jacobian consistency property if for any z∈ℝm, limμ⟶0distGμ′z,∂Gz=0.


## 3. Smoothing Function

In this section, we study the properties of the smoothing function (7).Definition 2 (see [8]).
For a nondifferentiable functionf:ℝm⟶ℝn, we consider a function fμ:ℝm⟶ℝn with a parameter μ>0 that has the following properties:(i)
fμ is differentiable for any μ>0(ii)
limμ⟶0fμx=fx for any x∈ℝm
Such a functionfμ is called a smoothing function of f.Lemma 1.
For anyw∈Kn and μ∈ℝ, one has(12)φμx,s,w=0⇔x∘s=w+μ2e,x∈Kn,s∈Kn.Proof.
We first suppose thatx∘s=w+μ2e,x∈Kn,s∈Kn. Then,(13)0=x∘s−w−μ2e=x+s2−x2+s2+x∘s+w+μ2e,and hence(14)x+s=x2+s2+x∘s+w+μ2e.
That is,φμx,s,w=0.
Conversely, suppose thatφμx,s,w=0; then, it follows from (7) that(15)x+s=x2+s2+x∘s+w+μ2e∈Kn.
Upon squaring both sides of it, we obtain(16)x∘s=w+μ2e∈Kn.
Let(17)ω≔x+s=x2+s2+x∘s+w+μ2e∈Kn,which implies(18)ω∈Kn,ω2=x2+s2+x∘s+w+μ2e∈Kn.
Therefore,(19)ω2−s2=x2+x∘s+w+μ2e∈Kn,ω2−x2=s2+x∘s+w+μ2e∈Kn.
Further, it follows from Proposition 3.4 [15] that(20)x=ω−s∈Kn,s=ω−x∈Kn.Letw=w0;w1∈Kn, μ∈ℝ, x=x0;x1,s=s0;s1∈ℝ×ℝn−1, and the mapping υμ:ℝ2n⟶ℝ×ℝn−1 be defined by(21)υμ=υ0μ;υ1μ=υμx,s,w≔x2+s2+x∘s+w+μ2e,For simplicity, we useυ to denote υμ when μ=0, that is,(22)υ=υ0;υ1=υx,s,w≔x2+s2+x∘s+w.By direct calculations, we have(23)υ0μ=x2+s2+xTs+w0+μ2=υ0+μ2,υ1μ=2x0x1+2s0s1+x0s1+s0x1+w1=υ1.Therefore,υμ=υ0μ;υ1. From the definition of spectral factorization, υμ can be decomposed as(24)υμ=λ1υμu1υ+λ2υμu2υ,where λ1υμ,λ2υμ, and u1υ,u2υ are the spectral values and the associated spectral vectors of υμ given by(25)λiυμ=x2+s2+xTs+w0+μ2+−1i2x0x1+2s0s1+x0s1+s0x1+w1,and(26)uiυ=121;−1iυ¯1,for i=1,2, where(27)υ¯1≔υ1υ1=2x0x1+2s0s1+x0s1+s0x1+w12x0x1+2s0s1+x0s1+s0x1+w1,if υ1≠0; otherwise, υ¯1 is any vector in ℝn−1 such that υ¯1=1. For any given w=w0;w1∈Kn and any x,s∈ℝn×ℝn, it can be verified that(28)υμ=x2+s2+x∘s+w+μ2e=x+s22+34s2+w+μ2e=s+x22+34x2+w+μ2e∈intKn,for any μ>0, and(29)υ=x2+s2+x∘s+w=x+s22+34s2+w=s+x22+34x2+w∈Kn.Givenμ∈ℝ and x=x0;x1,s=s0;s1∈ℝ×ℝn−1, we define(30)ωμ=ω0μ;ω1μ=ωμx,s,w≔x2+s2+x∘s+w+μ2e,and when μ=0,(31)ω=ω0;ω1=ωx,s,w≔x2+s2+x∘s+w.The spectral factorization ofωμ and ω is as follows:(32)ωμ=λ1υμu1υ+λ2υμu2υ,ω=λ1υu1υ+λ2υu2υ.By (29), we can partition ℝ2n as ℝ2n=O∪ℐ∪ℬ, where(33)O≔x,s∈ℝ2n:υ∈0=x,s∈ℝ2n:λ2υ=λ1υ=0,ℐ≔x,s∈ℝ2n:υ∈intKn=x,s∈ℝ2n:λ2υ≥λ1υ>0,ℬ≔x,s∈ℝ2n:υ∈bdKn/0=x,s∈ℝ2n:2υ0=λ2υ>λ1υ=0.Lemma 2.
For any givenw∈Kn and any μ,x,s∈ℝ×ℝn×ℝn, let φ and φμ be defined as (5) and (7), respectively. Then, we have(i)
The functionφμ is continuously differentiable everywhere with any μ>0, and its Jacobian is given by(34)φμ′x,s,w=I−Lx+s2L−1ωμI−Ls+x2L−1ωμ.HereL−1ωμ=1/υ0μI if υ1=0; otherwise,(35)L−1ωμ=L1υμ+L2υμ=bμcμυ¯1Tcμυ¯1aμI+bμ−aμυ¯1υ¯1T,with(36)L1υμ=12λ1υμ1−υ¯1T−υ¯1υ¯1υ¯1T,(37)L2υμ=12λ2υμ1υ¯1Tυ¯1υ¯1υ¯1T+aμ00T0I−υ¯1υ¯1T,where(38)aμ=2λ1υμ+λ2υμ,bμ=121λ1υμ+1λ2υμ,cμ=121λ2υμ−1λ1υμ.(ii)
For anyx,s∈ℝn×ℝn, we have limμ⟶0φμx,s,w=φx,s,w. Thus, φμ is a smoothing function of φ.(iii)
For anyμ,ν∈ℝ+,(39)φμx,s,w−φνx,s,w≤rμ−ν.Proof.
(i)
For anyx,s∈ℝn×ℝn and any μ>0, according to Corollary 5.4 [15] and (28), formula (34) holds. By Proposition 5.2 and its proof [15], we get formula (35).(ii)
Given anyx=x0;x1,s=s0;s1∈ℝ×ℝn−1. For any μ>0, we obtain from the spectral factorization of υμ and υ that(40)φμx,s,w=x+s−λ1υμu1υ+λ2υμu2υ,φx,s,w=x+s−λ1υu1υ+λ2υu2υ,where(41)λiυ=x2+s2+xTs+w0+−1i2x0x1+2s0s1+x0s1+s0x1+w1,andλiυμ and uiυ are, respectively, given by (25) and (26) for i=1,2. It is obvious that(42)λiυμ=λiυ+μ2,fori=1,2. Then,(43)limμ⟶0λ1υμu1υ+λ2υμu2υ=limμ⟶0λ1υ+μ2u1υ+λ2υ+μ2u2υ=λ1υu1υ+λ2υu2υ,andlimμ⟶0φμx,s,w=φx,s,w. Thus, by (i) and Definition 2, φμ 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 ofφ, which will be used in the subsequent analysis.Lemma 3.
For anyx=x0;x1,s=s0;s1,w˜=w˜0;w˜1∈ℝ×ℝn−1, let x2+s2+w˜2∈bdKn. Then, we have(44)x02=x12,s02=s12,w˜02=w˜12,x0s0=x1Ts1,x0w˜0=x1Tw˜1,s0w˜0=s1Tw˜1,x0s1=s0x1,x0w˜1=w˜0x1,s0w˜1=w˜0Ts1.Proof.
We can obtain the desired result by following the proof of Lemma2 [16].Lemma 4.
For anyx=x0;x1,s=s0;s1∈ℝ×ℝn−1, let υ=υ0;υ1=x2+s2+x∘s+w∈bdKn. Then, one has(45)x∘x′=0,s∘s′=0,(46)x∘s′=0,x∘w˜′=0,(47)s∘w˜′=0,w˜∘w˜′=0,(48)x∘υ′=0,s∘υ′=0,(49)x02+s02+x0s0+w02=x0x1+s0s1+x0s1+w12=x12+s12+x1Ts1+w12,wherew˜≔w. Moreover, the following equivalence holds:(50)υ0=0⇔υ1=0⇔υ=0⇔x0=s0=w0=0⇔x1=s1=w1=0⇔x,s,w=0,0,0.Proof.
Since(51)υ=x2+s2+x∘s+w=x+s22+34s2+w˜2=s+x22+34x2+w˜2∈bdKn,from Lemma 3, we have(52)x+s2∘x+s2′=0,s∘s′=0,x+s2∘s′=0,s∘w˜′=0,x+s2∘w˜′=0,w˜∘w˜′=0,s+x2x∘s+x2′=0,x∘x′=0,s+x2∘x′=0,x∘w˜′=0,s+x2∘w˜′=0.
It follows from these equalities that the results in (45)–(47) hold. Since υ∈bdKn, we have λ1υ=0, i.e.,(53)x2+s2+xTs+w0=2x0x1+2s0s1+x0s1+s0x1+w1.
By the last relation and (45)–(47), we obtain that (49) holds. To prove (48), we only need to verify x0υ1=υ0x1 and x1Tυ1=x0υ0 by the symmetry of x and s in υ. From (45)–(47) and (49),(54)x0υ1=x02x0x1+2s0s1+x0s1+s0x1+2w˜0w˜1=2x02+s02+x0s0+w˜02x1=2x12+s12+x1Ts1+w02x1=υ0x1,x1Tυ1=x1T2x0x1+2s0s1+x0s1+s0x1+2w˜0w˜1=2x0x12+s02+x1Ts1+w˜02=2x0x02+s02+x0s0+w02=x0υ0.
From (51), the equivalence is also true.


## 4. 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∈Kn, let z≔x,s,y∈ℝn×ℝn×ℝm. Based on smoothing function (7), we define Φμ:ℝn×ℝn×ℝm⟶ℝ2n+m by(55)Φz≔Fx,s,yφx,s,w,(56)Φμz≔Fx,s,yφμx,s,w.From (1) and (56) and Lemma 1,(57)Φμz=0⇔z=x,s,ysolves WSOCCP1.Since the functionΦz is typically nonsmooth, Newton’s method cannot be applied to the system Φz=0 directly. Thus, we can approximately solve the smooth system Φμz=0 at each iteration and make Φμz decrease gradually by reducing μ to zero. First, we show that the function Φμz satisfies the Jacobian consistency.Lemma 5.
For any arbitrary but fixed vectorw∈Kn, we have for any μ,x,s∈ℝ×ℝn×ℝn,(58)Jφ0x,s≔limμ⟶0φμ′x,s,w=I−Lx+s2JI−Ls+x2J,where(59)J≔L−1ω,ifx,s∈ℐ,122υ01υ¯1Tυ¯14I−3υ¯1υ¯1T,ifx,s∈ℬ,O,ifx,s∈O.Proof.
By (34) and the symmetry of x and s, it suffices to prove(60)limμ⟶0Lx+s2L−1ωμ=Lx+s2J.Case 1.
Ifx,s∈ℐ, it follows from (25) that(61)limμ⟶0ωμ=limμ⟶0λ1υμu1υ+λ2υμu2υ=limμ⟶0λ1υ+μ2u1υ+λ2υ+μ2u2υ=λ1υu1υ+λ2υu2υ=ω∈intKn.
Therefore,(62)limμ⟶0Lx+s2L−1ωμ=Lx+s2L−1ω.Case 2.
Ifx,s∈ℬ, it is easy to prove (51), and(63)2υ0=λ2υ>λ1υ=0,υ1=υ0=x+s22+34s2+w0>0.
Thus, we obtain the following from (25):(64)λ1υμ=λ1υ+μ2=μ2>0,(65)λ2υμ=λ2υ+μ2=2υ0+μ2>0.
For anyμ≠0, we may get from (35) that L−1ωμ=L1υμ+L2υμ. We first prove for any μ≠0,(66)Lx+s2L1υμ=O.
Let(67)ϑ≔1;υ¯1=1υ1υ0;υ1=υυ0.
Based on (36), (48), and (64), we have(68)Lx+s2L1υμ=12λ1υμLx+s2ϑ′ϑ′T=12μx+s2°ϑ′ϑ′T=12μυ02x+s2°υ′υ′T=O.
Next, we provelimμ⟶0L2υμ=J. From (37), (64), and (65), we have(69)limμ⟶0L2υμ=limμ⟶0122υ0+μ21υ¯1Tυ¯1υ¯1υ¯1T+limμ⟶02μ2+2υ0+μ200T0I−υ¯1υ¯1T=122υ01υ¯1Tυ¯14I−3υ¯1υ¯1T=J.
Combining (68) and (69) yields(70)limμ⟶0Lx+s2L−1ωμ=limμ⟶0Lx+s2L2υμ=Lx+s2J.Case 3.
Ifx,s∈O, it follows from Lemma 4 that x,s,w=0,0,0 and(71)ωμ=υμ=μe∈intKn,limμ⟶0Lx+s2L−1ωμ=limμ⟶0O⋅1μe=O=Lx+s2J.Lemma 6.
For any arbitrary but fixed vectorw∈Kn, we have for any x,s∈ℝn×ℝn,(72)I−UxI−Us∈∂Bφx,s,w,where(73)Ux=±12Z+Lx+s2J,Us=±Z+Ls+x2J,Z=O,ifx,s∈ℐ,121−υ¯1T−υ¯1υ¯1υ¯1T,ifx,s∈ℬ,I,ifx,s∈O,and J is defined by (59).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(74)φ′x,s,w=I−Lx+s2L−1ωI−Ls+x2L−1ω∈∂Bφx,s,w.
Thus, it suffices to consider the two cases:x,s∈ℬ and x,s∈O.
For anyx,s∈ℬ or x,s∈O, let x,s^=x,s+μe with sufficiently small μ≠0, and define(75)υ^=υ^0;υ^1≔x2+s^2+x∘s^+w,ω^=ω^0;ω^1≔υ^,ϑ^1≔υ^1υ^1,λ^i=λiυ^≔υ^0+−1iυ^1,i=1,2.
Then, we have(76)υ^=x2+s+μe2+x∘s+μe+w=υ+μx+2μs+μ2e,υ^0=υ0+μx0+2μs0+μ2,υ^1=υ1+μx1+2μs1,(77)λ^i=υ0+μx0+2μs0+μ2+−1iυ1+μx1+2μs1,i=1,2.
Obviously, whenμ⟶0, we have x,s^⟶x,s,υ^⟶υ,ω^⟶ωandλ^i⟶λiυ for i=1,2. Then by (7), it suffices to show(78)limμ⟶0Lx+s^2L−1ω^=Ux,limμ⟶0Ls^+x2L−1ω^=Us,if φ is differentiable at x,s^.Case 4.
Ifx,s∈ℬ, we obtain υ∈bdKn/0, and from (45), (46), and (48),(79)υ^12=υ1+μx1+2μs12=υ12+μ2x12+4μ2s12+4μυ1Ts1+2μυ1Tx1+4μ2x1Ts1=υ0+μx0+2μs02.
The last relation together withυ0>0 implies that for sufficiently small μ, we have(80)υ^1=υ0+μx0+2μs0>0.
For sufficiently smallμ≠0, we obtain from (77) and (80),(81)λ^1=υ0+μx0+2μs0+μ2−υ^1=μ2>0,(82)λ^2=υ0+μx0+2μs0+μ2+υ^1=2υ0+μx0+2μs0+μ2>0.
It follows from (81) and (82) that υ^∈intKn, and hence φ is differentiable at x,s^.
Now we will prove(83)limμ⟶0Lx+s^2L−1ω^=Ux,where L−1ω^=L1υ^+L2υ^, in which L1υ^ and L2υ^ are given by (36) and (37) with υ^ and ϑ^1 replacing υμ and υ¯1, respectively. By the expression of υ^1 and (80),(84)ϑ^≔1;ϑ^1=1υ^1υ^1;υ^1=1υ^1υ0+μx0+2μs0;υ1+μx1+2μs1=1υ^1υ+μx+2μs.
By (45), (46), (48), and (84), we have(85)x+s2°ϑ^′=1υ^1x+s2°υ+μx+2μs′=1υ^1x+s2°υ′+2μx+s2°x2+s′=0.
Thus, from (36) and (81),(86)Lx+s^2L1υ^=12λ^1x+s2+μe2°ϑ^′ϑ^′T=12μx+s2°ϑ^′ϑ^′T+μ2ϑ^′ϑ^′T=sgnμ4ϑ^′ϑ^′T.
It follows from (73)–(84) that as μ⟶0,(87)λ^1⟶λ1υ=0,λ^2⟶λ2υ=2υ0,ϑ^1⟶υ¯1,12ϑ^′ϑ^′T=121−ϑ^1T−ϑ^1−−ϑ^1ϑ^1T⟶Z.
Then, by following the proof of Case5 in Lemma 5, we have(88)limμ⟶0L2υ^=122υ01υ¯1Tυ¯14I−3υ¯1υ¯1T=J.
Therefore, we obtain from (86) and (88) that(89)limμ⟶±0Lx+s^2L−1ω^=limμ⟶±0Lx+s^2L1υ^+limμ⟶±0Lx+s^2L2υ^=limμ⟶±0sgnμ4ϑ^′ϑ^′T+Lx+s2J=±12Z+Lx+s2J=Ux.
Next we will prove(90)limμ⟶0Ls^+x2L−1ω^=Us.
By (45), (46), (48), (81), and (84), we have(91)s+x2°ϑ^′=1υ^1s+x2°υ+μx+2μs′=1υ^1s+x2°υ′+2μs+x2°x2+s′=0,and then(92)Ls^+x2L1υ^=12λ^1s+μe+x2°ϑ^′ϑ^′T=12μs+x2°ϑ^′ϑ^′T+μϑ^′ϑ^′T=sgnμ2ϑ^′ϑ^′T.
Therefore, we obtain from (88) and (92) that(93)limμ⟶±0Ls^+x2L−1ω^=limμ⟶±0Ls^+x2L1υ^+limμ⟶±0Ls^+x2L2υ^=limμ⟶±0sgnμ2ϑ^′ϑ^′T+Ls+x2J=±Z+Ls+x2J=Us.Case 5.
Ifx,s∈O, it follows from Lemma 4 that x,s,w=0,0,0. Thus, υ^=μ2e∈intKn, ω^=μe, and(94)limμ⟶±0Lx+s^2L−1ω^=limμ⟶±0μ2I⋅1μI=limμ⟶±0sgnμ2I=±12I=Ux,limμ⟶±0Ls^+x2L−1ω^=limμ⟶±0μI⋅1μI=limμ⟶±0sgnμI=±I=Us.
Now we show the Jacobian consistency of the functionΦμ (56) and then estimate an upper bound of the parameter μ>0 for the predicted accuracy of the distance between the gradient of Φμ (56) and the subgradient of Φ (55).Theorem 1.
The following results hold. (i) The functionΦμ defined by (56) with μ>0 satisfies the Jacobian consistency. (ii) For given τ>0 and any point z≔x,s,y∈ℝ2n+m, let ρx,s be any function such that(95)ρx,s≥Lx+s2JLs+x2J,and let μ¯:ℝ2n×ℝ+⟶ℝ+∪+∞ be defined by(96)μ¯x,s,τ≔λ1υτρ2x,s−λ1υτ2,ifx,s∈ℐandτ<ρx,s/λ1υ,υ0τ2ρx,s2ρx,s−τ2υ0,ifx,s∈ℬandτ<2ρx,s/2υ0,+∞,otherwise.
Then, for anyμ∈ℝ such that 0<μ≤μ¯x,s,τ, we have(97)distΦμ′z,∂Φz<τ.Proof.
By (56), it suffices to show the Jacobian consistency of φμ with μ>0. Define(98)Vi≔I−UxiI−Usi,where(99)Uxi=−1i12Z+Lx+s2J,Usi=−1iZ+Ls+x2J,for i=1,2, J and Z are defined by (59) and (73). Let(100)V≔12V1+V2=I−Lx+s2JI−Ls+x2J.
It follows from Lemma5 and Lemma 6 that(101)V=Jφ0x,s=limμ⟶0φμ′x,s,w,and V1,V2∈∂Bφx,s,w. Hence,(102)V=12V1+V2∈∂φx,s,w,which together with Definition 1 and Lemma 2 implies the Jacobian consistency of φμ with μ>0. (ii) For any z≔x,s,y∈ℝ2n+m, it follows from the proof of Theorem 1(i) that(103)Jφ0x,s=V∈∂φx,s,w,JΦ0z≔Jφ0x,sOFx,s′x,s,yFy′x,s,y∈∂Φx,s,y.
Thus, we obtain from (34) and (100) that(104)distΦμ′z,∂Φz≤Φμ′z−JΦ0z=φμ′z−Jφ0z=Lx+s2L−1ω^−JLs+x2L−1ω^−J.
Then, similar to the proof of Proposition 4.1 [13], we have(105)distΦμ′z,∂Φz≤g0x,s−gμx,s⋅Lx+s2JLs+x2J,where gμ:ℝ2n⟶ℝ+ is given by(106)gμx,s≔1λ1υ+μ2,ifx,s∈ℐ,22υ0+μ2+μ1λ1υ+μ2,ifx,s∈ℬ,0,ifx,s∈O.
Hence, by following the proof of Theorem 4.1 [13], the result holds.


## 5. Conclusions

In this paper, we show the Jacobian consistency of the smoothing functionφμ for WSOCCP, which will play a key role in analyzing the rapid convergence of smoothing methods. Moreover, in order to adjust a parameter appropriately in smoothing methods, we estimate the distance between the gradient of the smoothing function φμ and the subgradient of the weighted SOC complementarity function φ.



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*Source: 6674520-2021-01-23.xml*