to appear in Computational Optimization and Applications, 2011

### A Proximal Point Algorithm for the Monotone Second-Order Cone Complementarity Problem

Jia Wu

School of Mathematical Sciences Dalian University of Technology

Dalian 116024, China E-mail: jwu dora@mail.dlut.edu.cn

Jein-Shan Chen^{1}
Department of Mathematics
National Taiwan Normal University

Taipei 11677, Taiwan E-mail: jschen@math.ntnu.edu.tw

August 4, 2010

(revised on December 7, 2010) (2nd revised on December 21, 2010)

Abstract. This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems eﬃciently, we introduce a generalized Newton method and show that only one Newton step is even- tually needed to obtain a desired approximate solution that approximately satisfies the ap- propriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by [21], which confirm the theoretical results and the eﬀectiveness of the algorithm.

Key words.Complementarity problem, second-order cone, proximal point algorithm, ap- proximation criterion.

1Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Oﬃce. The author’s work is partially supported by National Science Council of Taiwan.

### 1 Introduction

The second-order cone complementarity problem (SOCCP) which is a natural extension of
*nonlinear complementarity problem (NCP), is to find x*∈ ℜ* ^{n}*satisfying

SOCCP(F) :*⟨F(x), x⟩ = 0, F(x) ∈ K, x ∈ K,* (1.1)
where*⟨·, ·⟩ is the Euclidean inner product, F is a mapping from ℜ** ^{n}* intoℜ

*, andK is the Cartesian product of second-order cones (SOC), in other words,*

^{n}K = K^{n}^{1} × · · · × K^{n}* ^{q}*, (1.2)

*where q, n*1*, . . . , n**q**≥ 1, n*1*+ · · · + n**q* *= n, and for each i ∈ {1, . . . , q}*

K^{n}* ^{i}* :

*= {(x*0

*, ¯x) ∈ ℜ × ℜ*

^{n}

^{i}^{−1}

*|∥ ¯x∥ ≤ x*0},

with∥ · ∥ denoting the Euclidean norm and K^{1} denoting the set of nonnegative reals ℜ_{+}.
IfK = ℜ^{n}_{+}, then (1.1) reduces to the nonlinear complementarity problem. Throughout this
paper, corresponding to the Cartesian structure of*K, we write F = (F*1*, . . . , F**q**) and x* =
*(x*_{1}*, . . . , x**q**) with F** _{i}* being mappings fromℜ

*toℜ*

^{n}

^{n}

^{i}*and x*

*∈ ℜ*

_{i}

^{n}

^{i}*, for each i*

*∈ {1, . . . , q}.*

*We also assume that the mapping F is continuously di*ﬀerentiable and monotone.

Until now, a variety of methods for solving SOCCP have been proposed and inves- tigated. They include interior-point methods [1, 2, 13, 18, 28, 31], the smoothing Newton methods [5, 10], the merit function method [4] and the semismooth Newton method [11], where the last three kinds of methods are all based on an SOC complementarity function or a merit function.

The proximal point algorithm (PPA) is known for its theoretically nice convergence
properties, which was first proposed by Martinet [16] and further studied by Rockafellar
*[24]. PPA is a procedure for finding a vector z satisfying 0∈ T (z), where T is a maximal*
monotone operator. Therefore, it can be applied to a broad class of problems such as
convex programming problems, monotone variational inequality problems, and monotone
complementarity problems.

In this paper, motivated by the work of Yamashita and Fukushima [29] for the NCPs, we
focus on introducing PPA for solving the SOC complementarity problems. For SOCCP(F),
*given the current point x*^{k}*, PPA obtains the next point x*^{k}^{+1} by approximately solving the
subproblem

SOCCP(F^{k}) :*⟨F*^{k}*(x), x⟩ = 0, F*^{k}*(x)∈ K, x ∈ K,* (1.3)
*where F** ^{k}* :ℜ

*→ ℜ*

^{n}*is defined by*

^{n}*F*^{k}*(x) := F(x) + c**k**(x− x** ^{k}*) (1.4)

*with c*

*k*

*> 0. It is obvious that F*

^{k}*is strongly monotone when F is monotone. Then,*

*by [8, Theorem 2.3.3], the subproblem SOCCP(F*

*), which is more tractable than the orig- inal problem, always has a unique solution. Thus, PPA is well defined. It was pointed out*

^{k}in [15, 24] that with appropriate criteria for approximate solutions of subproblems (1.3), PPA has global and superlinear convergence property under mild conditions. However, those criteria are usually not easy to check. Inspired by [29], we give a practical criterion based on a new merit function for SOCCP proposed by Chen in [3]. Another implemen- tation issue is how to solve subproblems eﬃciently and obtain an approximate solution such that the approximation criterion for the subproblem is fulfilled. We use a generalized Newton method proposed by De Luca et al. [14] which is used in [29] for the NCP case to solve subproblems. We also give the conditions under which the approximation criterion is eventually approximately fulfilled by a single Newton iteration of the generalized Newton method.

*The following notations and terminologies are used throughout the paper. I repre-*
sents an identity matrix of suitable dimension, ℜ^{n}*denotes the space of n*−dimensional
real column vectors, andℜ^{n}^{1} × · · · × ℜ^{n}* ^{q}* is identified withℜ

^{n}^{1}

^{+···+n}

^{q}*. Thus, (x*

_{1}

*, . . . , x*

*q*) ∈ ℜ

^{n}^{1}× · · · × ℜ

^{n}*is viewed as a column vector inℜ*

^{q}

^{n}^{1}

^{+···+n}

^{q}*. For any two vectors u and v, the*Euclidean inner product is denoted by

*⟨u, v⟩ := u*

^{T}*v and for any vector w, the norm*

*||w|| is*

*induced by the inner product which is called the Euclidean vector norm. For a matrix M,*the norm

*||M|| is denoted to be the matrix norm induced by the Euclidean vector norm, that*is the spectral norm. Given a di

*ﬀerentiable mapping F : ℜ*

*→ ℜ*

^{n}*, we denote by*

^{l}*JF(x)*

*the Jacobian of F at x and∇F(x) := JF(x)*

^{∗}, the adjoint of

*JF(x). For a symmetric matrix*

*M, we write M*

*≻ O (respectively, M ≽ O) if M is positive definite (respectively, positive*

*semidefinite). Given a finite number of square matrices Q*

_{1}

*, · · · , Q*

*q*, we denote the block

*diagonal matrix with these matrices as block diagonals by diag(Q*

_{1}

*, · · · , Q*

*q*

*) or by diag(Q*

*,*

_{i}*i= 1, . . . , q). If I and B are index sets such that I, B ⊆ {1, 2, . . . , q}, we denote by P*IBthe

*block matrix consisting of the sub-matrices P*

*∈ ℜ*

_{ik}

^{n}

^{i}

^{×n}

^{k}*of P with i∈ I, k ∈ B, and denote*

*by x*

_{B}

*a vector consisting of sub-vectors x*

*∈ ℜ*

_{i}

^{n}

^{i}*with i*∈ B.

The organization of this paper is as follows. In Section 2, we recall some notions and background materials. Section 3 is devoted to developing proximal point method to solve the monotone second-order cone complementarity problem with a practical approximation criterion based on a new merit function. In Section 4, a generalized Newton method is introduced to solve the subproblems and we prove that the proximal point algorithm in Section 3 has approximate genuine superlinear convergence under mild conditions, which is the main result of this paper. In Section 5, we report the numerical results for several test problems. Section 6 is to give conclusions.

### 2 Preliminaries

In this section, we review some background materials that will be used in the sequel. We first recall some mathematical concepts and the Jordan algebra associated with the SOC.

Then we talk about the complementarity functions and three merit functions for SOCCP.

Finally, we briefly mention the proximal point algorithm.

### 2.1 Mathematical concepts

Given a setΩ ∈ ℜ^{n}*locally closed around ¯x* *∈ Ω, define the regular normal cone to Ω at ¯x*
by

Nb_{Ω}*( ¯x) :*={

*v*∈ ℜ* ^{n}*lim sup

*x*−→Ω ^{¯x}

*⟨v, x − ¯x⟩*

*||x − ¯x||* ≤ 0}
.

The (limiting) normal cone to*Ω at ¯x is defined by*
NΩ*( ¯x) :*= lim sup

*x*

−→Ω ^{¯x}

NbΩ*(x)*,

where ”limsup” is the Painlev´e-Kuratowski outer limit of sets (see [25]).

We now recall definitions of monotonicity of a mapping which are needed for the as-
*sumptions throughout this paper. We say that a mapping G :* ℜ* ^{n}* → ℜ

*is monotone if*

^{n}*⟨G(ζ) − G(ξ), ζ − ξ⟩ ≥ 0, ∀ ζ, ξ ∈ ℜ** ^{n}*.

*Moreover, G is strongly monotone if there exists*ρ > 0 such that

*⟨G(ζ) − G(ξ), ζ − ξ⟩ ≥ ρ∥ζ − ξ∥*^{2}, ∀ ζ, ξ ∈ ℜ* ^{n}*.

*It is well known that, when G is continuously diﬀerentiable, G is monotone if and only if*

*∇G(ζ) is positive semidefinite for all ζ ∈ ℜ*^{n}*while G is strongly monotone if and only if*

*∇G(ζ) is positive definite for all ζ ∈ ℜ** ^{n}*. For more details about monotonicity, please refer
to [8].

There is another kind of concepts called Cartesian P-properties which have close re-
lationship with monotonicity concept and are introduced by Chen and Qi [6] for a lin-
ear transformation. Here we present the definitions of Cartesian P-properties for a matrix
*M*∈ ℜ^{n}* ^{×n}*and the nonlinear generalization in the setting ofK.

*A matrix M* ∈ ℜ^{n}* ^{×n}* is said to have the Cartesian P-property if for any 0

*, x =*

*(x*1

*, . . . , x*

*q*)∈ ℜ

^{n}*with x*

*i*∈ ℜ

^{n}*, there exists an index*

^{i}*ν ∈ {1, 2, . . . , q} such that ⟨x*ν

*, (Mx)*ν⟩ >

*0. And M is said to have the Cartesian P*_{0}-property if the above strict inequality becomes

*⟨x*ν*, (Mx)*ν*⟩ ≥ 0 where the chosen index ν satisfies x*ν , 0.

*Given a mapping G* *= (G*1*, . . . , G**q**) with G** _{i}* :ℜ

*→ ℜ*

^{n}

^{n}

^{i}*, G is said to have the uniform*

*Cartesian P-property if for any x*

*= (x*1

*, . . . , x*

*q*

*), y*

*= (y*1

*, . . . , y*

*q*) ∈ ℜ

*, there is an index*

^{n}*ν ∈ {1, 2, . . . , q} and a positive constant ρ > 0 such that*

*⟨x*ν*− y*ν*, G*ν*(x)− G*ν*(y)⟩ ≥ ρ∥x − y∥*^{2}.

*In addition, for a single-valued Lipschitz continuous mapping G :* ℜ* ^{n}* → ℜ

*, the B- subdi*

^{m}*ﬀerential of G at x denoted by ∂*

*B*

*G(x), is defined as*

∂*B**G(x) :*={

*k*lim→∞*JG(x** ^{k}*)

*x*

^{k}*→ x, G is diﬀerentiable at x*

*} .*

^{k}The convex hull of∂*B**G(x) is the Clarke’s generalized Jacobian of G at x, denoted by∂G(x),*
*see [7]. We say that G is strongly BD-regular at x if every element of*∂*B**G(x) is nonsingular.*

There is another important concept named semismoothness which was first introduced
*in [17] for functionals and was extended in [23] to vector-valued functions. Let G :*ℜ* ^{n}* →
ℜ

^{m}*be a locally Lipschitz continuous mapping. We say that G is semismooth at a point*

*x*∈ ℜ

^{n}*if G is directionally diﬀerentiable and for any ∆x ∈ ℜ*

^{n}*and V*

*∈ ∂G(x + ∆x) with*

*∆x → 0,*

*G(x+ ∆x) − G(x) − V(∆x) = o(||∆x||).*

*Furthermore, G is said to be strongly semismooth at x if G is semismooth at x and for any*

*∆x ∈ ℜ*^{n}*and V* *∈ ∂G(x + ∆x) with ∆x → 0,*

*G(x+ ∆x) − G(x) − V(∆x) = O(||∆x||*^{2}).

### 2.2 Jordan algebra associated with SOC

It is known thatK* ^{l}*is a closed convex self-dual cone with nonempty interior given by
int(K

*) :*

^{l}*= {x = (x*0

*, ¯x) ∈ ℜ × ℜ*

^{l}^{−1}

*| x*0

*> ∥ ¯x∥}.*

*For any x= (x*0*, ¯x) ∈ ℜ*^{l}*and y= (y*0*, ¯y) ∈ ℜ** ^{l}*, we define their Jordan product as

*x◦ y = (x*

^{T}*y, y*0

*¯x+ x*0

*¯y)*.

*We write x*^{2} *to mean x◦ x and write x + y to mean the usual componentwise addition of*
*vectors. Moreover, if x* ∈ K* ^{l}*, there exists a unique vector inK

^{l}*, which we denote by x*

^{1}

^{2},

*such that (x*

^{1}

^{2})

^{2}

*= x*

^{1}

^{2}

*◦ x*

^{1}

^{2}

*= x. And we recall that each x = (x*0

*, ¯x) ∈ ℜ × ℜ*

^{l}^{−1}admits a spectral factorization, associated withK

*, of the form*

^{l}*x*= λ1*(x)u*^{1}* _{x}*+ λ2

*(x)u*

^{2}

*,*

_{x}whereλ1*(x)*, λ2*(x) and u*^{1}_{x}*, u*^{2}*x* are the spectral values and the associated spectral vectors of
*x, respectively, defined by*

λ*i**(x)= x*0+ (−1)^{i}*∥ ¯x∥, u*^{i}* _{x}* = 1

2(1, (−1)^{i}*ω), i = 1, 2*

with*ω = ¯x/∥ ¯x∥ if ¯x , 0 and otherwise ω being any vector in ℜ** ^{l−1}*satisfying∥ω∥ = 1.

*For each x= (x*0*, ¯x) ∈ ℜ*^{l}*, define the matrix L**x*by
*L** _{x}* :=

[ *x*0 *¯x*^{T}

*¯x* *x*_{0}*I*
]

, (2.1)

which can be viewed as a linear mapping fromℜ* ^{l}*toℜ

*.*

^{l}*Lemma 2.1 The mapping L*_{x}*defined by (2.1) has the following properties.*

*(a) L**x**y= x ◦ y and L**x**+y* *= L**x**+ L**y**for any y*∈ ℜ^{l}*.*

*(b) x*∈ K^{l}*if and only if L*_{x}*≽ O. And x ∈ intK*^{l}*if and only if L*_{x}*≻ O.*

(c) *L*_{x}*is invertible whenever x*∈ intK^{l}*.*
Proof. Please see [4, 10]. *2*

### 2.3 Complementarity and merit functions associated with SOC

In this subsection, we discuss three reformulations of SOCCP that will play an important
role in the sequel of this paper. We deal with the problem SOCCP( ˆ*F), where ˆF :*ℜ* ^{n}* → ℜ

^{n}*is a certain mapping that has the same structure with F in Section 1, that is, ˆF*

*= ( ˆF*1

*, . . . , ˆF*

*q*) with ˆ

*F*

*:ℜ*

_{i}*→ ℜ*

^{n}

^{n}*.*

^{i}A mappingϕ : ℜ* ^{l}*× ℜ

*→ ℜ*

^{l}*is called an SOC complementarity function associated with the coneK*

^{l}*if*

^{l}*ϕ(x, y) = 0 ⇔ x ∈ K*^{l}*, y ∈ K*^{l}*, ⟨x, y⟩ = 0.* (2.2)
A popular choice ofϕ is the vector-valued Fischer-Brumeister (FB) function, defined by

ϕFB*(x, y) := (x*^{2}*+ y*^{2})^{1}^{2} *− x − y, ∀x, y ∈ ℜ** ^{l}*. (2.3)
The function was shown in [10] to satisfy the equivalence (2.2), and therefore its squared
norm

ψFB*(x, y) :=* 1

2∥ϕFB*(x, y)∥*^{2} (2.4)

is a merit function. The functionsϕFBandψFBwere studied in the literature [4,26], in which ϕFB was shown semismooth in [26] whereas ψFB was proved smooth everywhere in [4].

Due to these favorable properties, the SOCCP( ˆ*F) can be reformulated as the following*
nonsmooth system of equations

ΦFB*(x) :*=

ϕFB*(x*1*, ˆF*1*(x))*
ϕFB*(x*_{i}*, ˆF*... *i**(x))*

...

ϕFB*(x*_{q}*, ˆF**q**(x))*

= 0, (2.5)

whereϕFB *is defined as in (2.3) with a suitable dimension l. Moreover, its squared norm*
induces a smooth merit function, given by

*f*_{FB}*(x) :*= 1

2∥ΦFB*(x)*∥^{2} =

∑*q*
*i*=1

ψFB*(x**i**, ˆF**i**(x))*. (2.6)

*Lemma 2.2 The mappings*ΦFB *and f*_{FB} *defined in (2.5) and (2.6) have the following prop-*
*erties.*

*(a) If ˆF is continuously diﬀerentiable, then Φ*FB *is semismooth.*

*(b) If∇ ˆF is locally Lipschitz continuous, then Φ*FB *is strongly semismooth.*

*(c) If ˆF is continuously diﬀerentiable, then f*FB *is continuously diﬀerentiable everywhere.*

*(d) If ˆF is continuously diﬀerentiable and ∇ ˆF(x) at any x ∈ ℜ*^{n}*has the Cartesian P*_{0}*-*
*property, then every stationary point of f**F B**is a solution to the SOCCP( ˆF).*

(e) *If ˆF is strongly monotone and x*^{∗}*is a nondegenerate solution of SOCCP( ˆF), i.e., ˆF*_{i}*(x*^{∗})+
*x*^{∗}* _{i}* ∈ intK

^{n}

^{i}*for all i∈ {1, . . . , q}. Then Φ*FB

*is strongly BD-regular at x*

^{∗}

*.*

Proof. Items (a) and (b) come from [26, Corollary 3.3] and the fact that the composite of (strongly) semismooth functions is (strongly) semismooth by [9, Theorem 19]. Item (c) was shown by Chen and Tseng, which is an immediate consequences of [4, Proposition 2].

Item (d) is due to [20, Proposition 5.1].

For item (e), since*∇ ˆF(x*^{∗}) has Cartesian P-property and is positive definite, which can be
obtained from the strongly monotonicity of ˆ*F, it follows from [22, Proposition 2.1] that the*
conditions in [20, Theorem 4.1] are satisfied and hence (e) is proved. *2*

Since the complementarity functionΦFB *and its induced merit function f*_{FB} have many
useful properties described as in Lemma 2.2, especially when ˆ*F is strongly monotone,*
they play a crucial role in solving subproblems by using a generalized Newton method in
Section 4. On the other hand, in [3], Chen extended a new merit function for the NCP to the
SOCCP and studied conditions under which the new merit function provides a global error
bound and has property of bounded level sets, which play an important role in convergence
*analysis. In contrast, the merit function f*_{FB} lacks these properties. For this reason, we
utilize this new merit function to describe the approximation criterion.

Letψ0:ℜ* ^{l}*× ℜ

*→ ℜ*

^{l}_{+}be defined by ψ0

*(x, y) :=*1

2∥Π_{K}^{l}*(x◦ y)∥*^{2},

where the mappingΠ_{K}* ^{l}*(·) denotes the orthogonal projection onto the set K

*. After taking the fixed parameter as in [3], a new merit function is defined as*

^{l}*ψ(x, y) := ψ*0

*(x, y)+ψ*FB

*(x, y),*whereψFB is given by (2.4). Via the new merit function, it was shown that the SOCCP( ˆ

*F)*is equivalent to the following global minimization:

min

*x*∈ℜ^{n}*f (x) where f (x) :*=

∑*q*
*i*=1

*ψ( ˆF**i**(x), x**i*). (2.7)

Here*ψ is defined with a suitable dimension l.*

*The properties about the function f including the error bound property and the bound-*
edness of level sets which are given in [3] are summarized in the following three lemmas.

*Lemma 2.3 Let f be defined as in (2.7).*

*(a) If ˆF is smooth, then f is smooth and f*^{1}^{2} *is uniformly locally Lipschitz continuous on*
*any compact set.*

*(b) f*(*ζ) ≥ 0 for all ζ ∈ ℜ*^{n}*and f (ζ) = 0 if and only if ζ solves the SOCCP( ˆF).*

*(c) Suppose that the SOCCP( ˆF) has at least one solution, thenζ is a global minimization*
*of f if and only ifζ solves the SOCCP( ˆF).*

Proof. *From [3, Proposition 3.2], we only need to prove that f*^{1}^{2} is Lipschitz continuous on
the set*{y| f (y) = 0}. It follows from [3, Proposition 3.1] that if f (y) = 0, then y**i**◦ ˆF**i**(y)*= 0
*for all i= 1, 2, . . . , q. Thus, for any y ∈ {y| f (y) = 0}, we have*

* f (x)*^{1}^{2} *− f (y)*^{1}^{2}

*= f (x)*^{1}^{2}

≤ 1

√2

∑*q*
*i*=1

(∥ΠK^{ni}*(x*_{i}*◦ ˆF**i**(x))*∥ + ∥ϕFB*(x*_{i}*, ˆF**i**(x))*∥)

= 1

√2

∑*q*
*i*=1

(∥ΠK^{ni}*(x**i**◦ ˆF**i**(x))*− ΠK^{ni}*(y**i**◦ ˆF**i**(y))*∥ + ∥ϕFB*(x**i**, ˆF**i**(x))*− ϕ*F B**(y**i**, ˆF**i**(y))*∥)
.

*Noting that the functions x**i**◦ ˆF**i**(x) and*ϕFB*(x**i**, ˆF**i**(x)) are Lipschitz continuous provided that*
*F is smooth. Then from the Lipschitz continuity of*ˆ ϕFBand the nonexpansivity of projective
*mapping onto a convex set, we obtain that f*^{1}^{2} *is Lipschitz continuous at y.* *2*

Lemma 2.4 *[3, Proposition 4.1] Suppose that ˆF is strongly monotone with the modulus*
*ρ > 0 and ζ*^{∗}*is the unique solution of SOCCP ( ˆF). Then there exists a scalarτ > 0 such*
*that*

τ∥ζ − ζ^{∗}∥^{2} ≤ 3√

*2 f (*ζ)^{1}^{2}, ∀ζ ∈ ℜ* ^{n}*, (2.8)

*where f is given by (2.7) andτ can be chosen as*

τ := ρ

max{√

2*, ∥ ˆF(ζ*^{∗})∥, ∥ζ^{∗}∥}.

Lemma 2.5 *[3, Proposition 4.2] Suppose that ˆF is monotone and that SOCCP( ˆF) is*
*strictly feasible, i.e., there exists ˆ*ζ ∈ ℜ^{n}*such that ˆF( ˆζ), ˆζ ∈ intK. Then the level set*

*L(r) := {ζ ∈ ℜ*^{n}*| f (ζ) ≤ r}*

*is bounded for all r* *≥ 0, where f is given by (2.7).*

Another SOC complementarity function which we usually call it the natural residual mapping is defined by

ϕNR*(x, y) := x − Π*_{K}^{l}*(x− y), ∀x, y ∈ ℜ** ^{l}*,
based on which we define the mappingΦNR :ℜ

*→ ℜ*

^{n}*as*

^{n}ΦNR*(x) :*=

ϕNR*(x*_{1}*, ˆF*1*(x))*
ϕNR*(x**i**, ˆF*... *i**(x))*
ϕNR*(x*_{q}*, ˆF*... *q**(x))*

. (2.9)

Then it is straightforward to see that SOCCP( ˆ*F) is equivalent to the system of equations*
ΦNR*(x)*= 0.

*Lemma 2.6 The mapping*ΦNR *defined as in (2.9) has the following properties.*

*(a) If ˆF is continuously diﬀerentiable, then Φ*NR *is semismooth.*

*(b) If∇ ˆF is locally Lipschitz continuous, then Φ*NR *is strongly semismooth.*

*(c) If∇ ˆF(x) is positive definite, then every V ∈ ∂**B*ΦNR*(x) is nonsingular, i.e.,*ΦNR*is strongly*
*BD-regular at x.*

Proof. Items (a) and (b) are obvious after combining [5, Proposition 4.3] and [9, Theorem
19]. Note that these two items are also proved in [12] in a diﬀerent approach. The proof of
item (c) is similar to that in [27] and [30] for a more general setting, and we omit it. *2*

From Lemma 2.6(c) we know that the natural residual mapping ΦNR is strongly BD- regular under weaker conditions thanΦFB. In view of this, we will useΦNR to explore the condition of superlinear convergence of PPA in Section 3.

### 2.4 Proximal point algorithm

LetT : ℜ* ^{n}* ⇒ ℜ

*be a set-valued mapping defined by*

^{n}*T (x) := F(x) + N*_{K}*(x)*. (2.10)

ThenT is a maximal monotone mapping and SOCCP(F) defined by (1.1) is equivalent to
*the problem of finding a point x such that*

0*∈ T (x).*

*The proximal point algorithm generates, for any starting point x*^{0}, a sequence*{x** ^{k}*} by the
approximate rule:

*x*^{k+1}*≈ P**k**(x** ^{k}*),

*where P*

*:=(*

_{k}*I*+ _{c}^{1}* _{k}*T)

_{−1}

is a single-valued mapping fromℜ* ^{n}* toℜ

*,*

^{n}*{c*

*k*} is some sequence

*of positive real numbers, and x*

^{k}^{+1}

*≈ P*

*k*

*(x*

^{k}*) means that x*

^{k}^{+1}

*is an approximation to P*

_{k}*(x*

*).*

^{k}*Accordingly, for SOCCP(F), P**k**(x** ^{k}*) is given by

*P**k**(x** ^{k}*)=
(

*I*+ 1

*c*_{k}*(F*+ NK)
)_{−1}

*(x** ^{k}*),
from which we have

*P*_{k}*(x** ^{k}*)∈ SOL(SOCCP(F

^{k})),

*where F** ^{k}* is defined by (1.4) and SOL(SOCCP(F

^{k})) is the solution set of SOCCP(F

^{k}).

*Therefore, x** ^{k+1}* is given by an approximate solution of SOCCP(F

^{k}). Two general crite-

*ria for the approximate calculation of P*

*k*

*(x*

*) proposed by Rockafellar [24] are as follows:*

^{k}Criterion 1. *∥x*^{k+1}*− P**k**(x** ^{k}*)∥ ≤ ε

*k*, ∑

_{∞}

*k=0*ε*k* < ∞.

Criterion 2. *∥x*^{k}^{+1}*− P**k**(x** ^{k}*)∥ ≤ η

*k*

*∥x*

^{k}^{+1}

*− x*

*∥, ∑*

^{k}_{∞}

*k*=0η*k* < ∞.

Results on the convergence of the proximal point algorithm have already been stud- ied in [15, 24] from which we know that Criterion 1 guarantees global convergence while Criterion 2, which is rather restrictive, ensures superlinear convergence.

*Theorem 2.1 Let{x*^{k}*} be any sequence generated by the PPA under Criterion 1 with {c**k*}
*bounded. Suppose SOCCP(F) has at least one solution. Then{x*^{k}*} converges to a solution*
*x*^{∗}*of SOCCP(F).*

Proof. This can be proved by similar arguments as in [24, Theorem 1]. *2*

*Theorem 2.2 Suppose the solution set ¯X of SOCCP(F) is nonempty, and let* *{x*^{k}*} be any*
*sequence generated by PPA with Criteria 1 and 2 and c*_{k}*→ 0. Let us also assume that*

*∃ δ > 0, ∃ C > 0, s.t. dist (x, ¯X) ≤ C∥w∥ whenever x ∈ T*^{−1}(*ω) and ∥ω∥ ≤ δ.* (2.11)
*Then the sequence{dist(x*^{k}*, ¯X)} converges to 0 superlinearly.*

Proof. This can be also verified by similar arguments as in [15, Theorem 2.1]. *2*

### 3 A proximal point algorithm for solving SOCCP

Based on the previous discussion, in this section we describe PPA for solving SOCCP(F) as
*defined in (1.1) where F is smooth and monotone. We first illustrate the related mappings*
that will be used in the remainder of this paper.

The mappings ΦNR, ΦFB*, f*FB are defined by (2.9), (2.5) and (2.6), respectively, where
the mapping ˆ*F is substituted by F. And the function f*^{k}*, f*_{FB}* ^{k}* andΦ

^{k}_{FB}are defined by (2.7), (2.6) and (2.5), respectively, where the mapping ˆ

*F is replaced by F*

*which is given by (1.4), i.e.,*

^{k}ΦNR*(x) :*=

ϕNR*(x*_{1}*, F*1*(x))*
ϕNR*(x*_{i}*, F*... *i**(x))*
ϕNR*(x*_{q}*, F*... *q**(x))*

, Φ^{k}_{FB}*(x) :*=

ϕFB*(x*_{1}*, F*_{1}^{k}*(x))*
ϕFB*(x*_{i}*, F*... _{i}^{k}*(x))*
ϕFB*(x*_{q}*, F*... *q*^{k}*(x))*

, ΦFB*(x) :*=

ϕFB*(x*_{1}*, F*1*(x))*
ϕFB*(x*_{i}*, F*... *i**(x))*
ϕFB*(x*_{q}*, F*... *q**(x))*

,

*f*^{k}*(x) :*=

∑*q*
*i*=1

*ψ(x**i**, F*^{k}*i**(x))*, *f*_{FB}*(x) :*= 1

2∥ΦFB*(x)*∥^{2}, *f*^{k}

FB*(x) :*= 1

2∥Φ^{k}_{FB}*(x)*∥^{2}.

Now we are in a position to describe the proximal point algorithm for solving Problem (1.1).

Algorithm 3.1

Step 0. Choose parameters*α ∈ (0, 1), c*0 *∈ (0, 1) and an initial point x*^{0} ∈ ℜ^{n}*. Set k :*= 0.

Step 1. *If x*^{k}*satisfies f*_{F B}*(x** ^{k}*)= 0, then stop.

Step 2. *Let F*^{k}*(x)= F(x) + c**k**(x− x*^{k}*). Get an approximation solution x*^{k}^{+1}*of SOCCP(F** ^{k}*)
that satisfies the condition

*f*^{k}*(x*^{k}^{+1})≤ *c*^{6}* _{k}*min

*{1, ∥x*

^{k}^{+1}

*− x*

*∥*

^{k}^{4}} 18 max{√

2*, ∥F*^{k}*(P**k**(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}*

^{k}^{2}. (3.1)

Step 3. *Set c*_{k+1}*= αc**k**and k := k + 1. Go to Step 1.*

*Theorem 3.1 Let ¯X be the solution set of SOCCP(F). If ¯X* *, ∅, then the sequence {x** ^{k}*}

*generated by Algorithm 3.1 converges to a solution x*

^{∗}

*of SOCCP(F).*

Proof. From Theorem 2.1, it su*ﬃces to prove that such {x*^{k}*} satisfies Criterion 1. Since F*^{k}*is strongly monotone with modulus c*_{k}*> 0 and P**k**(x*^{k}*) is the unique solution of SOCCP(F** ^{k}*),
it follows from Lemma 2.4 that

*∥x*^{k}^{+1}*− P**k**(x** ^{k}*)∥

^{2}≤ 3√ 2

*c** _{k}* max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)*

^{k}*∥} f*

^{k}*(x*

^{k}^{+1})

^{1}

^{2}, (3.2) which together with (3.1) implies

*∥x*^{k}^{+1}*− P**k**(x** ^{k}*)

*∥ ≤ c*

*k*. (3.3)

*2*

To obtain superlinear convergence properties, we need to give the following assumption which will be connected to the condition (2.11) in Theorem 2.2.

Assumption 1. *∥x − Π*K*(x− F(x))∥ provides a local error bound for SOCCP(F), that is,*
there exist positive constants ¯*δ and ¯C such that*

*dist(x, ¯X) ≤ ¯C∥x − Π*_{K}*(x− F(x))∥, for all x with ∥x − Π*_{K}*(x− F(x))∥ ≤ ¯δ,* (3.4)
where ¯*X denotes the solution set of SOCCP(F).*

The following lemma can help us to understand Assumption 1 as it implies conditions under which Assumption 1 holds.

Lemma 3.1 *[19, Proposition 3] If a Lipschitz continuous mapping H is strongly BD-*
*regular at x*^{∗}*, then there is a neighborhood N of x*^{∗} *and a positive constant* *α, such that*

*∀x ∈ N and V ∈ ∂**B**H(x), V is nonsingular and∥V*^{−1}*∥ ≤ α. If, furthermore, H is semismooth*
*at x*^{∗} *and H(x*^{∗}) *= 0, then there exists a neighborhood N*^{′} *of x*^{∗} *and a positive constant*β
*such that∀x ∈ N*^{′}*,∥x − x*^{∗}*∥ ≤ β∥H(x)∥.*

Note that when*∇F(x) is positive definite at one solution x of SOCCP(F), Assumption*
1 holds by Lemma 2.6 and Lemma 3.1.

*Theorem 3.2 LetT be defined by (2.10). If ¯X , ∅, then Assumption 1 implies condition*
*(2.11), that is, there existδ > 0 and C > 0 such that*

*dist(x, ¯X) ≤ C∥ω∥,*
*whenever x*∈ T^{−1}(*ω) and ∥ω∥ ≤ δ.*

Proof. *For all x* ∈ T^{−1}(ω) we have

*w∈ T (x) = F(x) + N*_{K}*(x)*.

*Therefore there exists v* ∈ NK*(x) such that w* *= F(x) + v. Because K is a convex set, it is*
easy to obtain that

ΠK*(x+ v) = x.* (3.5)

Noting that the projective mapping onto a convex set is nonexpansive, we have from (3.5) that

*∥x − Π*_{K}*(x− F(x))∥ = ∥Π*_{K}*(x+ v) − Π*_{K}*(x− F(x))∥ ≤ ∥v + F(x)∥ = ∥ω∥.*

*From Assumption 1 and letting C* *= ¯C, δ = ¯δ yield the desired condition (2.11).* *2*
The following theorem gives the superlinear convergence of Algorithm 3.1, whose
proof is based on Theorem 3.2 and can be obtained in the same way as Theorem 3.1.

We omit the proof here.

*Theorem 3.3 Suppose that Assumption 1 holds. Let{x*^{k}*} be generated by Algorithm 3.1.*

*Then the sequence{dist(x*^{k}*, ¯X)} converges to 0 superlinearly.*

Although we have obtained the global and superlinear convergence properties of Al- gorithm 3.1 under mild conditions, this does not mean that Algorithm 3.1 is practically eﬃcient, as it says nothing about how to obtain an approximation solution of the strongly monotone second-order cone complementarity problem in Step 2 satisfying (3.1) and what is the cost. We will give the answer in the next section.

### 4 Generalized Newton method

In this section, we introduce the generalized Newton method proposed by De Luca, Facchinei,
and Kanzow [14] for solving the subproblems in Step 2 of Algorithm 3.1. As mentioned
*earlier, for each fixed k, Problem (1.3) is equivalent to the following nonsmooth equation*

Φ^{k}_{FB}*(x)*= 0. (4.1)

Now we describe as below the generalized Newton method for solving the nonsmooth sys- tem (4.1), which is employed from what was introduced in [29] for solving NCP.

Algorithm 4.1*(generalized Newton method for SOCCP(F** ^{k}*))
Step 0. Chooseβ ∈ (0,

^{1}

_{2}

*) and an initial point x*

^{0}∈ ℜ

^{n}*. Set j :*= 0.

Step 1. If∥Φ^{k}_{FB}*(x** ^{j}*)∥ = 0, then stop.

Step 2. *Select an element V** ^{j}* ∈ ∂

*B*Φ

^{k}_{FB}

*(x*

^{j}*). Find the solution d*

*of the system*

^{j}*V*^{j}*d* = −Φ^{k}_{FB}*(x** ^{j}*). (4.2)

Step 3. *Find the smallest nonnegative integer i** _{j}* such that

*f*

^{k}FB*(x** ^{j}*+ 2

^{−i}

^{j}*d*

*)≤ (1 − β2*

^{j}^{1}

^{−i}

^{j}*) f*

^{k}FB*(x** ^{j}*).
Step 4.

*Set x*

^{j}^{+1}:

*= x*

*+ 2*

^{j}

^{−i}

^{j}*d*

^{j}*and j := j + 1. Go to Step 1.*

*To guarantee the descent sequence of f*^{k}

FB must have an accumulation point, Pan and
*Chen [20] give the following condition under which the coerciveness of f*_{FB}^{k}*for SOCCP(F** ^{k}*)
can be established.

Condition 1.For any sequence*{x** ^{j}*} ⊆ ℜ

*satisfying*

^{n}*∥x*

^{j}*∥ → +∞, if there exists an index i ∈*

*{1, 2, . . . , q} such that {λ*1

*(x*

_{i}*)} and {λ1*

^{j}*(F*

_{i}*(x*

*))} are bounded below, and λ2*

^{j}*(x*

_{i}*),λ2*

^{j}*(F*

_{i}*(x*

*))→ +∞, then*

^{j}lim sup

*j→∞*

⟨ *x*_{i}^{j}

*∥x*_{i}* ^{j}*∥,

*F*

*i*

*(x*

*)*

^{j}*∥F**i**(x** ^{j}*)∥

⟩

> 0.

*As F** ^{k}* is strongly monotone, it then has the uniform Cartesian P-property. From [20], we
have the following theorem.

Theorem 4.1 *[20, Proposition 5.2] For SOCCP(F*^{k}*), if Condition 1 holds, then the merit*
*function f*^{k}

FB *is coercive.*

To obtain the quadratic convergence of Algorithm 4.1, we need the following two As- sumptions which are also essential in the follow-up work.

*Assumption 2. F*is continuously diﬀerentiable function with a local Lipschitz Jacobian.

Assumption 3. *The limit point x*^{∗} of the sequence *{x** ^{k}*} generated by Algorithm 3.1 is

*nondegenerate, i.e., x*

^{∗}

_{i}*+ F*

*i*

*(x*

^{∗})∈ intK

^{n}

^{i}*holds for all i∈ {1, . . . , q}.*

*Note that when k is large enough, the unique solution P*_{k}*(x*^{k}*) of SOCCP(F** ^{k}*) is nonde-

*generate, that is, (P*

_{k}*(x*

*))*

^{k}

_{i}*+ F*

_{i}

^{k}*(P*

_{k}*(x*

*))∈ intK*

^{k}

^{n}

^{i}*holds for all i*

*∈ {1, . . . , q}. Because F*

*is strongly monotone, we immediately have the following convergence theorem from Lemma 2.2 and [14, Theorem 3.1].*

^{k}*Theorem 4.2 If the sequence{x*^{j}*} generated by Algorithm 4.1 has an accumulation point*
*and Assumptions 2 and 3 hold. Then{x*^{j}*} globally converges to the unique solution P**k**(x** ^{k}*)

*and the rate is quadratic.*

Noting that the condition (3.1) in Algorithm 3.1 is equivalent to the following two criteria:

*Criterion 1. f*^{k}*(x*^{k}^{+1})≤ *c*^{6}* _{k}*
18 max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}*

^{k}^{2}.

*Criterion 2. f*

^{k}*(x*

*)≤*

^{k+1}*c*

^{6}

_{k}*∥x*

^{k}^{+1}

*− x*

*∥*

^{k}^{4}

18 max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}*

^{k}^{2}.

It follows from Section 3 that Criterion 1 guarantees global convergence, while Criterion 2,
which is rather restrictive, ensures superlinear convergence of PPA. Next, we give condi-
tions under which a single Newton step of generalized Newton method can generate a point
*eventually that satisfies the following two criteria for any given r* ∈ (0, 1), i.e., Criterion 1
and the following criterion:

Criterion 2(r). *f*^{k}*(x*^{k}^{+1})≤ *c*^{6}_{k}*∥x*^{k+1}*− x** ^{k}*∥

^{4(1−r)}18 max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}*

^{k}^{2}.

Thereby the PPA can be practically eﬃcient, which we call that Algorithm 3.1 has approx-
imate genuine superlinear convergence. Firstly, we have the following two lemmas, which
indicate the relationship between*∥x*^{k}*− P**k**(x** ^{k}*)

*∥ and dist(x*

^{k}*, ¯X).*

*Lemma 4.1 If* *SOCCP(F) is strictly feasible. Then, for suﬃciently large k, there exists a*
*constant B*_{1} *≥ 2 such that*

2≤ max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}*

^{k}^{2}

*≤ B*1. (4.3) Proof. From Lemma 2.5, we obtain that the solution set ¯

*X of SOCCP(F) is bounded, which*

*implies the boundedness of F( ¯X). Let m*

_{1}> 0 be such that

max{sup

*x∈ ¯X* *∥x∥, sup*

*x∈ ¯X* *∥F(x)∥} ≤ m*1.

*Since c*_{k}*→ 0, it follows from Theorem 3.1 that the two sequences {x*^{k}*} and {P**k**(x** ^{k}*)} have

*the same limit point x*

^{∗}

*∈ ¯X. Then there exists a positive constant m*2 such that

*∥P**k**(x** ^{k}*)

*− x*

^{∗}

*∥ ≤ m*2, and

*∥F(P**k**(x** ^{k}*))

*− F(x*

^{∗})

*∥ ≤ m*2,

*when k is large enough. Thus, the following two inequalities*

*∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥ = ∥F(P*

*k*

*(x*

*))*

^{k}*+ c*

*k*

*(P*

_{k}*(x*

*)*

^{k}*− x*

*)∥*

^{k}*≤ ∥F(P**k**(x** ^{k}*))

*∥ + c*

*k*

*∥P*

*k*

*(x*

*)*

^{k}*− x*

*∥*

^{k}*≤ ∥F(x*^{∗})*∥ + 2m*2

and

*∥P**k**(x** ^{k}*)

*∥ ≤ ∥x*

^{∗}

*∥ + m*2

hold for su*ﬃciently large k. Let B*1*= max{2, (m*1*+ 2m*2)^{2}}, we complete the proof. *2*

*Lemma 4.2 If* *SOCCP(F) is strictly feasible, then for suﬃciently large k, there exists a*
*constant B*_{2} *> 0 such that*

*∥x*^{k}*− P**k**(x** ^{k}*)∥ ≤

*B*2

√*c*_{k}*dist(x*^{k}*, ¯X)*^{1}^{2}

Proof. *Let ¯x** ^{k}* be the nearest point in ¯

*X from x*

*. From [8, Theorem 2.3.5] we know that ¯*

^{k}*X*is convex, and hence the mappingΠ

*X*

^{¯}(·) is nonexpansive. Therefore,

*∥ ¯x*^{k}*− x*^{∗}∥ = ∥Π*X*^{¯}*(x** ^{k}*)− Π

*X*

^{¯}

*(x*

^{∗})

*∥ ≤ ∥x*

^{k}*− x*

^{∗}∥.

Since*{x*^{k}*} is bounded, so is { ¯x*^{k}*}. Let ˆX be a bounded set containing {x*^{k}*} and { ¯x** ^{k}*}. From

*Lemma 2.3, we know that f*

^{1}

^{2}is uniformly Lipschitz continuous on ˆ

*X. Then there exists*

*L*

_{1}> 0 such that

(*f (x** ^{k}*))

^{1}

_{2}

=(
*f (x** ^{k}*))

^{1}

_{2}

−(
*f ( ¯x** ^{k}*))

^{1}

_{2}

*≤ L*^{2}1*∥x*^{k}*− ¯x*^{k}*∥ = L*^{2}1*dist(x*^{k}*, ¯X),*
which implies that

(*f (x** ^{k}*))

^{1}

_{4}

*≤ L*1*dist(x*^{k}*, ¯X)*^{1}^{2}.
It follows from Lemma 2.4 that

*∥x*^{k}*− P**k**(x** ^{k}*)∥

^{2}≤ 3√ 2 τ

*k*

*f*^{k}*(x** ^{k}*)

^{1}

^{2}, where

τ*k* = *c*_{k}

max{√

2*, ∥F*^{k}*(P*_{k}*(x** ^{k}*))

*∥, ∥P*

*k*

*(x*

*)∥}, which together with Lemma 4.1 yields*

^{k}√2
*c** _{k}* ≤ 1

τ*k*

≤

√*B*_{1}
*c** _{k}* .
Hence, we have

*∥x*^{k}*− P**k**(x** ^{k}*)∥ ≤
(3√

*2B*_{1}
*c*_{k}

)^{1}_{2} (

*f*^{k}*(x** ^{k}*))

^{1}

_{4}.

*On the other hand, since F*^{k}*(x** ^{k}*)

*= F(x*

^{k}*), we know f*

^{k}*(x*

*)*

^{k}*= f (x*

*) and hence*

^{k}*∥x*^{k}*− P**k**(x** ^{k}*)∥ ≤
(3√

*2B*_{1}
*c*_{k}

)^{1}_{2} (
*f (x** ^{k}*))

^{1}

_{4}

≤ (3√

*2B*_{1}
*c*_{k}

)^{1}_{2}

*L*_{1}*dist(x*^{k}*, ¯X)*^{1}^{2}.
*Then, letting B*_{2} = (3√

*2B*_{1})^{1}^{2}*L*_{1}leads to the desired inequality. *2*

The next three lemmas give the relationship between *∥x*^{k}_{N}*− P**k**(x** ^{k}*)

*∥ and ∥x*

^{k}*− P*

*k*

*(x*

*)∥ which is the key to the main result in this section. One attention we should pay to is*

^{k}that we will not be able to obtain the inequality in Lemma 4.3 without twice continuously diﬀerentiability. The reason is as explained in [29, Remark 4.1]. To this end, we make the following assumption.

*Assumption 4. F*is twice continuously diﬀerentiable.

*Lemma 4.3 Suppose that Assumption 3 and Assumption 4 hold. Then* Φ^{k}_{FB} *is twice con-*
*tinuously diﬀerentiable in a neighborhood of x*^{k}*for suﬃciently large k, and there exists a*
*positive constant B*_{3}*such that*

∥JΦ^{k}_{FB}*(x*^{k}*)(x*^{k}*− P**k**(x** ^{k}*))− Φ

^{k}_{FB}

*(x*

*)+ Φ*

^{k}

^{k}_{FB}

*(P*

_{k}*(x*

*))*

^{k}*∥ ≤ B*3

*∥x*

^{k}*− P*

*k*

*(x*

*)∥*

^{k}^{2}.

Proof. *It is obvious that when F is twice continuously di*ﬀerentiable and Assumption 3
holds,Φ^{k}_{FB} is twice continuously di*ﬀerentiable near x*^{k}*and P*_{k}*(x*^{k}*) when k is large enough.*

Then from the second order Taylor expansion and the Lipschitz continuity of∇Φ^{k}_{FB}*near x** ^{k}*,

*there exist positive constants m*3

*, m*4

*such that when k is su*ﬃciently large,

∥Φ^{k}_{FB}*(x** ^{k}*)− Φ

^{k}_{FB}

*(P*

_{k}*(x*

*))− JΦ*

^{k}

^{k}_{FB}

*(P*

_{k}*(x*

^{k}*))(x*

^{k}*− P*

*k*

*(x*

*))*

^{k}*∥ ≤ m*3

*∥x*

^{k}*− P*

*k*

*(x*

*)∥*

^{k}^{2},

∥JΦ^{k}_{FB}*(x*^{k}*)(x*^{k}*− P**k**(x** ^{k}*))− JΦ

^{k}_{FB}

*(P*

_{k}*(x*

^{k}*))(x*

^{k}*− P*

*k*

*(x*

*))*

^{k}*∥ ≤ m*4

*∥x*

^{k}*− P*

*k*

*(x*

*)∥*

^{k}^{2}.

*Let B*

_{3}

*= m*3

*+ m*4, we have for su

*ﬃciently large k*

∥JΦ^{k}_{FB}*(x*^{k}*)(x*^{k}*− P**k**(x** ^{k}*))− Φ

^{k}_{FB}

*(x*

*)+ Φ*

^{k}

^{k}_{FB}

*(P*

_{k}*(x*

*))*

^{k}*∥ ≤ B*3

*∥x*

^{k}*− P*

*k*

*(x*

*)∥*

^{k}^{2}.

*2*

Now let us denote

*x*^{k}* _{N}* :

*= x*

^{k}*− V*

*k*

^{−1}Φ

^{k}_{FB}

*(x*

*)*

^{k}*, V*

*k*∈ ∂

*B*Φ

^{k}_{FB}

*(x*

*). (4.4)*

^{k}*Then x*

^{k}*is a point produced by a single Newton iteration of Algorithm 4.1 with the initial*

_{N}*point x*

*.*

^{k}*Lemma 4.4 Suppose that Assumption 3 holds, then*Φ^{k}_{FB} *is diﬀerentiable at x*^{k}*and the Ja-*
*cobian*JΦ^{k}_{FB}*(x*^{k}*) is nonsingular for suﬃciently large k.*

Proof. Let

*z*_{i}*(x)= (x*^{2}_{i}*+ (F*_{i}^{k}*(x))*^{2})^{1}^{2}, (4.5)
*for each i* *∈ {1, . . . , q}. From Assumption 3 and [20, Lemma 4.2], we have that for every*
*i∈ {1, . . . , q},*

*x*^{k}_{i}*+ F*^{k}*i**(x** ^{k}*)∈ intK

^{n}*, and*

^{i}*(x*^{k}* _{i}*)

^{2}

*+ (F*

*i*

^{k}*(x*

*))*

^{k}^{2}∈ intK

^{n}*,*

^{i}