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

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 Chen¹ 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∈ ℜⁿsatisfying

SOCCP(F) :⟨F(x), x⟩ = 0, F(x) ∈ K, x ∈ K, (1.1) where⟨·, ·⟩ is the Euclidean inner product, F is a mapping from ℜⁿ intoℜⁿ, andK is the Cartesian product of second-order cones (SOC), in other words,

K = Kⁿ¹ × · · · × K^nq, (1.2)

where q, n1, . . . , nq≥ 1, n1+ · · · + nq = n, and for each i ∈ {1, . . . , q}

Kⁿⁱ := {(x0, ¯x) ∈ ℜ × ℜⁿⁱ⁻¹|∥ ¯x∥ ≤ x0},

with∥ · ∥ denoting the Euclidean norm and K¹ denoting the set of nonnegative reals ℜ₊. IfK = ℜⁿ₊, then (1.1) reduces to the nonlinear complementarity problem. Throughout this paper, corresponding to the Cartesian structure ofK, we write F = (F1, . . . , Fq) and x = (x₁, . . . , xq) with F_i being mappings fromℜⁿ toℜⁿⁱ and x_i ∈ ℜⁿⁱ, for each i ∈ {1, . . . , q}.

We also assume that the mapping F is continuously differentiable 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 :ℜⁿ → ℜⁿ is defined by

F^k(x) := F(x) + ck(x− x^k) (1.4) with ck > 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^k), which is more tractable than the orig- inal problem, always has a unique solution. Thus, PPA is well defined. It was pointed out

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, ℜⁿ denotes the space of n−dimensional real column vectors, andℜⁿ¹ × · · · × ℜ^nq is identified withℜ^n1+···+nq. Thus, (x₁, . . . , xq) ∈ ℜⁿ¹ × · · · × ℜ^nq is viewed as a column vector inℜ^n1+···+nq. For any two vectors u and v, the Euclidean inner product is denoted by⟨u, v⟩ := u^Tv 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 differentiable mapping F : ℜⁿ → ℜ^l, we denote byJF(x) the Jacobian of F at x and∇F(x) := JF(x)^∗, the adjoint ofJF(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 Q₁, · · · , Qq, we denote the block diagonal matrix with these matrices as block diagonals by diag(Q₁, · · · , Qq) 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 PIBthe block matrix consisting of the sub-matrices P_ik ∈ ℜ^ni×nk of P with i∈ I, k ∈ B, and denote by x_Ba vector consisting of sub-vectors x_i ∈ ℜⁿⁱ 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 ∈ ℜⁿlocally closed around ¯x ∈ Ω, define the regular normal cone to Ω at ¯x by

Nb_Ω( ¯x) :={

v∈ ℜⁿlim sup

x−→Ω ^¯x

⟨v, x − ¯x⟩

||x − ¯x|| ≤ 0} .

The (limiting) normal cone toΩ 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 : ℜⁿ → ℜⁿ is monotone if

⟨G(ζ) − G(ξ), ζ − ξ⟩ ≥ 0, ∀ ζ, ξ ∈ ℜⁿ. Moreover, G is strongly monotone if there existsρ > 0 such that

⟨G(ζ) − G(ξ), ζ − ξ⟩ ≥ ρ∥ζ − ξ∥², ∀ ζ, ξ ∈ ℜⁿ.

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 ∈ ℜⁿ while G is strongly monotone if and only if

∇G(ζ) is positive definite for all ζ ∈ ℜⁿ. 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×nand 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 = (x1, . . . , xq)∈ ℜⁿwith xi ∈ ℜⁿⁱ, there exists an indexν ∈ {1, 2, . . . , q} such that ⟨xν, (Mx)ν⟩ >

0. And M is said to have the Cartesian P₀-property if the above strict inequality becomes

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

Given a mapping G = (G1, . . . , Gq) with G_i :ℜⁿ → ℜⁿⁱ, G is said to have the uniform Cartesian P-property if for any x = (x1, . . . , xq), y = (y1, . . . , yq) ∈ ℜⁿ, there is an index ν ∈ {1, 2, . . . , q} and a positive constant ρ > 0 such that

⟨xν− yν, Gν(x)− Gν(y)⟩ ≥ ρ∥x − y∥².

In addition, for a single-valued Lipschitz continuous mapping G : ℜⁿ → ℜ^m, the B- subdifferential of G at x denoted by ∂BG(x), is defined as

∂BG(x) :={

klim→∞JG(x^k)x^k → x, G is differentiable at x^k} .

The convex hull of∂BG(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∂BG(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 :ℜⁿ → ℜ^m be a locally Lipschitz continuous mapping. We say that G is semismooth at a point x ∈ ℜⁿ if G is directionally differentiable and for any ∆x ∈ ℜⁿ 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 ∈ ℜⁿand 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^lis a closed convex self-dual cone with nonempty interior given by int(K^l) := {x = (x0, ¯x) ∈ ℜ × ℜ^l−1| x0 > ∥ ¯x∥}.

For any x= (x0, ¯x) ∈ ℜ^l and y= (y0, ¯y) ∈ ℜ^l, we define their Jordan product as x◦ y = (x^Ty, y0¯x+ x0¯y).

We write x² 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 x¹², such that (x¹²)² = x¹² ◦ x¹² = x. And we recall that each x = (x0, ¯x) ∈ ℜ × ℜ^l−1 admits a spectral factorization, associated withK^l, of the form

x= λ1(x)u¹_x+ λ2(x)u²_x,

whereλ1(x), λ2(x) and u¹_x, u²x are the spectral values and the associated spectral vectors of x, respectively, defined by

λi(x)= x0+ (−1)ⁱ∥ ¯x∥, uⁱ_x = 1

2(1, (−1)ⁱω), i = 1, 2

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

For each x= (x0, ¯x) ∈ ℜ^l, define the matrix Lxby L_x :=

[ x0 ¯x^T

¯x x₀I ]

, (2.1)

which can be viewed as a linear mapping fromℜ^ltoℜ^l.

Lemma 2.1 The mapping L_xdefined by (2.1) has the following properties.

(a) Lxy= x ◦ y and Lx+y = Lx+ Lyfor 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_xis 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( ˆF), where ˆF :ℜⁿ → ℜⁿ is a certain mapping that has the same structure with F in Section 1, that is, ˆF = ( ˆF1, . . . , ˆFq) with ˆF_i :ℜⁿ → ℜⁿⁱ.

A mappingϕ : ℜ^l× ℜ^l → ℜ^l is called an SOC complementarity function associated with the coneK^lif

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) := (x²+ y²)¹² − 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( ˆF) can be reformulated as the following nonsmooth system of equations

ΦFB(x) :=











ϕFB(x1, ˆF1(x)) ϕFB(x_i, ˆF... i(x))

...

ϕFB(x_q, ˆFq(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(xi, ˆFi(x)). (2.6)

Lemma 2.2 The mappingsΦFB and f_FB defined in (2.5) and (2.6) have the following prop- erties.

(a) If ˆF is continuously differentiable, then ΦFB is semismooth.

(b) If∇ ˆF is locally Lipschitz continuous, then ΦFB is strongly semismooth.

(c) If ˆF is continuously differentiable, then fFB is continuously differentiable everywhere.

(d) If ˆF is continuously differentiable and ∇ ˆF(x) at any x ∈ ℜⁿ has the Cartesian P₀- property, then every stationary point of fF Bis 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 ∈ intKⁿⁱ for all i∈ {1, . . . , q}. Then ΦFB 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 ˆF, 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 ˆF 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^l → ℜ₊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^l. After taking the fixed parameter as in [3], a new merit function is defined asψ(x, y) := ψ0(x, y)+ψFB(x, y), whereψFB is given by (2.4). Via the new merit function, it was shown that the SOCCP( ˆF) is equivalent to the following global minimization:

min

x∈ℜⁿ f (x) where f (x) :=

∑q i=1

ψ( ˆFi(x), xi). (2.7)

Hereψ 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 f¹² is uniformly locally Lipschitz continuous on any compact set.

(b) f(ζ) ≥ 0 for all ζ ∈ ℜⁿand 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 f¹² is Lipschitz continuous on the set{y| f (y) = 0}. It follows from [3, Proposition 3.1] that if f (y) = 0, then yi◦ ˆFi(y)= 0 for all i= 1, 2, . . . , q. Thus, for any y ∈ {y| f (y) = 0}, we have

 f (x)¹² − f (y)¹²

= f (x)¹²

≤ 1

√2

∑q i=1

(∥ΠKⁿⁱ(x_i◦ ˆFi(x))∥ + ∥ϕFB(x_i, ˆFi(x))∥)

= 1

√2

∑q i=1

(∥ΠKⁿⁱ(xi◦ ˆFi(x))− ΠKⁿⁱ(yi◦ ˆFi(y))∥ + ∥ϕFB(xi, ˆFi(x))− ϕF B(yi, ˆFi(y))∥) .

Noting that the functions xi◦ ˆFi(x) andϕFB(xi, ˆFi(x)) are Lipschitz continuous provided that F is smooth. Then from the Lipschitz continuity ofˆ ϕFBand the nonexpansivity of projective mapping onto a convex set, we obtain that f¹² 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)¹², ∀ζ ∈ ℜⁿ, (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 ˆζ ∈ ℜⁿ such that ˆF( ˆζ), ˆζ ∈ intK. Then the level set

L(r) := {ζ ∈ ℜⁿ| 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 :ℜⁿ → ℜⁿas

ΦNR(x) :=











ϕNR(x₁, ˆF1(x)) ϕNR(xi, ˆF... i(x)) ϕNR(x_q, ˆF... q(x))











. (2.9)

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

Lemma 2.6 The mappingΦNR defined as in (2.9) has the following properties.

(a) If ˆF is continuously differentiable, then ΦNR is semismooth.

(b) If∇ ˆF is locally Lipschitz continuous, then ΦNR is strongly semismooth.

(c) If∇ ˆF(x) is positive definite, then every V ∈ ∂BΦNR(x) is nonsingular, i.e.,ΦNRis 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 : ℜⁿ ⇒ ℜⁿbe a set-valued mapping defined by

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 x⁰, a sequence{x^k} by the approximate rule:

x^k+1≈ Pk(x^k), where P_k :=(

I+ _c¹_kT)₋₁

is a single-valued mapping fromℜⁿ toℜⁿ,{ck} is some sequence of positive real numbers, and x^k+1 ≈ Pk(x^k) means that x^k+1 is an approximation to P_k(x^k).

Accordingly, for SOCCP(F), Pk(x^k) is given by

Pk(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 Pk(x^k) proposed by Rockafellar [24] are as follows:

Criterion 1. ∥x^k+1− Pk(x^k)∥ ≤ εk, ∑_∞

k=0εk < ∞.

Criterion 2. ∥x^k+1− Pk(x^k)∥ ≤ ηk∥x^k+1− x^k∥, ∑_∞

k=0ηk < ∞.

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 {ck} 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, fFB are defined by (2.9), (2.5) and (2.6), respectively, where the mapping ˆF 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 ˆF is replaced by F^k which is given by (1.4), i.e.,

ΦNR(x) :=











ϕNR(x₁, F1(x)) ϕNR(x_i, F... i(x)) ϕNR(x_q, F... q(x))











, Φ^k_FB(x) :=











ϕFB(x₁, F₁^k(x)) ϕFB(x_i, F... _i^k(x)) ϕFB(x_q, F... q^k(x))











, ΦFB(x) :=











ϕFB(x₁, F1(x)) ϕFB(x_i, F... i(x)) ϕFB(x_q, F... q(x))











,

f^k(x) :=

∑q i=1

ψ(xi, F^ki(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 parametersα ∈ (0, 1), c0 ∈ (0, 1) and an initial point x⁰ ∈ ℜⁿ. Set k := 0.

Step 1. If x^ksatisfies f_{F B}(x^k)= 0, then stop.

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

f^k(x^k+1)≤ c⁶_kmin{1, ∥x^k+1− x^k∥⁴} 18 max{√

2, ∥F^k(Pk(x^k))∥, ∥Pk(x^k)∥}². (3.1)

Step 3. Set c_k+1 = αckand 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 suffices to prove that such {x^k} satisfies Criterion 1. Since F^k is strongly monotone with modulus c_k > 0 and Pk(x^k) is the unique solution of SOCCP(F^k), it follows from Lemma 2.4 that

∥x^k+1− Pk(x^k)∥²≤ 3√ 2

c_k max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(x^k)∥} f^k(x^k+1)¹², (3.2) which together with (3.1) implies

∥x^k+1− Pk(x^k)∥ ≤ ck. (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 − ΠK(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 ∈ ∂BH(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 constantβ 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 ∈ (0,¹₂) and an initial point x⁰ ∈ ℜⁿ. Set j := 0.

Step 1. If∥Φ^k_FB(x^j)∥ = 0, then stop.

Step 2. Select an element V^j ∈ ∂BΦ^k_FB(x^j). Find the solution d^j of the system

V^jd = −Φ^k_FB(x^j). (4.2)

Step 3. Find the smallest nonnegative integer i_j such that f^k

FB(x^j+ 2^−ijd^j)≤ (1 − β2^1−ij) f^k

FB(x^j). Step 4. Set x^j+1:= x^j+ 2^−ijd^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} ⊆ ℜⁿsatisfying∥x^j∥ → +∞, if there exists an index i ∈ {1, 2, . . . , q} such that {λ1(x_i^j)} and {λ1(F_i(x^j))} are bounded below, and λ2(x_i^j),λ2(F_i(x^j))→ +∞, then

lim sup

j→∞

⟨ x_i^j

∥x_i^j∥, Fi(x^j)

∥Fi(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. Fis 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 + Fi(x^∗)∈ intKⁿⁱ 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^k))∈ intKⁿⁱ holds for all i ∈ {1, . . . , q}. Because F^k is strongly monotone, we immediately have the following convergence theorem from Lemma 2.2 and [14, Theorem 3.1].

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 Pk(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)≤ c⁶_k 18 max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(x^k)∥}². Criterion 2. f^k(x^k+1)≤ c⁶_k∥x^k+1− x^k∥⁴

18 max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(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)≤ c⁶_k∥x^k+1− x^k∥^4(1−r) 18 max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(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− Pk(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 B₁ ≥ 2 such that

2≤ max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(x^k)∥}² ≤ B1. (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 m₁ > 0 be such that

max{sup

x∈ ¯X ∥x∥, sup

x∈ ¯X ∥F(x)∥} ≤ m1.

Since c_k → 0, it follows from Theorem 3.1 that the two sequences {x^k} and {Pk(x^k)} have the same limit point x^∗ ∈ ¯X. Then there exists a positive constant m2 such that

∥Pk(x^k)− x^∗∥ ≤ m2, and

∥F(Pk(x^k))− F(x^∗)∥ ≤ m2, when k is large enough. Thus, the following two inequalities

∥F^k(P_k(x^k))∥ = ∥F(Pk(x^k))+ ck(P_k(x^k)− x^k)∥

≤ ∥F(Pk(x^k))∥ + ck∥Pk(x^k)− x^k∥

≤ ∥F(x^∗)∥ + 2m2

and

∥Pk(x^k)∥ ≤ ∥x^∗∥ + m2

hold for sufficiently large k. Let B1= max{2, (m1+ 2m2)²}, we complete the proof. 2

Lemma 4.2 If SOCCP(F) is strictly feasible, then for sufficiently large k, there exists a constant B₂ > 0 such that

∥x^k− Pk(x^k)∥ ≤ B2

√c_kdist(x^k, ¯X)¹²

Proof. Let ¯x^k be the nearest point in ¯X from x^k. From [8, Theorem 2.3.5] we know that ¯X is convex, and hence the mappingΠX^¯(·) is nonexpansive. Therefore,

∥ ¯x^k− x^∗∥ = ∥ΠX^¯(x^k)− ΠX^¯(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 f¹² is uniformly Lipschitz continuous on ˆX. Then there exists L₁ > 0 such that

(f (x^k))¹₂

=( f (x^k))¹₂

−( f ( ¯x^k))¹₂

≤ L²1∥x^k− ¯x^k∥ = L²1dist(x^k, ¯X), which implies that

(f (x^k))¹₄

≤ L1dist(x^k, ¯X)¹². It follows from Lemma 2.4 that

∥x^k− Pk(x^k)∥²≤ 3√ 2 τk

f^k(x^k)¹², where

τk = c_k

max{√

2, ∥F^k(P_k(x^k))∥, ∥Pk(x^k)∥}, which together with Lemma 4.1 yields

√2 c_k ≤ 1

τk

≤

√B₁ c_k . Hence, we have

∥x^k− Pk(x^k)∥ ≤ (3√

2B₁ c_k

)¹₂ (

f^k(x^k))¹₄ .

On the other hand, since F^k(x^k)= F(x^k), we know f^k(x^k)= f (x^k) and hence

∥x^k− Pk(x^k)∥ ≤ (3√

2B₁ c_k

)¹₂ ( f (x^k))¹₄

≤ (3√

2B₁ c_k

)¹₂

L₁dist(x^k, ¯X)¹². Then, letting B₂ = (3√

2B₁)¹²L₁leads to the desired inequality. 2

The next three lemmas give the relationship between ∥x^k_N − Pk(x^k)∥ and ∥x^k − Pk(x^k)∥ which is the key to the main result in this section. One attention we should pay to is

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. Fis 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 B₃such that

∥JΦ^k_FB(x^k)(x^k− Pk(x^k))− Φ^k_FB(x^k)+ Φ^k_FB(P_k(x^k))∥ ≤ B3∥x^k− Pk(x^k)∥².

Proof. It is obvious that when F is twice continuously differentiable and Assumption 3 holds,Φ^k_FB is twice continuously differentiable 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_FBnear x^k, there exist positive constants m3, m4such that when k is sufficiently large,

∥Φ^k_FB(x^k)− Φ^k_FB(P_k(x^k))− JΦ^k_FB(P_k(x^k))(x^k− Pk(x^k))∥ ≤ m3∥x^k− Pk(x^k)∥²,

∥JΦ^k_FB(x^k)(x^k− Pk(x^k))− JΦ^k_FB(P_k(x^k))(x^k − Pk(x^k))∥ ≤ m4∥x^k− Pk(x^k)∥². Let B₃= m3+ m4, we have for sufficiently large k

∥JΦ^k_FB(x^k)(x^k− Pk(x^k))− Φ^k_FB(x^k)+ Φ^k_FB(P_k(x^k))∥ ≤ B3∥x^k− Pk(x^k)∥². 2

Now let us denote

x^k_N := x^k− Vk⁻¹Φ^k_FB(x^k), Vk ∈ ∂BΦ^k_FB(x^k). (4.4) Then x^k_N is a point produced by a single Newton iteration of Algorithm 4.1 with the initial point x^k.

Lemma 4.4 Suppose that Assumption 3 holds, thenΦ^k_FB is differentiable at x^k and the Ja- cobianJΦ^k_FB(x^k) is nonsingular for sufficiently large k.

Proof. Let

z_i(x)= (x²_i + (F_i^k(x))²)¹², (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^ki(x^k)∈ intKⁿⁱ, and

(x^k_i)²+ (Fi^k(x^k))² ∈ intKⁿⁱ,