A regularization smoothing method for second-order cone complementarity problem

Nonlinear Analysis: Real World Applications 12 (2011) 731–740

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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

A regularization smoothing method for second-order cone complementarity problem✩ Xiangsong Zhang a,∗ , Sanyang Liu a , Zhenhua Liu a,b a

Applied Mathematics Department, Xidian University, Xi’an, Shaanxi 710071, PR China

b

Network and Data Security Key Laboratory of Sichuan Province, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China

article

info

Article history: Received 26 February 2009 Accepted 3 August 2010 Keywords: Second-order cone complementarity problem Regularization method Smoothing Newton method Global convergence Local superlinear convergence

abstract In this paper, the second-order cone complementarity problem is studied. Based on the Fischer–Burmeister function with a perturbed parameter, which is also called smoothing parameter, a regularization smoothing Newton method is presented for solving the sequence of regularized problems of the second-order cone complementarity problem. Under proper conditions, the global convergence and local superlinear convergence of the proposed algorithm are obtained. Moreover, the local superlinear convergence is established without strict complementarity conditions. Preliminary numerical results suggest the effectiveness of the algorithm. © 2010 Published by Elsevier Ltd

1. Introduction The second-order cone complementarity problem (SOCCP in short) is to find a vector z ∈ Rn such that

⟨f (z ), z ⟩ = 0 and f (z ) ∈ K ,

z ∈ K,

(1) n

n

where ⟨·, ·⟩ represents the Euclidean inner product, f : R → R is a continuously differentiable mapping, and K is the Cartesian product of second-order cones, that is K = K n1 ×K n2 ×· · ·×K nm with n1 +n2 +· · ·+nm = n and n1 , n2 , . . . , nm ≥ 1. The ni -dimensional second-order cone K ni is defined by K ni := {(z1 , z2T )T ∈ R × Rni −1 |z1 ≥ ‖z2 ‖}, where ‖ · ‖ denotes the Euclidean norm and K 1 denotes the set of nonnegative reals R+ (the nonnegative orthant in R). When n1 = n2 = · · · = nm = 1, it can be seen that SOCCP is equivalent to the nonlinear complementarity problem (NCP). Additionally, the Karush–Kuhn–Tucker(KKT) conditions for any second-order cone programming with continuously differentiable functions can also be written in the form of SOCCP; see Ref. [1]. Recently, the second-order cone complementarity problem has drawn a lot of attention partially due to its wide applications [1–4]. Analogous to the nonlinear complementarity problem and the semidefinite complementarity problem, the second-order cone complementarity problem can be employed for a reformulation of (1) as an unconstrained smooth minimization problem or a system of nonlinear equations to solve. Some methods have been developed to treat it [5–7], but most of their algorithms depend on the assumptions of monotone or strict complementarity. Moreover, there is little work for solving the singular second-order cone complementarity problem, in which the derivative of the mapping may be

✩ This work is supported by the National Science Foundation of China (Grant No. 60674108) and Basic Science Research Fund in Xidian University.

∗

Corresponding author. E-mail address: [email protected] (X. Zhang).

1468-1218/$ – see front matter © 2010 Published by Elsevier Ltd doi:10.1016/j.nonrwa.2010.08.001

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X. Zhang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 731–740

seriously ill-conditioned. These motivated us to study this class of problems and obtain a method to try to circumvent one or several difficulties in their algorithms. As we know, there are two classes of methods to handle the singular nonlinear complementarity problems: regularization methods [8,9] and proximal point methods [10,11], see the report [12] and the references therein for details. In this paper, we discuss the class of regularization methods for the second-order cone complementarity problem. This class of methods can deal with the singularity problem by considering a sequence of perturbed problems which possibly have better conditions. The simplest regularization technique is the so-called Tikhonov regularization, which involves solving a sequence of complementarity problems, i.e.,

⟨fµ (z ), z ⟩ = 0 and fµ (z ) ∈ K ,

z ∈ K,

where µ is a positive parameter tending to zero and fµ (z ) = f (z ) + µz. Based on this regularization method, Sun proposed a regularization Newton method for solving the nonlinear complementarity problems with a P0 function in [9]. It was shown that the proposed algorithm [9] does not require the strict complementarity condition in the local superlinear (quadratic) convergence. Lately, Chen and Ma gave a new regularization smoothing method for the nonlinear complementarity problem in [13], i.e., fµ ( z ) = f ( z ) +

1 2

µesin z .

This new regularization method can smooth the complementarity function, but the Tikhonov-regularization method cannot. Considering the virtues of the regularization technique of Chen et al. and the algorithm of Sun, we shall combine and extend them to solve the second-order cone complementarity problem. In this paper, by using a new regularization method, we reformulate the SOCCP into a system of nonlinear equations based on the Fischer–Burmeister function, and present a regularization smoothing Newton method for solving the sequence of problems approximately. The proposed algorithm only solves a linear system of equations and performs only one line search at each iteration. We prove the global convergence and local superlinear convergence of the algorithm. Furthermore, in the absence of a strict complementarity condition, we establish the local superlinear convergence of the algorithm under the assumption of nonsingularity. To evaluate the efficiency of the algorithm, we conduct some numerical experiments. This paper is organized as follows. In the next section, some preliminaries with second-order cones are introduced first, then a complementarity function is studied and some definitions are included. In Section 3, a regularization smoothing Newton algorithm is presented. In Section 4, the global convergence and local convergence of the algorithm are discussed. Numerical results are reported in Section 5. Some conclusions are given in Section 6. Throughout this paper, all vectors are column vectors, T denotes√ transpose, I represents an identity matrix of suitable dimension, and ‖ · ‖ denotes the Euclidean norm defined by ‖x‖ := xT x for a vector x. R++ means the positive orthant of R. For any differentiable function f : Rn → Rn , ∇ f (x) denotes the gradient of f at x. Let intK denote the interior of K . x ≽ y or x ≻ y means that x − y ∈ K or x − y ∈ intK , respectively. For simplicity, we use x = (x1 , x2 ) ∈ R × Rn−1 for the column vector x = (x1 , xT2 )T . 2. Preliminaries 2.1. Jordan algebra associated with SOC In this subsection, we shall give the basic facts concerning Euclidean–Jordan algebra [1,14], which provides a useful methodology of dealing with second-order cone (SOC for short). A Euclidean–Jordan algebra is a triple (V , ⟨·, ·⟩, ◦) (V for short), where (V , ⟨·, ·⟩) is a finite-dimensional inner product space over R and (x, y) → x ◦ y : V × V → V is a bilinear mapping which satisfies the following conditions: (a) x ◦ y = y ◦ x, for any x, y ∈ V . (b) x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y) for all x, y ∈ V where x2 = x ◦ x. (c) ⟨(x ◦ y, z )⟩ = ⟨(x, y ◦ z )⟩ for all x, y, z ∈ V . In this paper, we consider Rn with the Euclidean–Jordan algebra ⟨·, ·⟩ and norm ‖ · ‖. In this algebra, the SOC K is the cone of squares, i.e., K = {x2 : x ∈ (Rn , ⟨·, ·⟩, ◦)}. For any x = (x1 , x2 ), y = (y1 , y2 ) ∈ Rn , their Jordan product associated with K is defined by x ◦ y := (xT y, x1 y2 + y1 x2 ). Some of the prominent relations involving the binary operation ◦ are as follows, (a) (b) (c) (d)

the vector e¯ = (1, 0, . . . , 0)T ∈ Rn is the unique identity element: x ◦ e¯ = x. Write x2 to mean x ◦ x and x + y for the usual componentwise addition of vectors. x2 ∈ K , for all x ∈ Rn . If x ∈ K , there exists a unique vector in K , denoted by x1/2 , such that (x1/2 )2 = x1/2 ◦ x1/2 = x.

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(e) For any vector x = (x1 , x2 ) ∈ R × Rn−1 , the following symmetrical matrix can be used to denote it,

 Lx :=

x1 x2

xT2 x1 I



,

where I represents the (n − 1) × (n − 1) identity matrix. It is easy to verify that x ◦ s = Lx s for any s ∈ Rn . Moreover, Lx is positive definite (hence invertible) if and only if x ∈ intK . Lemma 2.1 (Lemma X.2.2 [14] or Proposition 2.1 [2] and Proposition 6 [15]). The following conditions are equivalent. (a) x ∈ K , y ∈ K and x ◦ y = 0. (b) x ∈ K , y ∈ K and ⟨x, y⟩ = 0. Next, the spectral factorization of vectors in Rn associated with SOC shall be recalled. Spectral factorization is one of the basic concepts in Euclidean–Jordan algebras with respect to SOC, which has been introduced in [14]. Theorem 2.1 (Spectral Factorization). For any vector x = (x1 , x2 ) ∈ R × Rn−1 , its spectral factorization with respect to the second-order cone K is defined as x = λ1 u(1) + λ2 u(2) , where λ1 , λ2 are the spectral values given by

λi = x1 + (−1)i ‖x2 ‖, and u

(1)

,u

u(i)

(2)

i = 1, 2,

(2)

are the spectral vectors given by

   x2 1   1, (−1)i ,  ‖x 2 ‖ = 2  1   (1, (−1)i ω), 2

x2 ̸= 0

i = 1, 2,

(3)

x2 = 0

for any ω ∈ Rn−1 such that ‖ω‖ = 1. For any x = (x1 , x2 ) ∈ Rn , the spectral factorizations of x, x2 and x1/2 have some basic properties as below, whose proof can be found in [2]. These properties will be used in the analysis of the smoothing function. Proposition 2.1. For any x = (x1 , x2 ) ∈ R × Rn−1 with the spectral values λ1 , λ2 given by (2) and spectral vectors u(1) , u(2) given by (3), we have (a) u(i) ◦ u(i) = u(i) , for i = 1, 2. (b) u(1) ◦ u(2) = 0, u1 + u2 = e¯ and ‖u(1) ‖ = ‖u(2) ‖ = √1 . 2 (c) x ∈ K if and only if λ1 > 0, and x ∈ intK if and only if λ1 > 0. 2 (1) 2 (2) 2 (d) x = λ1√ u + λ2 u √ ∈ K . (e) x1/2 = λ1 u(1) + λ2 u(2) ∈ K , if x ∈ K . 2.2. The regularization smoothing function In this subsection, we first recall some concepts for our discussion in what follows, and then investigate a regularization smoothing function for SOCCP. Firstly, we give the concepts of P-properties for a nonlinear mapping on Cartesian products and with SOCs. The concepts of P-properties on Cartesian products in Rn were first established by Facchinei and Pang [16]. Recently, Chen and Qi [17] and Kong et al. [18] extended the concepts of Cartesian P-properties to the setting of positive semidefinite cones and the general Euclidean–Jordan algebra, respectively. Definition 2.1. A nonlinear mapping f = (f1 , . . . , fm ) with fi : Rn → Rni is said to have (a) the Cartesian P-property, if for any x, y ∈ Rn with x ̸= y, there exists an index i ∈ {1, 2, . . . , m} such that

⟨xi − yi , fi (x) − fi (y)⟩ > 0; (b) the Cartesian P0 -property, if for any x, y ∈ Rn with x ̸= y, there exists an index i ∈ {1, 2, . . . , m} such that xi ̸= yi

and ⟨xi − yi , fi (x) − fi (y)⟩ ≥ 0.

The concepts of P-properties for a nonlinear mapping in the setting of SOC are special cases of those given by Tao and Gowda [19]. Definition 2.2 ([19]). A nonlinear mapping f : Rn → Rn is said to have (a) the Jordan P-property if (x − y) ◦ (f (x) − f (y)) ∈ −K ⇒ x = y. (b) the P-property if the condition that Lxi −yi Lfi (x)−fi (y) = Lfi (x)−fi (y) Lxi −yi , i = 1, 2, . . . , m and (x − y) ◦ (f (x) − f (y)) ∈ −K implies x = y.

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Proposition 2.2 ([20]). (a) If a mapping f : Rn → Rn has the Cartesian P-property, then it must have the Jordan P-property. (b) If a mapping f : Rn → Rn has the Cartesian P0 -property, then it must have the P0 -property. If the continuously differentiable mapping f has the Cartesian P-property (or P0 -property), then its transposed Jacobian matrix ∇ f (x) at any x ∈ Rn has the corresponding Cartesian P-properties. Without loss of generality, we may assume that m = 1 and n1 = n in the following analysis, since our analysis can be easily extended to the general case. If G is locally Lipschitz continuous on Rn , then according to Rademacher’s theorem, G is differentiable almost everywhere. Let DG denote the set of points at which G is differentiable and let ∂B G(x) be defined by

 ∂B G(x) = J =



lim

xi →x,xi ∈DG

G′ (xi ) .

We denote by ∂ G(x) the generalized derivative in the sense of Clarke [21], i.e.,

∂ G(x) = the convex hull of ∂B G(x). Semismoothness is a generalization concept of smoothness, which was originally introduced by Mifflin in [22] for functions and extended to vector-valued functions in [23]. Convex functions, smooth functions, and piecewise linear functions are examples of semismooth functions, and the composition of (strongly) semismooth functions is still a (strongly) semismooth function [22]. Definition 2.3 ([23]). Suppose that ϕ : Rl → Rn is a locally Lipschitz function. Then (a) ϕ(·) is said to be semismooth at x, if for any V ∈ ∂ϕ(x + h), h → 0,

‖ϕ(x + h) − ϕ(x) − Vh‖ = o(‖h‖). (b) ϕ(·) is said to be strongly semismooth at x, if for any V ∈ ∂ϕ(x + h), h → 0,

‖ϕ(x + h) − ϕ(x) − Vh‖ = o(‖h‖2 ), where ∂ϕ(·) stands for the generalized Jacobian of ϕ in the sense of Clarke [21]. Next we give the concept of a smoothing function of a non-differentiable function introduced by Hayashi et al. in [24]. Definition 2.4 ([24]). For a non-differentiable function h : Rn → Rℓ , we consider a function hµ : Rn → Rℓ with a parameter µ > 0 that has the following properties: (a) hµ is differentiable for any µ > 0, (b) limµ→0 hµ (x) = h(x) for any x ∈ Rn . Such a function hµ is called a smoothing function of h. Now, we introduce a regularization method for solving SOCCP. It is known that the regularization method has been widely studied to deal with ill-posed problems [25,8,26,27]. Here we refer to the regularization method in [13]. By introducing a regularization parameter µ ∈ R into the function f (x), we have f µ ( x) = f ( x) +

1

µesin x , (4) 2 where µ is a parameter converging to zero. Based on the perturbed function (4), we study the vector-valued Fischer–Burmeister function φ0 : Rk × Rk → Rk , which is defined by φ0 (x, y) = x + y −



x2 + y2 .

This complementarity function satisfies

φ0 (x, y) = 0 ⇔ x ◦ y = 0,

x ∈ K, y ∈ K.

We note that φ0 is typically nonsmooth. To hurdle this shortcoming, we let y = fµ (x). Then, from (4), we obtain the following vector-valued function φµ : R × Rn → Rn defined as

φµ (x, fµ (x)) = x + fµ (x) −



x2 + fµ (x)2 ,

(5)

where µ can also be called a smoothing parameter. For simplicity, we write φ(µ, x) to denote φµ (x, fµ (x)). Then, as we will show, φ(µ, x) defined by (5) is a smooth approximation for the Fischer–Burmeister function. This property plays a fundamental role in the analysis of the local convergence rate of our regularization smoothing Newton method. Finally, we will show the properties of the functions fε given in (4) and φ given in (5) for our later analysis.

X. Zhang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 731–740

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Proposition 2.3 ([20]). For any ε > 0, let fε : Rn → Rn be given as in (4). If f has the Cartesian P0 -property, then fε will have the Cartesian P-property. Lemma 2.2. Let φ(µ, x) be given in (5). Then for any (µ, x) ∈ R × Rn , (a) φ(µ, x) is globally Lipschitz continuous and strongly semismooth everywhere. Moreover, φ is continuously differentiable at any (µ, x) ∈ R++ × Rn with its Jacobian

   ∇µ φ(µ, x) ∇φ(µ, x) = = ∇x φ(µ, x)

1 2

1 sin x (I − L− w Lfµ )e



1 T −1 (I − Lx L− w ) + ∇ fµ (x) (I − Lfµ Lw )

(6)

where w := w(µ, x) = x2 + fµ (x)2 and I denotes an proper dimensional identity matrix. (b) For any (µ, x) ∈ R × Rn , φ(µ, x) is a smoothing function of φ0 (x, f (x)).



Proof. From theorem 3.2 in [28], it is not difficult to show that φ is globally Lipschitz continuous, strongly semismooth everywhere and continuously differentiable at any (µ, x) ∈ R++ × Rn . Furthermore, by the definition of w , we have w ≻ 0 for any (µ, x) ∈ R++ × Rn , which implies that Lw is invertible. Then, from the definition of φ(µ, x), we immediately obtain the desired Jacobian formula (6) by the chain rule for differentiation. Next, we prove (b). From the definition of w , we get w2 = x2 + fµ (x)2 . It follows from the spectral factorization of w and Proposition 2.1 that

  φ(µ, x) = x + fµ (x) − ( λ1 (µ)u(1) + λ2 (µ)u(2) ), where λi (µ) = ‖x‖2 + ‖f (x) + 21 µesin x ‖2 + 2(−1)i ‖v(µ)‖, i = 1, 2.

u(i)

   v(µ) 1   1, (−1)i ,  ‖v(µ)‖ = 2  1   (1, (−1)i ω),

if v(µ) ̸= 0

i = 1, 2,

if v(µ) = 0

2 v(µ) = x1 x2 + fµ1 (x)fµ2 (x),

with ω ∈ Rn−1 being an arbitrary vector satisfying ‖ω‖ = 1 and fµ (x) = (fµ1 (x), fµ2 (x)) ∈ R × Rn−1 . On the other hand, using Theorem 2.1 then yields that

  φ0 (x, f (x)) = x + f (x) − ( λ1 u(1) + λ2 u(2) ), where λi = ‖x‖2 + ‖f (x)‖2 + 2(−1)i ‖v‖, i = 1, 2,

u(i)

   1  i v  1, (−1) ,  ‖v‖ = 2   1 (1, (−1)i ω),  2

if v ̸= 0

i = 1, 2,

if v = 0

v = x1 x2 + f1 (x)f2 (x) with f (x) = (f1 (x), f2 (x)) ∈ R × Rn−1 . Then we can get that limµ→0 φ(µ, x) = φ0 (x, f (x)). It then follows from Definition 2.4 that φ(µ, x) is a smoothing function of φ0 (x, f (x)).  3. Algorithm description Based on function (5) introduced in the previous section, we present a regularization smoothing Newton method for SOCCP. First we give the following notations   for algorithm description. µ

Let z = (µ, x) ∈ R × Rn , H (z ) = φ(z ) , where φ(z ) is defined by (5). Let θ (z ) = ‖H (z )‖2 = µ2 + ‖φ(z )‖2 . The function β : Rn+1 → R+ is defined by

β(z ) = γ min{1, ‖H (z )‖}. Then the algorithm for SOCCP can be described as follows. Algorithm 3.1. Step 0. Choose constants δ, η ∈ (0, 1), σ ∈ (0, 1/2), η0 ∈ (0, η] and µ0 ∈ R++ . Let z¯ = (µ0 , 0) ∈ R++ × Rn and an arbitrary starting point z 0 = (µ0 , x0 ) ∈ R++ × Rn . Choose γ ∈ (0, 1) to satisfy γ ‖H (z 0 )‖ ≤ 1, γ µ0 < 1. Set k := 0. Step 1. If H (z k ) = 0, stop. Otherwise, let β k := β(z k ) = γ min{1, ‖H (z k )‖}.

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X. Zhang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 731–740

Step 2. Compute 1z k := (1µk , 1xk ) ∈ R × Rn by H (z k ) + ∇ H (z k )1z k = β k z¯ .

(7)

Step 3. If ‖H (z + 1z )‖ ≤ η , then let z := z + 1z , η the maximum of the values 1, δ, δ 2 , . . . such that k

k

k+1

k

k

k

k+1

:= η γ 0

k+1

, set k := k + 1, go to Step 1; Otherwise, let δ

θ (z k + δ m 1z k ) ≤ [1 − 2σ (1 − γ µ0 )δ m ]θ (z k ).

mk

be

(8)

Step 4. Set z k+1 := z k + δ mk 1z k and k := k + 1. Go to Step 1. To analyze the convergence of the algorithm, we shall introduce the following Lemmas. Lemma 3.1. (a) H is continuously differentiable at any z = (µ, x) ∈ R++ × Rn , and is strongly semismooth everywhere on z = (µ, x) ∈ Rn+1 , with its Jacobian

∇ H (z ) =



1

∇µ φ(z )

0

∇x φ(z )



.

(9)

(b) If f is a Cartesian P0 function, then ∇ H (z ) is nonsingular for any µ ∈ R++ . Proof. Since the composition of the strongly semismooth function is strongly semismooth [22], by Lemma 2.2, it is easy to show that H (z ) is continuously differentiable and strongly semismooth at any z = (µ, x) ∈ R++ × Rn , we have (a). In order to prove (b), it suffices to prove that ∇x φ(µ, x) is nonsingular. By using the definition of w , we have w 2 = x2 + fµ (x)2 , which implies that w ≻ 0, w 2 ≻ x2 and w 2 ≽ fµ (x)2 for any µ > 0, x ∈ Rn . It follows from Lemma 3.1 in [2] that Lw ≻ Lx and Lw ≽ Lfµ (x) . Then we readily obtain that Lw ≻ 0,

Lw − Lx ≻ 0,

Lw − Lfµ (x) ≽ 0,

(10)

which means that Lw and Lw − Lx are nonsingular. On the other hand, by the definition of ∇ H (z ) and the expression of ∇φ(µ, x), we have 1 T −1 ∇x φ(µ, x) = (I − Lx L− w ) + ∇ fµ (x) (I − Lfµ (x) Lw ).

(11)

Applying Lw to both sides of (11) yields that

∇x φ(µ, x)Lw = (Lw − Lx ) + ∇ fµ (x)T (Lw − Lfµ (x) ).

(12)

Since f is a P0 function, using Propositions 2.2 and 2.3, then for any µ ∈ R++ we yield that fµ (x) is a P function. Thus, we obtain that ∇ fµ (x) is a P matrix for all µ ∈ R++ . Next, we show the nonsingularity of ∇x φ(µ, x) the following cases. (i) If Lw − Lfµ (x) ≻ 0, it then follows from Theorem 3.3 in [29] that the right side of Eq. (12) is nonsingular. So ∇x φ(µ, x) is nonsingular. (ii) If Lw − Lfµ (x) = 0, then the right side of Eq. (12)

(Lw − Lx ) + ∇ fµ (x)T (Lw − Lfµ (x) ) = Lw − Lx ≻ 0, which implies that ∇x φ(µ, x)Lw ≻ 0. From the nonsingularity of Lw , it then follows that ∇x φ(µ, x) is nonsingular. Combining these two cases, we get that ∇x φ(µ, x) is nonsingular. This completes the proof.  Lemma 3.2. If z k = (µk , xk ) ∈ R++ × Rn for any k ≥ 0, then z k+1 can be generated by Algorithm 3.1 and z k+1 ∈ R++ × Rn . Proof. From the notation of H (·) and the definition of β k , it follows that H (z k ) ̸= 0 and β k = β(z k ) > 0 for any µk > 0. In addition, it follows from Eq. (7) that 1µk = −µk + β k µ0 . Then, for any α ∈ [0, 1],

µk+1 = µk + α 1µk = (1 − α)µk + αβ k µ0 > 0,

(13)

this clearly means that µ ∈ R++ . From Step 3 of Algorithm 3.1, if ‖H (z k + 1z k )‖ ≤ ηk , then z k+1 = z k + 1z k can be generated and z k+1 ∈ R++ × Rn . Otherwise, if ‖H (z k + 1z k )‖ ≤ ηk not be accepted. From the definition of β(·) and γ µ0 < 1, it follows that β(z ) ≤ γ ‖H (z )‖, β(z ) ≤ γ < 1 and µk ≤ ‖H (z k )‖. Thus k+1

(µk + α 1µk )2 = ((1 − α)µk + αβ k µ0 )2 = ((1 − α)µk )2 + 2α(1 − α)µk β k µ0 + (αβ k µ0 )2 ≤ ((1 − α)µk )2 + 2α(1 − α)γ ‖H (z k )‖2 µ0 + (αγ ‖H (z k )‖µ0 )2 ≤ (1 − 2α)(µk )2 + 2αγ θ (z k )µ0 + O(α 2 ).

(14)

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737

We define h(α) = G(z k + α 1z k ) − G(z k ) − α(∇ G(z k ))T 1z k , where G(z k ) = ‖φ(z k )‖2 . Note that from the continuity of φ(·), we have h(α) = o(α), and (∇ G(z k ))T 1z k = −2‖φ(z k )‖2 = −2G(z k ) for any k ≥ 0, µk > 0. Then

‖φ(z k + α 1z k )‖2 = G(z k + α 1z k ) = G(z k ) + α(∇ G(z k ))T 1z k + h(α) = G(z k ) − 2α G(z k ) + o(α) = (1 − 2α)G(z k ) + o(α).

(15)

From (14) and (15), we have

θ (z k + α 1z k ) = = ≤ =

‖H (z k + α 1z k )‖2 (µk + α 1µk )2 + ‖φ(z k + α 1z k )‖2 (1 − 2α)(µk )2 + 2αγ µ0 θ (z k ) + (1 − 2α)G(z k ) + o(α) 2αγ µ0 θ (z k ) + (1 − 2α)θ (z k ) + o(α),

which implies that there exists a positive number α¯ ∈ (0, 1], such that for all α ∈ (0, α] ¯ and σ ∈ (0, 1),

θ (z k + α 1µk ) ≤ [1 − 2σ (1 − γ µ0 )α]θ (z k ). Then a nonnegative mk , that satisfied (8) can be found. By Lemma 3.1 and (13), z k+1 can be generated by Algorithm 3.1 and z k+1 ∈ R++ × Rn .  Lemma 3.3. Suppose that the infinite sequence {z k } is generated by Algorithm 3.1. Then the algorithm is well defined and z k ∈ Γ for all k ≥ 0, where

Γ = {z = (µ, x) ∈ R++ × Rn |µ ≥ β(z )µ0 }. Proof. By Lemmas 3.1 and 3.2, we know that Algorithm 3.1 is well defined and generates an infinite sequence {z k } with µk > 0 for all k ≥ 0. Now we prove z k ∈ Γ for all k ≥ 0. Since β(z 0 ) ≤ γ ‖H (z 0 )‖ < 1, it is easy to see that z 0 ∈ Γ . Suppose that z k ∈ Γ , then µk ≥ β k µ0 . We prove that z k+1 ∈ Γ by considering the following two cases: Case (i) If ‖H (z k )‖ ≥ 1, then β k = γ . Since β k+1 = γ min{1, ‖H (z k+1 ‖)} ≤ γ , we have for any α ∈ (0, 1],

µk+1 − β k+1 µ0 = ≥ ≥ =

µk + α 1µk − β(z k + α 1z k )µ0 (1 − α)µk + αβ k µ0 − γ µ0 (1 − α)β k µ0 + αβ k µ0 − γ µ0 0.

(16)

Case (ii) If ‖H (z )‖ < 1, then β = γ ‖H (z )‖. It follows from (8) that ‖H (z k

k

k

k+1

)‖ ≤ ‖H (z )‖ < 1 and β k

k+1

= γ ‖H (z

k+1

)‖.

Then for any α ∈ (0, 1],

µk+1 − β k+1 µ0 = ≥ = =

(1 − α)µk + αβ k µ0 − γ ‖H (z k+1 )‖µ0 (1 − α)β k µ0 + αβ k µ0 − γ ‖H (z k )‖µ0 β k µ0 − γ ‖H (z k )‖µ0 0.

Combining (16) and (17) yields that z ∈ Γ for any k ≥ 0. k

(17)



4. Convergence analysis In this section, a lemma is given in the first, then the global convergence and local convergence of Algorithm 3.1 are analyzed, respectively. Lemma 4.1. Suppose that the infinite sequence {z k = (µk , xk )} is generated by Algorithm 3.1, then 0 < µk+1 ≤ µk for all k ≥ 0. Proof. From Lemma 3.1, it follows that z k ∈ Γ , which implies that µk ≥ β k µ0 . Thus for any α ∈ (0, 1],

µk+1 = µk + α 1µk = (1 − α)µk + αβ k µ0 ≤ (1 − α)µk + αµk = µk . With the addition of (13), we have 0 < µk+1 ≤ µk for all k ≥ 0.



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Theorem 4.1. Suppose that the solution set of SOCCP is nonempty and bounded. Let {z k } be the iteration sequence generated by the algorithm. Then any accumulation point z ∗ = (µ∗ , x∗ ) of the sequence {z k } is a solution of H (z ) = 0. Proof. From Lemma 3.3, we know that an infinite sequence {z k } is generated by the algorithm and {z k } ∈ Γ . Without loss of generality, we assume that the infinite sequence {z k } converges to z ∗ = (µ∗ , x∗ ) as k → ∞. Since {‖H (z k )‖} is monotonically decreasing and bounded from by zero, it follows from the continuity of H (·) that {‖H (z k )‖} converges to a nonnegative number ‖H (z ∗ )‖. Then we obtain that {β k } converges to β ∗ = γ ‖H (z ∗ )‖ by the definition of β(·). On account of (13) and Lemma 4.1, we have that {µk } converges to µ∗ . According to Algorithm 3.1 in Section 3, if ‖H (z k+1 )‖ ≤ ηk , we have limk→∞ ηk = 0 by the definition of ηk . Therefore, from ‖H (·)‖ ≥ 0, it follows that {‖H (z k )‖} converges to ‖H (z ∗ )‖ = 0. Otherwise, if ‖H (z k+1 )‖ ≤ ηk cannot be accepted, we prove ‖H (z ∗ )‖ = 0 as below. If ‖H (z ∗ )‖ = 0, we obtain the desired result. Suppose ‖H (z ∗ )‖ > 0, which implies that θ (z ∗ ) > 0. Since 0 < β ∗ µ0 ≤ µ∗ , from Lemma 3.1, it follows that ∇ H (z ∗ ) is nonsingular. Thus there exists a closed neighborhood N (z ∗ ) of z ∗ such that for any z ∈ N (z ∗ ) we have µ ∈ R++ and ∇ H (z ) is invertible. For any z ∈ N (z ∗ ), let 1z = (1µ, 1x) ∈ R × Rn be the unique solution of the system of equations H (z ) + ∇ H (z )1z = β(z )¯z . Then by following the proof of Lemma 5 in [30], we can find a positive number α¯ ∈ [0, 1] such that

θ (z k + α 1µk ) ≤ [1 − 2σ (1 − γ µ0 )α]θ (z k ) for all α ∈ [0, α] ¯ and z ∈ N (z ∗ ). This implies that for all z k ∈ N (z ∗ ) there exists a nonnegative integer m such that δ m ∈ (0, α] ¯ and

θ (z k + δ m 1µk ) ≤ [1 − 2σ (1 − γ µ0 )δ m ]θ (z k ). Thus, for every sufficiently large k, mk ≤ m, we have δ mk ≥ δ m , and

θ (z k+1 ) ≤ [1 − 2σ (1 − γ µ0 )δ mk ]θ (z k ) ≤ [1 − 2σ (1 − γ µ0 )δ m ]θ (z k ). This contradicts the fact that θ (z ∗ ) > 0. Consequently, the proof is complete.



Next, we analyze the local convergence of the algorithm. In the analysis, we suppose that z ∗ is an accumulation point of the iteration sequence {z k } generated by Algorithm 3.1, and assume that z ∗ satisfies the nonsingularity condition but may not satisfy the strict complementarity. Let β(z ) = γ min{1, ‖H (z )‖1+t } (where t > 0), then we have the following result. Theorem 4.2. Suppose that the solution set of SOCCP (f ) is nonempty and bounded. If all V ∈ ∂ H (z ∗ ) are nonsingular, then the sequence {z k } generated by Algorithm 3.1 is superlinear convergent to its accumulation z ∗ . Proof. It follows from the nonsingularity of V ∈ ∂ H (z ∗ ) that there exists C > 0 such that ‖∇ H (z k )−1 ‖ ≤ C for all k ≥ 0. Since H (·) is global Lipschitz continuous and strongly semismooth, by using Definition 2.3, for all z k sufficiently close to z ∗ yields that

‖H (z k )‖ = ‖H (z k ) − H (z ∗ )‖ = O(‖z k − z ∗ ‖), ‖H (z k ) − H (z ∗ ) − ∇ H (z k )(z k − z ∗ )‖ = O(‖z k − z ∗ ‖2 ), and β k µ0 ≤ γ µ0 ‖H (z k )‖1+t = O(‖z k − z ∗ ‖1+t ). Then for all z k sufficiently close to z ∗ ,

‖z k + 1 z k − z ∗ ‖ = = ≤ =

‖z k + ∇ H (z k )−1 (−H (z k ) + β k µ0 ) − z ∗ ‖ ‖∇ H (z k )−1 ‖ ‖H (z k ) − H (z ∗ ) − ∇ H (z k )(z k − z ∗ ) + β k µ0 ‖ C ‖H (z k ) − H (z ∗ ) − ∇ H (z k )(z k − z ∗ )‖ + O(‖z k − z ∗ ‖1+t ) O(‖H (z k ) − H (z ∗ ) − ∇ H (z k )(z k − z ∗ )‖) + O(‖z k − z ∗ ‖1+t )

= O(‖z k − z ∗ ‖1+t ).

(18) ∗

k

Consequently, for all z sufficiently close to z ,

‖H (z k+1 )‖ = ‖H (z k + 1z k )‖ = O(‖z k + 1z k − z ∗ ‖) = O(‖z k − z ∗ ‖1+t ) = O(‖H (z k ) − H (z ∗ )‖1+t ) = O(‖H (z k )‖1+t ). ∗

k

For all z sufficiently close to z , we have z

‖z

k+1

− z ‖ = O(‖z − z ‖ ∗

k

∗ 1 +t

).

k+1

(19)

= z + 1z . From (18), it follows that k

k

X. Zhang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 731–740

739

Table 1 Numerical results of Algorithm 3.1 for SOCCP of various problem sizes (n). n

Iter

CPU (s)

Val

10 20 50 100 200 400

9 8 8 11 14 12

0.156 0.174 0.204 0.581 0.985 2.746

8.69e−6 3.97e−5 4.68e−5 7.27e−5 5.12e−6 1.32e−5

Thus, for all sufficiently large k, µk+1 = µk + 1µk = β k µ0 = γ µ0 ‖H (z k )‖1+t , which together with (19) obtain that

µk+1 ‖H (z k )‖1+t O((‖H (z k−1 )‖1+t )1+t ) = = . µk ‖H (z k−1 )‖1+t (‖H (z k−1 )‖1+t )1+t Therefore, for all z k sufficiently close to z ∗ , this implies that µk+1 = O((µk )1+t ).



5. Numerical results In this section, we implement some numerical experiments to confirm the effectiveness of Algorithm 3.1. All experiments were performed on a desktop computer with 1.86 GHz CPU and 1 GB memory. The operating system was Windows XP and the implementations were done in MATLAB 7.0.1. We consider the following second-order cone complementarity problems. Example 1. Linear case Find a vector x ∈ Rn such that

⟨f (x), x⟩ = 0,

x ∈ K,

n×n

f (x) = Mx + q ∈ K ,

n

where M ∈ R and q ∈ R . The matrix M was generated by setting M = N T N, where N is a square matrix. The elements of q are chosen randomly from the interval [−1, 1]. In the experiments, the initial points of the algorithm were chosen randomly from the interval [−1, 1]. The parameters used in our algorithm were as follows: σ = 0.25, η = 0.2, η0 = 0.01, δ = 0.7, µ0 = 0.001, and γ = 0.8 min{1, 1/‖H (z 0 )‖}. The stopping criterion was set as ‖H (z )‖ ≤ 10−5 . Numerical results are summarized in Table 1. Val represents the value of |f (x)T x| at final iteration in the trial, Iter denotes the number of iterations for the convergence of each trial, and the CPU time is in seconds. In this experiment, we implement on the problems with different size n. From the results in Table 1, we may observe that the algorithm is promising. Example 2. Nonlinear case Find a vector x ∈ Rn such that

⟨x, f (x)⟩ = 0, 3

x ∈ K,

2

where K = K × K and f : R5 → R5 is given by 24(2x1 − x2 )3 + exp( x1 − x3 ) − 4x4 + x5





−12(2x − x )3 + 3(3x + 5x )/ 1 + (3x + 5x )2 − 6x − 7x  1 2 2 3  2 3 4 5   f ( x) =   − exp(x1 − x3 ) + 5(3x2 + 5x3 )/ 1 + (3x2 + 5x3 )2 − 3x4 + 5x5  .   4x1 + 6x2 + 3x3 − 1 −x1 + 7x2 − 5x3 + 2 In view of the KKT conditions for SOCP, SOCCP is equivalent to the following SOCP (see [24]): Minimize

exp(z1 − z3 ) + 3(2z1 − z2 )4 + z1 z2 z3

  subject to

∈ K 3,



4

−1

6 7



1 + 3(z2 + 5z3 )2  z1    3 −1 z2 + ∈ K 2. −5 2 z3

Since the objective function of this SOCP is convex, we can easily see that the function f is monotone. We test this problem 10 times. In detail, we conduct the experiments with the initial points x0 = 0.1e and x0 = 1.0e, respectively, where e is the unit element in K . Set the parameters in the experiments as: σ = 0.2, δ = 0.5, η = 0.1, η0 = 0.01, µ0 = 0.001 and γ = τ min{1, 1/‖H (z 0 )‖}, where τ ∈ (0, 1). The stopping criterion is as follows: ‖H (z )‖ ≤ 10−5

740

X. Zhang et al. / Nonlinear Analysis: Real World Applications 12 (2011) 731–740

Table 2 Numerical results of Algorithm 3.1 for nonlinear SOCCP. x0

τ

Iter

Val

x0

τ

Iter

Val

0.1e 0.1e 0.1e 0.1e

0.1 0.2 0.5 0.8

6 7 6 6

7.31e−3 1.48e−2 5.07e−2 1.42e−2

1.0e 1.0e 1.0e 1.0e

0.1 0.2 0.5 0.8

7 7 7 7

1.29e−2 1.36e−2 1.52e−2 1.64e−2

or the steplength is lower than 10−12 . Numerical results are summarized in Table 2, where Val represents the value of |f (x)T x| at the final iteration and Iter denotes the number of iterations for each trial. From the results in Table 2, we observe that the Val is smaller when τ is away from 0.5 when x0 = 0.1e, and the Val decreases with the parameter τ when x0 = 1.0e. But the change of the number of iterations with the initial points of the algorithm and the parameter τ is slight. It would be recommended to choose a value of τ other than 0.5 and suitably small initial points. 6. Conclusions In this paper, we discussed the second-order cone complementarity problems in detail. By using a new regularization method, which was shown to be beneficial for smoothing the complementarity function, we proposed a regularization smoothing Newton method for solving the second-order cone complementarity problem. The proposed algorithm was proved to be well defined, and was shown to be globally convergent and locally superlinear convergent under proper conditions. Especially, the local superlinear convergence was obtained without strict complementarity. The numerical results show that the algorithm performs well. References [1] F. Alizadeh, D. Goldfarb, Second-order cone programming, Math. Program. 95 (2003) 3–51. [2] M. Fukushima, Z.Q. Luo, P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM J. Optim. 12 (2001) 436–460. [3] S. Hayashi, N. Yamashita, M. Fukushima, Robust Nash equilibria and second order cone complementarity problems, J. Nonlinear Convex Anal. 6 (2005) 283–296. [4] Y. Kanno, J.A.C. Martins, A. Pintoda Costa, Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem, in: BGE Research Report, Department of Urban and Environmental Engineering, Kyoto University, February, 2004. [5] X.D. Chen, D. Sun, J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order cone complementarity problems, Comput. Optim. Appl. 25 (2003) 39–56. [6] J. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293–327. [7] S. Pan, J.-S. Chen, A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions, Comput. Optim. Appl. (2008) doi:10.1007/s10589-008-9166-9. [8] A.L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems, in: Lecture Notes in Mathematics, vol. 1543, Springer-Verlag, Berlin, 1993. [9] D. Sun, A regularization Newton method for solving nonlinear complementarity problems, Appl. Math. Optim. 40 (1999) 315–339. [10] R.T. Rockafellar, Monotone operators and the proximal point algorithms, SIAM J. Control Optim. 14 (1976) 877–898. [11] R.T. Rockafellar, Monotone operators and augmented Lagrangian methods in nonlinear programming, in: O.L. Mangasarian, R.R. Meyer, S.M. Robinson (Eds.), in: Nonlinear Programming, vol. 3, Academic Press, New York, 1978, pp. 1–25. [12] J. Eckstein, M.C. Ferris, Smooth methods of multipliers for complementarity problems, in: RUCTOR Research Report 27–96, Faculty of Management and RUTCOR, Rutgers Univesity, New Brunswick, NJ08903, February, 1997. [13] X.H. Chen, C.F. Ma, A regularization smoothing Newton method for solving nonlinear complementarity problem, Nonlinear Anal. RWA (2008) doi:10.1016/j.nonrwa.2008.02.010. [14] J. Faraut, A. Korányi, Analysis on Symmetric Cones, Oxford University Press, Oxford, UK, 1994. [15] M.S. Gowda, R. Sznajder, J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra Appl. 393 (2004) 203–232. [16] F. Facchinei, F. Pang, Finite Dimensional Variational Ineqaulities and Complementarity Problems, Springer-Verlag, New York, 2003. [17] X. Chen, H. Qi, Cartesian P-property and its applications to the semidefinite linear complementarity problem, Math. Program. 106 (2006) 177–201. [18] L.C. Kong, L. Tuncel, N.H. Xiu, Vector-valued implicit Lagrangian for symmetric cone complementarity problems, CORR (2006) 2006-24. [19] J.Y. Tao, M.S. Gowda, Some P-properties for nonlinear transformations on Euclidean Jordan algebras, Math. Oper. Res. 30 (2005) 985–1004. [20] S. Pan, J.-S. Chen, A regularization method for the second-order cone complementarity problem with the Cartesian P0 -property, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.02.028. [21] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [22] R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim. 15 (1977) 957–972. [23] L. Qi, J. Sun, A nonsmooth version of Newton’s method, Math. Program. 58 (1993) 353–367. [24] S. Hayashi, N. Yamashita, M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM J. Control Optim. (2005) 593–615. [25] A. Bakushinsky, A. Goncharsky, Ill-Posed Problem: Theory and Appliations, in: Mathematics and Its Applications, vol. c301, Kluwer Academic Publishers, Dordrecht, 1994. [26] F. Facchinei, C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control Optim. 37 (1999) 1150–1161. [27] H.D. Qi, A regularized smoothing Newton method for box constrained variational inequality problems with P0 -functions, SIAM J. Optim. 10 (1999) 315–330. [28] D. Sun, J. Sun, Strong semismoothness of Fischer–Burmeister SDC and SOC complementarity functions, Math. Program. 103 (2005) 575–581. [29] G. Ravindran, M.S. Gowda, Regularization of P0 -functions in box variational inequality problems, SIAM J. Optim. 11 (2001) 748–760. [30] L. Qi, D. Sun, G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program. 87 (2000) 1–35.