A Smoothing Inexact Newton Method for Generalized Nonlinear Complementarity Problem

Abstract

Based on the smoothing function of penalized Fischer-Burmeister NCP-function, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving the generalized nonlinear complementarity problem. We view the smoothing parameter as an independent variable. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the generalized nonlinear complementarity problem. We also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Preliminary numerical experiments indicate the feasibility and efficiency of the proposed algorithm.

1. Introduction

Consider the generalized nonlinear complementarity problem, denoted by GNCP (𝐹,𝐺,𝒦), which is to find a vector 𝑥∈𝑅^𝑛 such that𝐹(𝑥)∈𝒦,𝐺(𝑥)∈𝒦^∘,𝐹(𝑥)^⊤𝐺(𝑥)=0, (1.1) where 𝐹,𝐺∶𝑅^𝑛→𝑅^𝑚 are continuously differentiable mappings. 𝒦 is a nonempty closed convex cone in 𝑅^𝑚 and 𝒦^∘ denotes its polar cone.

GNCP (𝐹,𝐺,𝒦) finds important applications in many fields, such as engineering and economics, and is a wide class of problems that contains the classical nonlinear complementarity problem (abbreviated as NCP); see [1–3] and references therein. To solve it, one usually reformulates it as a minimization problem over a simple set or an unconstrained optimization problem; see [4] for the case that 𝐾 is a general cone, and see [3, 5] for the case that 𝒦=𝑅⁺_𝑛. The conditions under which a stationary point of the reformulated optimization is a solution of the GNCP (𝐹,𝐺,𝒦) were also provided in the literature.

In this paper, we consider the GNCP (𝐹,𝐺,𝒦) for the case that 𝑚=𝑛, and 𝒦 is a polyhedral cone in 𝑅^𝑛; that is, there exist 𝐴∈𝑅^𝑠×𝑛 and 𝐵∈𝑅^𝑡×𝑛 such that𝒦=𝑣∈𝑅^𝑛∣𝐴𝑣≥0,𝐵𝑣=0. (1.2) It is easy to verify that its polar cone 𝒦^∘ has the following representation:𝒦^∘=𝑢∈𝑅^𝑛∣𝑢=𝐴^⊤𝜆+𝐵^⊤𝜇,𝜆∈𝑅^𝑠,𝜆≥0,𝜇∈𝑅^𝑡. (1.3)

In the following, the GNCP (𝐹,𝐺,𝒦) is specialized over a polyhedral cone, and in the subsequent analysis, we abbreviate it as GNCP for simplicity. In [1], Andreani et al. reformulated the problem as a smooth optimization problem with simple constraints and presented the sufficient conditions under which a stationary point of the optimization problem is a solution of the concerned problem. Wang et al. [6] reformulated the problem as a system of nonlinear and nonsmooth equations and proposed a nonsmooth L-M method to solve this problem and proved that the algorithm is both globally and superlinearly convergent under mild assumptions. Zhang et al. [7] rearranged the GNCP over a polyhedral as a smoothing system of equations, then developed a smoothing Newton-type method for solving it, and proved that their method has local superlinear (quadratic) convergence under certain conditions. There are a lot of computations to decide whether the linear system is solvable or not, which is in Step 2 of the algorithm presented in [7]. The inexact approach is one way to overcome this difficulty. Inexact Newton methods have been proposed for solving NCP [8–10]. Their main idea is to solve the linear system only approximately. It seems reasonable to ask if this kind of method can be applied to the GNCP, and this actually constitutes the main motivation of this paper. In this paper, we propose a new smoothing inexact Newton algorithm with non-monotone line search for solving GNCP by using a new type of smoothing function. We view the smoothing parameter as an independent variable. The forcing parameter of inexact Newton method links the norm of residual vector to the norm of mapping at the current iterate. Under suitable conditions, we show that any accumulation point of the generated sequence is a solution of the GNCP, and we also establish the local superlinear (quadratic) convergence of the proposed algorithm under the BD-regular assumption. Some numerical examples indicate the feasibility and efficiency of the proposed algorithm.

The rest of this paper is organized as follows. In Section 2, we show some preliminaries. Stationary point and nonsingularity conditions of the GNCP are discussed in Section 3. In Section 4, we propose a kind of algorithm for solving GNCP and obtain the convergence properties of this kind of algorithm. Numerical experiments are exhibited in Section 5 and the conclusion is stated in the last section.

At the end of this section, we indicate some standard notions used in this paper. For a continuously differentiable function 𝐹∶𝑅^𝑛→𝑅^𝑚, we denote the Jacobian of 𝐹 at 𝑥∈𝑅^𝑛 by 𝐹^′(𝑥)∈𝑅^𝑚×𝑛, whereas the transposed Jacobian is denoted by ∇𝐹(𝑥). In particular, if 𝑚=1, ∇𝐹(𝑥) is a column vector. We use 𝑥^⊤𝑦 to denote the inner product of vectors 𝑥,𝑦∈𝑅^𝑛, and use [𝑥]_𝑖 or 𝑥_𝑖 to denote the 𝑖th component of the vector 𝑥∈𝑅^𝑛. The null space of a matrix 𝐵 is denoted by 𝒩(𝐵).

2. Preliminaries

In this section, we first review some definitions and basic results.

Definition 2.1. A matrix 𝑀∈𝑅^𝑛×𝑛 is said to be a 𝑃₀-matrix if every principle minor of 𝑀 is nonnegative.

Definition 2.2. A function 𝐹∶𝑅^𝑛→𝑅^𝑛 is said to be a 𝑃₀-function if for all 𝑥,𝑦∈𝑅^𝑛 with 𝑥≠𝑦, there exists an index 𝑖₀∈{1,2,…,𝑛} such that 𝑥_𝑖0≠𝑦_𝑖0,𝑥_𝑖0−𝑦_𝑖0𝐹_𝑖0(𝑥)−𝐹_𝑖0𝑦≥0. (2.1)
For vector 𝑎∈𝑅^𝑛, we use 𝐷_𝑎 for diag(𝑎). For a 𝑃₀-matrix, the following conclusion holds [11].

Lemma 2.3. If 𝑀∈𝑅^𝑛×𝑛 is a 𝑃₀-matrix, then every matrix of the form 𝐷_𝑎+𝐷_𝑏𝑀 (2.2) is nonsingular for all positive definite diagonal matrices 𝐷_𝑎,𝐷_𝑏∈𝑅^𝑛×𝑛.

Definition 2.4. Suppose that 𝐺∶𝑅^𝑛→𝑅^𝑚 is locally Lipschitz function. 𝐺 is said to be semismooth at 𝑥 if 𝐺 is directionally differentiable at 𝑥 and lim_{𝑉∈𝜕𝐺𝑥+𝑡ℎ′,ℎ′→ℎ,𝑡→0+}𝑉ℎ^′ (2.3) exist for any ℎ∈𝑅^𝑛, where 𝜕𝐺(𝑥) denotes the generalized derivative in [12].

Definition 2.5. Suppose that 𝐺∶𝑅^𝑛→𝑅^𝑚 is locally Lipschitz function. 𝐺 is said to be strongly semismooth at 𝑥∈𝑅^𝑛 if 𝐺(𝑥) is semismooth at 𝑥 and for any 𝑉∈𝜕𝐺(𝑥+ℎ) and ℎ→0, it holds that 𝐺(𝑥+ℎ)−𝐺(𝑥)−𝑉ℎ=O‖ℎ‖². (2.4)
The concept of semismoothness was originally introduced by Mifflin for functionals [13]. Qi and Sun extended the definition of semismooth function to vector-valued functions [14]. Convex functions, smooth functions, and piecewise linear functions are examples of semismooth functions. A function is semismooth at 𝑥 if and only if all its component functions are semismooth. The composition of semismooth functions is still a semismooth function.

Lemma 2.6 (see [14]). Suppose that 𝜑∶𝑅^𝑛→𝑅^𝑚 is a locally Lipschitz function and semismooth at 𝑥. Then(a)for any 𝑉∈𝜕𝜑(𝑥+ℎ),ℎ→0,𝑉ℎ−𝜑^′(𝑥;ℎ)=𝑜(‖ℎ‖), (2.5)(b) for any ℎ→0,𝜑(𝑥+ℎ)−𝜑(𝑥)−𝜑^′(𝑥;ℎ)=𝑜(‖ℎ‖). (2.6)

In [6], Wang et al. reformulated the GNCP as a system of nonlinear equations based on the following Fischer function [15] 𝜙(𝑎,𝑏)=√𝑎²+𝑏²−𝑎−𝑏 for 𝑎,𝑏∈𝑅. In [7], Zhang et al. established a type of smoothing reformulation of GNCP based on the following smoothing approximation function to the Fischer function 𝜙(𝜖,𝑎,𝑏)=√𝑎²+𝑏²+𝛼𝜖²−𝑎−𝑏 for 𝑎,𝑏,𝜖∈𝑅, where 𝛼>0 is a constant.

In this paper, we use the following smoothing approximation function:𝜙_𝛼(𝜖,𝑎,𝑏)=𝑎+𝑏−√𝑎²+𝑏²+2𝜖²+𝛼4𝑎+√𝑎²+4𝜖²𝑏+√𝑏²+4𝜖² (2.7) of the penalized Fischer-Burmeister NCP-function:𝜙_𝛼(𝑎,𝑏)=𝑎+𝑏−√𝑎²+𝑏²+𝛼𝑎₊𝑏₊, (2.8) where 𝛼>0,𝑎,𝑏,𝜖∈𝑅, and 𝑡₊=max{0,𝑡}. This NCP-function has turned out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously (see [16]). The latter term penalizes violations of the complementarity condition and plays a significant role from both a theoretical and a practical point of view.

It is easy to see that the following Lemma is true from [6].

Lemma 2.7. 𝑥∈𝑅^𝑛 is a solution of the GNCP if and only if there exist 𝜆∈𝑅^𝑠 and 𝜇∈𝑅^𝑡, such that Φ_𝛼((𝑥))(𝑥)𝐺(𝑥)𝐴𝐹,𝜆=0,𝐵𝐹=0,−𝐴^⊤𝜆−𝐵^⊤𝜇=0, (2.9) where Φ_𝛼(𝑎,𝑏)=(𝜙_𝛼(𝑎₁,𝑏₁),𝜙_𝛼(𝑎₂,𝑏₂),…,𝜙_𝛼(𝑎_𝑠,𝑏_𝑠))^⊤ for 𝑎,𝑏∈𝑅^𝑠.

Based on the relation between 𝜙_𝛼(⋅,⋅) and 𝜙_𝛼(⋅,⋅,⋅), we can establish the following smoothing function to the GNCP.

DenoteΦ_𝛼((𝑥))=𝜙𝜖,𝐴𝐹,𝜆^𝛼𝜖,[𝐴𝐹(𝑥)]₁,[𝜆]₁𝜙_𝛼𝜖,[𝐴𝐹(𝑥)]₂,[𝜆]₂⋮𝜙_𝛼𝜖,[𝐴𝐹(𝑥)]_𝑠,[𝜆]_𝑠,𝜓_𝛼=Φ𝜖,𝑥,𝜆,𝜇^𝛼((𝑥))(𝑥)𝐺(𝑥)𝜖,𝐴𝐹,𝜆𝐵𝐹−𝐴^⊤𝜆−𝐵^⊤𝜇,𝐻_𝛼=𝜖𝜓𝜖,𝑥,𝜆,𝜇_𝛼.𝜖,𝑥,𝜆,𝜇 (2.10)

For simplicity, we let 𝑦=(𝑥,𝜆,𝜇) and 𝑧=(𝜖,𝑦) and denote𝑓_𝛼(𝑧)=12𝜓_𝛼(𝑧)²,𝑇_𝛼(𝑧)=12𝐻_𝛼(𝑧)²=12𝜖²+𝑓_𝛼(𝑧). (2.11)

From Lemma 2.7, we can see that 𝑥^∗ is a solution of the GNCP if and only if there exist 𝜆^∗∈𝑅^𝑠,𝜇^∗∈𝑅^𝑡 such that 𝑧^∗=(0,𝑥^∗,𝜆^∗,𝜇^∗) is a global minimizer with zero objective function value of the unconstrained optimization problem:min_{𝑧∈𝑅1+𝑛+𝑠+𝑡}𝑇_𝛼(𝑧). (2.12)

By simple calculation, we can see that the following lemma is true.

Lemma 2.8. (1) The function Φ_𝛼(𝜖,𝐴𝐹(𝑥),𝜆) is continuously differentiable for 𝜖>0 and it holds that 𝜕Φ_𝛼(𝜖,𝐴𝐹(𝑥),𝜆)⊆𝚌,𝐷₁(𝑥)+𝐷₃(𝑥)𝐴𝐹^′(𝑥),𝐷₂(𝑥)+𝐷₄(𝑥), (2.13)
where 𝚌=vec𝑐_𝑖{},𝑐∶𝑖∈𝑁,𝑁=1,2,…,𝑛_𝑖=−2𝜖[𝐴𝐹(𝑥)]²_𝑖+[𝜆]²_𝑖+2𝜖²+𝛼4[𝐴𝐹(𝑥)]_𝑖+[𝐴𝐹(𝑥)]²_𝑖+4𝜖²4𝜖[𝜆]²_𝑖+4𝜖²+[𝜆]_𝑖+[𝜆]²_𝑖+4𝜖²4𝜖[𝐴𝐹(𝑥)]²_𝑖+4𝜖²,𝐷₁(𝑥)[(𝑥=diag1−𝐴𝐹)]_𝑖[𝐴𝐹(𝑥)]²_𝑖+[𝜆]²_𝑖+2𝜖²,𝐷∶𝑖∈𝑁₂(𝑥)[𝜆]=diag1−^𝑖[𝐴𝐹(𝑥)]²_𝑖+[𝜆]²_𝑖+2𝜖²,𝐷∶𝑖∈𝑁₃(𝑥)𝛼=diag4[(𝑥1+𝐴𝐹)]_𝑖[𝐴𝐹(𝑥)]²_𝑖+4𝜖²[𝜆]_𝑖+[𝜆]²_𝑖+4𝜖²,𝐷∶𝑖∈𝑁₄(𝑥)𝛼=diag4[𝜆]1+^𝑖[𝜆]²_𝑖+4𝜖²[𝐴𝐹(𝑥)]_𝑖+[𝐴𝐹(𝑥)]²_𝑖+4𝜖²∶𝑖∈𝑁. (2.14)
(2) 𝐻_𝛼(𝑧) is semismooth on 𝑅^{1+𝑛+𝑠+𝑡} and is strongly semismooth on 𝑅^{1+𝑛+𝑠+𝑡} if 𝐹^′(𝑥) and 𝐺^′(𝑥) are both Lipschitz continuous on 𝑅^𝑛.
(3) 𝑇_𝛼(𝑧) is continuously differentiable on 𝑅^{1+𝑛+𝑠+𝑡} with ∇𝑇_𝛼(𝑧)=𝑉^⊤𝐻_𝛼(𝑧) for any 𝑉∈𝜕𝐻_𝛼(𝑧) and 𝑓_𝛼(𝜖,𝑦) is continuously differentiable with ∇𝑓_𝛼(0,𝑦)=𝑉^⊤𝜓_𝛼(0,y) for any 𝑉∈𝜕𝜓_𝛼(0,𝑦).

3. Stationary Point and Nonsingularity Conditions

Generally, for an optimization problem, we can obtain its stationary point when we use the existing optimization methods to solve it. Now we should study how to guarantee that every stationary point of (2.12) is a solution of the GNCP. In the following, we will discuss the conditions.

Theorem 3.1. Let 𝑧=(𝜖,𝑥,𝜆,𝜇) be a stationary point of (2.12). If 𝜖≥0 and ∇𝐹(𝑥)⁻¹∇𝐺(𝑥) is positive definite in 𝒩(𝐵), then 𝑥 is a solution of the GNCP.

Proof. Define 𝑈=Φ_𝛼((𝑥)),(𝑥),(𝑥)𝜖,𝐴𝐹,𝜆𝑉=𝐵𝐹𝑊=𝐺−𝐴^⊤𝜆−𝐵^⊤𝜇. (3.1) Since 𝑧=(𝜖,𝑥,𝜆,𝜇) is a stationary point of (2.12), one has ∇𝑇_𝛼(𝑧)=0. (3.2) From Lemma 2.8, we have 1𝚌^⊤(𝑥)𝐴000∇𝐹^⊤𝐷₁(𝑥)+𝐷₃(𝑥)∇𝐹(𝑥)𝐵^⊤(𝑥)0𝐷∇𝐺₂(𝑥)+𝐷₄(𝑥𝜖𝑈𝑉𝑊)0−𝐴000−𝐵=0, (3.3) that is, 𝜖+𝚌^⊤(𝑥)𝐴𝑈=0,∇𝐹^⊤𝐷₁(𝑥)+𝐷₃(𝑥)𝑈+∇𝐹(𝑥)𝐵^⊤(𝑥)𝐷𝑉+∇𝐺𝑊=0,₂(𝑥)+𝐷₄(𝑥)𝑈−𝐴𝑊=0,−𝐵𝑊=0. (3.4)
From the forth equation of (3.4), we have 𝑊∈𝒩(𝐵). Premultiplying the second equation of (3.4) by 𝑊^⊤∇𝐹(𝑥)⁻¹, one has 𝑊^⊤𝐴^⊤𝐷₁(𝑥)+𝐷₃(𝑥)𝑈+𝑊^⊤𝐵^⊤𝑉+𝑊^⊤∇𝐹(𝑥)⁻¹∇𝐺(𝑥)𝑊=0. (3.5)
Combining (3.5) with the third and forth equations of (3.4), we obtain 𝑈^⊤𝐷₂(𝑥)+𝐷₄(𝑥)𝐷₁(𝑥)+𝐷₃(𝑥)𝑈+𝑊^⊤∇𝐹(𝑥)⁻¹∇𝐺(𝑥)𝑊=0. (3.6)
From Lemma 2.8, we have that [𝐷₂(𝑥)+𝐷₄(𝑥)][𝐷₁(𝑥)+𝐷₃(𝑥)] is positive definite. Since 𝑊∈𝒩(𝐵) from the forth equation of (3.4), together with the positive definiteness of ∇𝐹(𝑥)⁻¹∇𝐺(𝑥) in 𝒩(𝐵), we have 𝑈=0,𝑊=0. (3.7)
Subtituting 𝑈=0 into the first equation of (3.4), we have 𝜖=0. (3.8)
Together the second equation of (3.4) with (3.8) and (3.7), we have ∇𝐹(𝑥)𝐵^⊤𝑉=0. (3.9)
Since ∇𝑓(𝑥) is nonsingular, premultiplying (3.9) by 𝐹(𝑥)^⊤∇𝐹(𝑥)⁻¹, one has ‖𝑉‖²=0. (3.10) Hence, 𝑉=0. We complete the proof.

Theorem 3.2. Let 𝑧=(𝜖,𝑥,𝜆,𝜇) be a stationary point of (2.12), 𝜖>0,𝐵 has full row rank, and ∇𝐹(𝑥)∇𝐺(𝑥)^⊤ is positive definite, then V is nonsingular for any 𝑉∈𝜕𝐻_𝛼(𝑧).

Proof. By Lemma 2.8, we know that any element 𝑉∈𝜕𝐻_𝛼(𝑧) can be written as 𝑉^⊤=1𝚌^⊤(𝑥)𝐴000∇𝐹^⊤𝐷₁(𝑥)+𝐷₃(𝑥)∇𝐹(𝑥)𝐵^⊤(𝑥)0𝐷∇𝐺₂(𝑥)+𝐷₄(𝑥)0−𝐴000−𝐵, (3.11) where 𝐷₁(𝑥),𝐷₂(𝑥),𝐷₃(𝑥), and 𝐷₄(𝑥) are defined in Lemma 2.8.
In order to complete the proof, we only need to prove that (𝑥)𝐴𝐷=∇𝐹^⊤𝐷₁(𝑥)+𝐷₃(𝑥)∇𝐹(𝑥)𝐵^⊤(𝑥)𝐷∇𝐺₂(𝑥)+𝐷₄(𝑥)0−𝐴00−𝐵 (3.12) is nonsingular.
That is to say, if and only if 𝐷^⊤=𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝐷₂(𝑥)+𝐷₄(𝑥)0𝐵∇𝐹(𝑥)^⊤(𝑥)00∇𝐺^⊤−𝐴^⊤−𝐵^⊤ (3.13) is nonsingular.
Suppose that 𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝐷₂(𝑥)+𝐷₄(𝑥)0𝐵∇𝐹(𝑥)^⊤(𝑥)00∇𝐺^⊤−𝐴^⊤−𝐵^⊤𝑝𝑞𝑟=0, (3.14) which means that 𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝑝+𝐷₂(𝑥)+𝐷₄(𝑥(𝑥))𝑞=0,𝐵∇𝐹^⊤(𝑥)𝑝=0,∇𝐺^⊤𝑝−𝐴^⊤𝑞−𝐵^⊤𝑟=0. (3.15)
According to the first equation of (3.15), we have 𝑞=−𝐷₂(𝑥)+𝐷₄(𝑥)⁻¹𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝑝. (3.16) Combining (3.16) with the third equation of (3.15), one has ∇𝐺(𝑥)^⊤+𝐴^⊤𝐷₂(𝑥)+𝐷₄(𝑥)⁻¹𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝑝−𝑏^⊤𝑟=0. (3.17) Premultiplying by 𝑝^⊤∇𝐹(𝑥) and recalling the second equation of (3.15), we get 𝑝^⊤∇𝐹(𝑥)∇𝐺(𝑥)^⊤+𝐴∇𝐹(𝑥)^⊤⊤𝐷₂(𝑥)+𝐷₄(𝑥)⁻¹𝐷₁(𝑥)+𝐷₃(𝑥)𝐴∇𝐹(𝑥)^⊤𝑝=0. (3.18) By using the assumption that ∇𝐹(𝑥)∇𝐺(𝑥)^⊤ is positive, we obtain 𝑝=0, (3.19) which, combining with (3.16), gives 𝑞=0. (3.20) By using the assumption that 𝐵 has full row rank and the third equation of (3.15), we obtain 𝑟=0, (3.21) which completes the proof.

4. Algorithm and Convergence Property

In this section, we formally present our smoothing inexact Newton-type algorithm with nonmontone line search for solving 𝐻_𝛼(𝑧)=0 by using the smoothing penalized FB function 𝜙_𝛼(𝜖,𝑎,𝑏). This nonmontone line search method was used to solve the NCP problem in [17]. Furthermore, we show the local superlinear (quadratic) convergence properties of the algorithm.

Algorithm 4.1. Step 1. Take constants 𝛿,𝛾∈(0,1),𝛼>0, and 𝜖⁰>0 such that 𝛾𝜖⁰<1. Choose 𝜎∈(0,1/2).
Let 𝜖=(𝜖⁰,0)∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡} and 𝑦⁰=(𝑥⁰,𝜆⁰,𝜇⁰)∈𝑅^{𝑛+𝑠+𝑡} be an arbitary point.
Let 𝑧⁰=(𝜖⁰,𝑦⁰),𝐶₀=𝑇_𝛼(𝑧⁰), and 𝑄₀=1. Let 𝜂_min and 𝜂_max be two constants such that 0≤𝜂_min<𝜂_max<1. Choose 𝜂₀∈[𝜂_min,𝜂_max]. Set 𝜌(𝑧⁰)=𝛾min{1,𝑇_𝛼(𝑧⁰)} and 𝑘∶=0.Step 2. If 𝐻_𝛼(𝑧^𝑘)=0, then stop; otherwise, let 𝜌_𝑘=𝜌(𝑧^𝑘).Step 3. Compute Δ𝑧^𝑘=(Δ𝜖^𝑘,Δ𝑥^𝑘,Δ𝜆^𝑘,Δ𝜇^𝑘)∈𝑅^{1+𝑛+𝑠+𝑡} by 𝐻_𝛼𝑧^𝑘+𝐻^′_𝛼𝑧^𝑘Δ𝑧^𝑘=𝜌_𝑘𝜖. (4.1)Step 4. Let 𝜉_𝑘=𝛿^𝑚, where 𝑚 is the smallest nonnegative integer such that 𝑇_𝛼𝑧^𝑘+𝛿^𝑚Δ𝑧^𝑘≤1−2𝜎1−𝛾𝜖₀𝛿^𝑚𝐶_𝑘. (4.2)Step 5. Set z^𝑘+1=𝑧^𝑘+𝜉_𝑘Δ𝑧^𝑘, 𝑘=𝑘+1, and 𝜌𝑧^𝑘=min𝛾,𝛾𝑇_𝛼𝑧^𝑘,𝜌𝑧^𝑘−1. (4.3)Step 6. Choose 𝜂_𝑘∈[𝜂_min,𝜂_max]; set 𝑄_𝑘=𝜂_𝑘−1𝑄_𝑘−1+1,𝐶_𝑘=𝜂_𝑘−1𝑄_𝑘−1𝐶_𝑘−1+𝑇_𝛼𝑧^𝑘𝑄_𝑘. (4.4)
Go to Step 2.
From Algorithm 4.1 and [17], it is easy to see that the following remark is true.

Remark 4.2. Let the sequence {𝐶_𝑘} and {𝑧^𝑘=(𝜖^𝑘,𝑥^𝑘,𝜆^𝑘,𝜇^𝑘)} be generated by Algorithm 4.1.(i)𝐶_𝑘+1≤𝐶_𝑘 for any 𝑘.(ii)𝑇_𝛼(𝑧^𝑘)≤𝐶_𝑘 for any 𝑘.(iii)𝜌(𝑧^𝑘+1)≤𝜌(𝑧^𝑘) for any 𝑘.(iv)𝜖⁰𝜌(𝑧^𝑘)≤𝜖^𝑘 for any 𝑘.(v)𝜖^𝑘>0 and 𝜖^𝑘+1≤𝜖^𝑘 for any 𝑘.

Theorem 4.3. Suppose that {𝑧^𝑘=(𝜖^𝑘,𝑥^𝑘,𝜆^𝑘,𝜇^𝑘)} is a sequence generated by Algorithm 4.1, 𝐵 has full row rank, and ∇𝐹(𝑥^𝑘)∇𝐺(𝑥^𝑘)^⊤ is positive definite, 𝜖^𝑘>0. Then Algorithm 4.1 is well defined.

Proof. From Theorem 3.2, we know that 𝐻^′_𝛼(𝑧^𝑘) is nonsingular. Hence, Step 3 is well defined at the kth iteration.
In the following, we show that the Step 4 is well defined. For 𝜉∈(0,1), we let 𝑅^𝑘(𝜉)=𝑇_𝛼𝑧^𝑘+𝜉Δ𝑧^𝑘−𝑇_𝛼𝑧^𝑘−𝜉𝑇^′_𝛼𝑧^𝑘Δ𝑧^𝑘, (4.5) and then 𝑇_𝛼𝑧^𝑘+𝜉Δ𝑧^𝑘=𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘+𝜉𝑇^′_𝛼𝑧^𝑘Δ𝑧^𝑘=𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘+2𝜉𝐻_𝛼𝑧^𝑘⊤𝐻^′_𝛼𝑧^𝑘Δ𝑧^𝑘=𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘+2𝜉𝐻_𝛼𝑧^𝑘⊤−𝐻_𝛼𝑧^𝑘+𝜌_𝑘𝜖=𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘−2𝜉𝐻_𝛼𝑧^𝑘⊤𝐻_𝛼𝑧^𝑘+2𝜉𝐻_𝛼𝑧^𝑘⊤𝜌_𝑘𝜖≤𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘−2𝜉𝐻_𝛼𝑧^𝑘⊤𝐻_𝛼𝑧^𝑘+2𝜉𝜌_𝑘𝐻_𝛼𝑧^𝑘𝜖⁰=𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘+2𝜉−𝑇_𝛼𝑧^𝑘+𝜖⁰𝜌_𝑘𝐻_𝛼𝑧^𝑘. (4.6)
If 𝑇_𝛼(𝑧^𝑘)≤1, then ‖𝐻_𝛼(𝑧^𝑘)‖≤1, which, together with (4.3), implies that 𝜖⁰𝜌_𝑘𝐻_𝛼𝑧^𝑘≤𝜖⁰𝛾𝑇_𝛼𝑧^𝑘. (4.7) If 𝑇_𝛼(𝑧^𝑘)>1, then ‖𝐻_𝛼(𝑧^𝑘)‖≤𝑇_𝛼(𝑧^𝑘). Combining with (4.3), we have 𝜖⁰𝜌_𝑘𝐻_𝛼𝑧^𝑘≤𝜖⁰𝛾𝑇_𝛼𝑧^𝑘. (4.8) As a result, for any 𝑘, 𝜖⁰𝜌_𝑘𝐻_𝛼𝑧^𝑘≤𝜖⁰𝛾𝑇_𝛼𝑧^𝑘. (4.9)
From (4.6) and (4.9), one has 𝑇_𝛼𝑧^𝑘+𝜉Δ𝑧^𝑘≤𝑅^𝑘(𝜉)+𝑇_𝛼𝑧^𝑘+2𝜉−𝑇_𝛼𝑧^𝑘+𝜖⁰𝛾𝑇_𝛼𝑧^𝑘≤𝑅^𝑘(𝜉)+1−2𝜉1−𝜖⁰𝛾𝑇_𝛼𝑧^𝑘. (4.10)
According to Lemma 2.8, we know that 𝑇_𝛼(𝑧) is continuously differentiable. Hence, there exists 𝜉>0 such that for any 𝜉∈(0,𝜉),1−2𝜉(1−𝜖⁰𝛾)>0 and 𝑇_𝛼𝑧^𝑘+𝜉Δ𝑧^𝑘≤1−2𝜉1−𝜖⁰𝛾𝑇_𝛼𝑧^𝑘≤1−2𝜉1−𝜖⁰𝛾𝐶_𝑘, (4.11) which completes the proof.

Theorem 4.4. Suppose that {𝑧^𝑘=(𝜖^𝑘,𝑥^𝑘,𝜆^𝑘,𝜇^𝑘)} is an infinite sequence generated by Algorithm 4.1. Then the arbitrary accumulation point 𝑧 of the sequence {𝑧^𝑘} is a stationary point of (2.12).

Proof. (i) Suppose that 𝑧=(𝜖̂,𝑦) is an arbitrary accumulation point of {𝑧^𝑘}, then there exists an infinite subsequence 𝐾₁⊆{1,2,…} such that {𝑧^𝑘}_𝐾1→𝑧 as 𝑘∈𝐾₁ and 𝑘→∞. Without loss of generality, we assume that 𝐾₁={1,2,…}. That is, lim_k→∞𝑧^𝑘=𝑧. (4.12) According to Remark 4.2 (i), (iii), and (v), we obtain that the sequence {𝐶_𝑘},{𝜌_𝑘}, and {𝜖^𝑘} are convergent. Without loss of generality, we assume that lim_𝑘→∞𝐶_𝑘=𝐶,lim_𝑘→∞𝜌_𝑘=𝜌,lim_𝑘→∞𝜖^𝑘=𝜖̂. (4.13)
It is easy to see that 𝐶≥0,𝜌≥0, and 𝜖̂≥0. By Remark 4.2 (i) and (ii), we know that 0≤𝑇_𝛼𝑧^𝑘+1≤𝐶_𝑘+1≤𝐶_𝑘≤𝐶₀, (4.14) which means that {𝑇_𝛼(𝑧^𝑘)} is bounded.
In the following, suppose that 𝜌≠0. Without loss of generality, we assume that {𝑇_𝛼(𝑧^𝑘)} is convergent and denote 𝑇=lim_𝑘→∞𝑇_𝛼𝑧^𝑘. (4.15)
Combining (4.3) with the assumption 𝜌≠0, we have 𝑇>0. Furthermore, by using (4.14) and Remark 4.2 (iv), we obtain that 𝐶>0 and 𝜖̂>0. Hence, we can deduce that 𝑇=𝐻_𝛼(𝑧)^⊤𝐻_𝛼(𝑧),𝐶𝛾𝑇≤,𝜌=𝛾,if𝑇>1,𝑇,if𝑇≤1. (4.16)(4.17)
We now break up the proof of (i) into two cases.
(1) Assume that 𝛼_𝑘≥𝑒>0 for all 𝑘, where 𝑒 is a constant. In this case, by (4.4) and (4.2), it follows that 𝐶_𝑘+1=𝜂_𝑘𝑄_𝐾𝐶_𝑘+𝑇_𝛼𝑧^𝑘𝑄_𝑘+1≤𝜂_𝑘𝑄_𝐾𝐶_𝑘+𝐶_𝑘−2𝜎1−𝛾𝜖⁰𝛼_𝑘𝐶_𝑘𝑄_𝑘+1=𝐶_𝑘−2𝜎1−𝛾𝜖⁰𝛼_𝑘𝑄_𝑘+1𝐶_𝑘≤𝐶_𝑘−2𝜎1−𝛾𝜖⁰𝑒𝑄_𝑘+1𝐶_𝑘, (4.18) for any 𝑘, which, together with the boundness of {𝐶_𝑘}, yields ^∞_𝑛=12𝜎1−𝛾𝜖⁰𝑒𝑄_𝑘+1𝐶_𝑘<∞. (4.19)
On the other hand, by 𝜂_max∈[0,1) and the definition of 𝑄_𝑘 given in (4.4), we have 𝑄_𝑘+1=1+^𝑘𝑖_𝑖=0𝑗=0𝜂_𝑘−𝑗≤1+^𝑘_𝑖=0𝜂^𝑖+1_max≤^∞_𝑖=0𝜂^𝑖_max=11−𝜂_max, (4.20) for any 𝑘.
As a result, we obtain that lim_𝑘→∞𝐶_𝑘=0, which is a contradiction with 𝐶>0.
(2) Assume that lim_𝑘→∞𝛼_𝑘=0. The stepsize 𝛼_𝑘=𝛼_𝑘/𝛿 does not satisfy the line search condition (4.2) for any sufficiently large 𝑘, that is, 𝑇_𝛼𝑧^𝑘+𝛼_𝑘Δ𝑧^𝑘>1−2𝜎1−𝛾𝜖⁰𝛼_𝑘𝐶_𝑘 (4.21) holds for sufficiently large 𝑘. Since 𝑇_𝛼(𝑧^𝑘)≤𝐶_𝑘, the aforementioned inequality becomes 𝑇_𝛼𝑧^𝑘+𝛼_𝑘Δ𝑧^𝑘−𝑇_𝛼𝑧^𝑘𝛼_𝑘>2𝜎1−𝛾𝜖⁰𝐶_𝑘. (4.22)
Since 𝜖̂>0 and 𝑇_𝛼(⋅) is continuously differentiable at 𝑧, from Lemma 2.8, we have −2𝜎1−𝛾𝜖⁰𝑇=−2𝜎1−𝛾𝜖⁰𝐶≤𝑇^′_𝛼(𝑧)Δ𝑧=2𝐻_𝛼(𝑧)^⊤−𝐻_𝛼(𝑧)+𝜌𝜖≤2−𝑇+𝜖⁰𝜌𝐻_𝛼(𝑧−)≤2𝑇+𝜖⁰𝛾𝑇=2−1+𝜖⁰𝛾𝑇, (4.23) where the first equality holds from (4.4) (in the form of limit), the first inequality holds from (4.22) (in the form of limit); the second equality holds from (4.1) (in the form of limit), and the third inequality holds from 𝜌‖𝐻_𝛼(𝑧)‖≤𝛾𝑇 by using (4.17). Hence it follows from (4.23) and 𝑇>0 that −1+𝛾𝜖⁰≥−𝜎(1−𝛾𝜖⁰), which contradicts the fact that 𝜎∈(0,1/2) and 𝛾𝜖⁰<1. We complete the proof.

Similar to the proof of Theorem  4.3 in [7], the following theorem holds.

Theorem 4.5. Suppose that {𝑧^𝑘=(𝜖^𝑘,𝑥^𝑘,𝜆^𝑘,𝜇^𝑘)} is an infinite sequence generated by Algorithm 4.1, 𝑧 is the arbitrary accumulation point of the sequence {𝑧_𝑘}, and 𝑧 is a BD-regular solution of 𝐻(𝑧)=0. Then(1)the point 𝑥 is a solution of the GNCP;(2)the sequence {𝑧^𝑘} converges to 𝑧 superlinearly. In particular, if 𝐹′ and 𝐺′ are locally Lipschitz continuous at 𝑧, then {𝑧^𝑘} converges to 𝑧 Q-quadratically.

5. Numerical Experiments

In this section, we implement Algorithm 4.1 for solving some GNCPs in order to see the behavior of Algorithm 4.1. The parameters used in the algorithm are chosen as follows:𝛼=0.01,𝜖⁰=2.25,𝜎=0.29,𝛾=0.02,𝛿=0.2. (5.1)

The following problems are tested in [5–7].

Example 5.1. Consider the implicit complementarity problems with the following form: find 𝑦∈𝑅^𝑛 such that 𝑦−𝑚𝑦≥0,𝐹𝑦≥0,𝐹𝑦^𝑇𝑦−𝑚𝑦=0, (5.2) where 𝑚_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛, and 𝐹𝑦111⋮11=𝐴𝑦+𝑏=2−100⋯00−12−10⋯000−12−1⋯00⋯⋯0000⋯2−10000⋯−12𝑦+, (5.3) and 𝑚(𝑦)=𝜓(𝐴𝑦+𝑏) with 𝜓∶𝑅^𝑛→𝑅^𝑛 being twice continuously differentiable. The following choices of function 𝜓 define our test problems.(1)𝜓_𝑖(𝑥)=−0.5−𝑥_𝑖,𝑖=1,2,…,𝑛, (2)𝜓_𝑖(𝑥)=−1.5𝑥_𝑖+0.25𝑥²_𝑖,𝑖=1,2,…,𝑛.

The numerical results are shown in Tables 1, 2, and 3 with the following four kinds of initial points:

Table 1 

Numerical results of Example 5.1 with 𝑛=4.

Table 2 

Numerical results of Example 5.1 with 𝑛=8.

Table 3 

Numerical results of Example 5.1 with 𝑛=12.

(a) (0,0,…,0)^⊤, (b)(−0.5,−0.5,…,−0.5)^⊤,

(c) (−1,−1,…,−1)^⊤, (d)(0.5,0.5,…,0.5)^⊤.

In Tables 1, 2, and 3, ST denotes initial point, IT is the iterative number, TV is the final value of T when the algorithm terminates, and CPU denotes the computing time in the computer, respectively. For the starting point, we choose 𝜆⁰=(0.5,0.5,…,0.5)^⊤∈𝑅^𝑛 and the termination criterion for the Algorithm is ‖𝑇(𝑧^𝑘)‖≤10⁻⁶.

From Tables 1, 2, and 3, we can see that Algorithm 4.1 is efficient in solving this kind of problem.

In the following, we will compare Algorithm 4.1 (denoted by Inexact) with the exact Newton algorithm with nonmontone line search (denoted by Exact). Here, our test problem is the previous problem (1), that is,𝜓_𝑖(𝑥)=−0.5−𝑥_𝑖,𝑖=1,2,…,𝑛. (5.4)

The initial point is (a) or (b). The numerical results are shown in Table 4.

Table 4 

Comparison of numerical results with 𝑛=800.

From Table 4, we can see that Algorithm 4.1 is prior to the exact smoothing Newton method when the GNCP is of relatively large scale.

6. Conclusion

In this paper, we combine the smoothing function of penalized Fischer-Burmeister NCP-function with nonmontone line search in [17] to present a new smoothing algorithm to solve the generalized nonlinear complementarity problem. We obtain that the iteration sequence generated by Algorithm 4.1 converges to a solution of the generalized nonlinear complementarity problem locally superlinear (quadratic). Preliminary numerical results show the efficiency of the algorithm.

Acknowledgments

The authors are gratefully indebted to the anonymous referees for their valuable suggestions and remarks that allowed them to essentially improve the original presentation of the paper. This paper is supported by the National Nature Science Foundation of China (Grant nos. 10901096 and 10971118) and Shandong Provincial Natural Science Foundation, China (Grant no. ZR2009AL019) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J09LA53).