A Penalized-Equation-Based Generalized Newton Method for Solving Absolute-Value Linear Complementarity Problems

We consider a class of absolute-value linear complementarity problems.We propose a new approximation reformulation of absolute value linear complementarity problems by using a nonlinear penalized equation. Based on this approximation reformulation, a penalized-equation-based generalized Newton method is proposed for solving the absolute value linear complementary problem. We show that the proposed method is globally and superlinearly convergent when the matrix of complementarity problems is positive definite and its singular values exceed 1. Numerical results show that our proposed method is very effective and efficient.


Introduction
Let  : R  → R  be a given function.The complementarity problems, CP() for short, is to find a solution of the system  ≤ 0,  () ≤ 0, ()   = 0.
The CP() is called the linear complementarity problems (for short LCP) if  is an affine mapping of the form where  ∈ R × and  ∈ R  .Otherwise, the CP() is called the nonlinear complementarity problems (NCP()).
The systematic study of the finite-dimensional CP() began in the mid-1960s; in a span of five decades, the subject has developed into a very fruitful discipline in the field of mathematical programming.The developments include a rich mathematical theory, a host of effective solution algorithms, a multitude of interesting connections to numerous disciplines, and a wide range of important applications in engineering and economics (see, e.g., [1][2][3][4] and the references therein).
Generalized Newton method (semismooth Newton method) is one of efficacious algorithms for solving CP().
The main idea of semismooth Newton method is based on an equivalent reformulation of the complementarity problems consisting of a nonsmooth equation and then solving the nonsmooth equation by Newton type method (see, e.g., [5,6]).Most reformulations of the CP() are based on the Fischer-Burmeister function [7] (see, e.g., [8][9][10] and the references therein).Chen et al. [11] introduced a penalized Fischer-Burmeister function and proposed a new semismooth Newton method based on this new NCP function.Kanzow and Kleinmichel [12] proposed a new, one-parametric class of NCP functions based on Fischer-Burmeister function and gave a semismooth Newton method via these NCP functions.Kanzow [13] researched an inexact semismooth Newton method based on Fischer-Burmeister function and penalized Fischer-Burmeister function.Ito and Kunisch [14] studied a semismooth Newton method based on the max-type NCP function.
All generalized Newton methods mentioned above involve the continuously differentiable assumption on  in CP().The existed generalized Newton methods proposed are based on the equivalent reformulation via NCP functions.To the best of our knowledge, until now, there exist very few literature resources to study the complementary problems when the involved function  is not differentiable.However,

Preliminaries
For convenience, we will now briefly explain some of the terminologies that will be used in the next section.R  denotes the -dimensional Euclidean space.All vectors in R  are column vectors.Let  = (  ) ∈ R × be an  ×  real matrix.The scalar product of two vectors  and  is denoted by   .For  > 1, the -norm of  = ( 1 ,  2 , . . .,   )  ∈ R  is defined as ‖‖  = (∑  =1 |  |  ) 1/ .When  = 2, the -norm becomes the 2-norm ‖‖ = (  ) 1/2 .For any  = ( 1 ,  2 , . . .,   ) ∈ R  and  ∈ R,   = (  1 ,   2 , . . .,    ).|| stands for the vector in R  of absolute values of components of .sign() will denote the vector with components equal to 1, 0, or −1 depending on whether the corresponding component of  is positive, zero, or negative.diag(sign()) will denote a diagonal matrix corresponding to sign().The plus function [] + , which replaces the negative components of  by zeros, is a projection operator that projects  onto the nonnegative orthant; namely, [] + = max{, 0}.For a solvable matrix equation  = , we will use the MATLAB backslash  \  to denote a solution .The generalized Jacobians || of || and [] 1/ + of [] 1/ + based on a subgradient [20] of their components are given by the diagonal matrices () and   (), respectively, where () = diag(sign()) and   () is the the diagonal matrix whose diagonal entries are equal to (1/) (1/)−1  , 0, or a real number  ∈ [0, 1] depending on whether the corresponding component of  is positive, negative, or zero.
Lemma 3 (see [15]).Let K be a closed and convex set in R  .A vector  solves the AVLCP (, ) if and only if  solves the following absolute-value variational inequalities: where ‖‖  denotes the Frobenius norm Lemma 5 (see [22]).The singular values of the matrix  ∈ R × exceed 1 if and only if the minimum eigenvalue of    exceeds 1.

A Penalized-Equation Approximation
Reformulation of AVLCP(, ) In this section, we construct a nonlinear penalized equation corresponding to absolute-value linear complementarity problem (3).
Find   ∈ R  such that where  > 1 is the penalized parameter and ℓ ≥ 1.
We will prove that the solution to the penalized equation (10) converges to that of the AVLCP(, ).Thus, we make the following assumptions on the system matrix : (A1)  is positive definite and  > 1 in Definition 1; (A2) the entries of  satisfy   > 0,   ≤ 0 for all ,  = 1, 2, . . .,  with  ̸ = .
Under assumption (A1), the solution of AVLCP(, ) is unique [15].Our main results in this section are as follows.First, we start our discussion with the following lemma.Lemma 9. Let   be the solution to nonlinear penalized equation (10).Then there exists a positive constant  0 , independent of ,   , and  such that where  and ℓ are parameters used in (10).
Using Lemma 9, we can establish the relationship between solutions of penalized equation (10) and solutions of the AVLCP(, ).Theorem 10.Let  * and   be the solution to AVLCP(, ) and nonlinear penalized equation (10), respectively.Then there exists a positive constant , independent of ,  * ,   , and  such that where  and ℓ are parameters used in (10).

A Penalized-Equation-Based Generalized Newton Method and Its Convergence
In this section, we present a generalized Newton method for solving nonlinear penalized equation (10).We begin by defining the vector function specified by the nonlinear penalized equation (10) as follows: Let   = ( 1 ,  2 , . . .,   )  ; a generalized Jacobian (  ) of (  ) is given by where + is a diagonal matrix whose diagonal entries are equal to (1/ℓ) (1/ℓ)−1  , 0, or a real number  ∈ [0, 1] depending on whether the corresponding component of   is positive, negative, or zero.The generalized Newton method for finding a solution of the equation (  ) = 0 consists of the following iteration: Replacing (   ) by its definition (40) and setting (   ) by ( 41) give Thus, solving for  +1  gives which is our final generalized Newton iteration for solving the nonlinear penalized equation (10).In the following, we can establish the penalized-equation-based generalized Newton method for solving AVLCP(, ).
Step 2. Calculate  +1   from the generalized Newton equation starting from  0   associated with   . Step giving the result.
Furthermore, we have the following results.
Theorem 15.Suppose that the singular values of  ∈ R × exceed 1 and   is the unique solution of the nonlinear penalized equation (10).Then, for any    such that Let   be a solution of the nonlinear penalized equation (10).To simplify notation, let  = (  ),   = (    ),   =   (  ), and    =   (    ).Noting that where the first inequality follows from Hence, one has that Thus Letting  → ∞ and taking limits in (61), we can see that the sequence {    } +∞ =1 generated by generalized Newton iteration (45) converges superlinearly to a solution   .

Convergence of Penalized-Equation-Based Generalized
Newton Method.In this subsection, we will focus on the convergence of Algorithm 11.We first present the global convergence.
Theorem 20.Let the singular values of  ∈ R × exceed 1.Then the sequence {  } generated by Algorithm 11 is bounded.Consequently, there exists an accumulation point  * of the nonlinear penalized equation (10).
Proof.Since   is an accumulation point of {    } +∞ =1 , it follows from Theorem 13 that the sequence {    } +∞ =1 is bounded, we can thus obtain the boundness of {  }.Hence there exists an accumulation point  * of {  } such that the nonlinear penalized equation (10) holds, giving the results.
We then establish the linear convergence of Algorithm 11.
Proof.Taking into account "match property" (52), the theorem can be proved in a similar way to that of Theorem 15.
Finally, we establish the superlinear convergence of Algorithm 11.
Proof.Taking into account "match property" (52), the theorem can be proved in a similar way to that of Theorems 15 and 17.

Numerical Results
In this section, we consider several examples to show the efficiency of the proposed method by running in MATLAB 7.5 with Intel(R) Core (TM) of 2 × 2.70 GHz and RAM of 2.0 GB.Throughout these computational experiments, the parameters used in the algorithm are set as  = 10 −6 ,  0 = 10,  = 2, and ℓ = 1.The accumulation point of Algorithm 11 is written as  * .
From Table 1 to Table 4, we can see that our method has some nice convergence which coincides with our results.

Conclusion
In this paper, we propose a new approximation to absolutevalue linear complementarity problems (3) by using the nonlinear penalized equation (10), based on which a generalized Newton method is proposed for solving this penalized equation.Under suitable assumptions, the algorithm is shown to be both globally and superlinearly convergent.The numerical results presented showed that the generalized Newton method proposed by us is efficient.The results and ideas of this paper may be used to solve the   absolute variational inequalities and related optimization problems.

Table 1 :
The numerical results of Example 1.

Table 2 :
The numerical results of Example 2.
Example 3. Let the matrix  of AVLCP(, ) be given by  = (

Table 3 :
The numerical results of Example 3.

Table 4 :
Computational results of Example 4.