An Improved Filter Method for Nonlinear Complementarity Problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming whose objective function may be nonsmooth. For this case, we use decomposition strategy to decompose the nonsmooth function into a smooth one and a nonsmooth one. Together with filter method, we present an improved filter algorithm with decomposition strategy for solving nonlinear complementarity problem, which has better numerical results compared to the method that without the filter technique. Under mild conditions, the global convergent property is shown. In the end, the numerical example is reported.


Introduction
The nonlinear complementarity problem (NCP) is to find a point  ∈   , such that  ≥ 0,  () ≥ 0,    () = 0, where  ∈   ,  :   →   is an  1 function and ∇ is local Lipschitzian. is the dimension of the variables.The nonlinear complementarity problem has been utilized as a general framework for quadratic programming, linear complementarity, and other mathematical programming.A variety of methods have been proposed for solving it.One of the powerful approaches is reformulating the nonlinear complementarity problem as an equivalent unconstrained optimization problem [1,2] or as an equivalent system of nonlinear equation [3,4].For this case, a merit function for NCP is needed, whose global minima are coincide with the solutions of the NCP.
To construct a merit function, many kinds of NCP functions appear to lead to a system of equations.A function  :   →   is called an NCP function if it satisfies  (, ) = 0 ⇐⇒  ≥ 0,  ≥ 0,  = 0. ( Then we get that the NCP function with the form  (  ,   ()) = 0,  = 1, 2, . . ., is equivalent to (1) to a certain degree.Many algorithms based on (3) have been proposed, for example, Newton's method and generalized Newton's approaches [5,6].
In this paper, we use Fischer-Burmeister as NCP function, which is called F-B NCP function and given by  FB (, ) = √  2 +  2 −  − .
We remark that, in order to find a solution of (1), one has to seek global solutions of (8), while usual unconstrained minimization algorithm will compute the derivatives of Φ(), such as Newton's method or quasi-Newton's approach.But in many cases, the derivatives of Φ() are not available for the nonsmooth function Φ().For this case, there are some so-called derivative-free methods [3,4] appeared to avoid computing the derivatives of Φ().But they always demand that () is a monotone function.In this paper, we use decomposition technique to decompose the function  into a smooth part and a nonsmooth part; moreover, we have no demand on the monotonicity assumption on .Also, integer with the trust region filter technique, we present a new algorithm to solve (8) and find that any accumulation point of the sequence generated by the algorithm is a solution of (1).
This paper is organized as follows.In Section 2, we review some definitions and preliminary results that will be used in the latter sections.The algorithm is presented in Section 3. In Section 4, the global convergence theory is proved.The numerical results are reported in the last section.

Preliminaries
In this section, we recall some definitions and preliminary results about decomposition of NCP function and the filter algorithm, which will be used in the sequential analysis.
In smooth case, for solving the nonlinear equation ( 5), the Levenberg-Marquardt method can be viewed as a method for generating a sequence {  } of iterates where the step   between iterates is a solution to the problem for some bound Δ  > 0. The norm ‖ ⋅ ‖ 2 denotes the  2 -norm.But in the nonsmooth case, ∇Φ(  ) may not exist at some special points.However, in many cases, one may decompose the nonsmooth function Φ() into Φ() = ()+(), where  :   →   is smooth and  :   →   is nonsmooth, while () is relatively small compared to function ().We call such a decomposition a smooth plus nonsmooth (SPN) decomposition.In a certain sense,  can be regarded as the perturbation to the system.We now use to replace (13).
Definition 5. We say that Φ =  +  is a regular SPN decomposition of Φ if and only if  :   →   is smooth, and for any  ∈   , it holds as long as Φ() ̸ = 0.
Remark 6.In fact, for some given  > 0, define function   by Let So, we can see Then it is easy to see that () obtained by the previous decomposition is continuously differential, while () is nondifferential, and it holds Mathematical Problems in Engineering 3 2.2.Filter Algorithm.Filter algorithms are efficient algorithms for nonlinear programming without a penalty function [9,10].Recently, filter technique has been extended to solve nonlinear equations and nonlinear least square [11].In this paper, it will be used to find a solution to nonlinear complementarity problem.
As traditional filter technique, we define the objective function and violation constrained function as follows: The trial step should either reduce the value of the constraint violation function ℎ or the objective value of the function .To ensure sufficient decrease of at least one of two criteria, we say that a point   dominates a point   whenever for  ̸ = , where ℎ  = ℎ(  ),   = (  ).We thus aim to accept a new iterate   only if it is not dominated by any other iterate in the filter.
A filter set F is a set of points in   , such that no point dominates any other.
In practical computation, a trial point   is acceptable to the filter if and only if for all   ∈ F, where 0 <  < 1 is a small positive constant and   = (ℎ  ,   ), (  ,   ) = min{‖  ‖, ‖  ‖}.
As the algorithm progresses, we may want to add   to the filter.If an iteration   is acceptable for filter F, we do this by adding the point   to the filter and removing those   satisfying We also refer to this operation as "adding   to the filter."

An Improved Filter Algorithm for NCPs
In this section, we will present a decomposition filter method for the nonlinear complementarity problem and prove that it is well defined.

Global Convergence Property
In this section, we will give the global property of Algorithm A.
Lemma 9. Suppose that there are infinite many points entered into the filter F, then lim  → ∞ ‖(  )‖ = 0; that is, any accumulation point of {  } is a solution to the nonlinear complementarity problem.
Proof.Suppose   index the subsequence of iteration at which    is added to the filter F. Now suppose by contradiction that there exist a constant  1 > 0 and a subsequence From Assumption (A2), we have lim By the definition of {  },    is acceptable for the filter, which implies that Together with (29) and the definition of (⋅), we deduce that there exists a constant  2 > 0, such that (   ,   −1 ) >  2 .
Then by (30), it holds Let  → ∞, and it is easy to see that which is a contradiction.Hence lim Consider now any  ∉ {  }, and let  () be the last iteration before , such that   () was added to the filter.By the construction of Algorithm A, if   is not included in the filter, it must result in the decrease of the objective function ().Hence for all  ∉ {  }, it holds   (37) So there exists  3 = max{ 1 ,  2 }, such that the sequence (  ) is decreasing monotonically for  >  3 .In other hand, it is below bounded by Assumption (A2).Hence, we have (  ) − (  +   ) → 0 for  >  3 .Consequently, it follows which contradicts to (36).The desired conclusion holds.

A Numerical Example
In this section, we give a numerical example to test Algorithm A. We use the example as following.
In order to show the good numerical results of Algorithm A, we compare Algorithm A with the traditional method that without filter technique (see Table 1).
In Table 1,  denotes the iteration number, and CPU denotes the cpu's time in computing.
From Figure 1 and Table 1, we can see that Algorithm A is better than traditional algorithm that without filter technique whether from the iteration times or CPU time.Since we use filter technique in Algorithm A, the objective function () fluctuates to a certain degree, but there exists  1 , such that () is also monotone for  >  1 .Just by the filter technique, we have less iteration times and shorter CPU time compared to traditional method without filter technique.Hence, the decomposition filter method is effective.

Figure 1 Together
Figure 1

Table 1
Proof.It is natural by the previous results.