An Improved Filter Method for Nonlinear Complementarity Problem

Abstract

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.

1. Introduction

The nonlinear complementarity problem (NCP) is to find a point 𝑥∈𝑅^𝑛, such that 𝑥≥0,𝐹(𝑥)≥0,𝑥^𝑇𝐹(𝑥)=0, (1) where 𝑥∈𝑅^𝑛,  𝐹∶𝑅^𝑛→𝑅^𝑛 is an 𝐿𝐶¹ function and ∇𝐹 is local Lipschitzian. 𝑛 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 𝜙∶𝑅^𝑛→𝑅^𝑛 is called an NCP function if it satisfies 𝜙(𝑎,𝑏)=0⟺𝑎≥0,𝑏≥0,𝑎𝑏=0. (2)

Then we get that the NCP function with the form 𝜙𝑥_𝑖,𝐹_𝑖(𝑥)=0,𝑖=1,2,…,𝑛 (3) 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 𝜙_FB(𝑎,𝑏)=𝑎²+𝑏²−𝑎−𝑏. (4)

It is easy to check that 𝜙_FB(𝑎,𝑏)=0⇔𝑎≥0,  𝑏≥0,  and  𝑎𝑏=0. Hence we can reformulate problem (1) as a nonlinear system of equation Φ(𝑥)=0, (5) where the nonsmooth mapping Φ∶𝑅^𝑛→𝑅^𝑛 is defined by Φ(𝑥)=𝜙_FB𝑥₁,𝐹₁(𝑥),𝜙_FB𝑥₂,𝐹₂(𝑥)𝜙,…,_FB𝑥_𝑛,𝐹_𝑛(𝑥)^𝑇, (6) where 𝜙_FB(𝑥_𝑖,𝐹_𝑖(𝑥))=𝑥²_𝑖+𝐹²_𝑖(𝑥)−𝑥_𝑖−𝐹_𝑖(𝑥). For convenience, we write 𝜙_FB(𝑥_𝑖,𝐹_𝑖(𝑥)) as 𝜙_𝑖, and Φ(𝑥)=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇. We can now associate this system with its natural merit function; that is, 𝑓(𝑥)=12Φ^𝑇(𝑥)Φ(𝑥)=12‖Φ(𝑥)‖² (7) so that solving (1) is equivalent to find a global solution of the problem min𝑓(𝑥). (8)

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(𝑥), such as Newton’s method or quasi-Newton’s approach. But in many cases, the derivatives of Φ(𝑥) are not available for the nonsmooth function Φ(𝑥). For this case, there are some so-called derivative-free methods [3, 4] appeared to avoid computing the derivatives of Φ(𝑥). But they always demand that 𝐹(𝑥) is a monotone function. In this paper, we use decomposition technique to decompose the function 𝜙 into a smooth part and a nonsmooth part; moreover, we have no demand on the monotonicity assumption on 𝐹. 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.

2. 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.

2.1. Decomposition of NCP Function

If Φ∶𝑅^𝑛→𝑅^𝑛 is continuous and local Lipschitzian, then the  𝐵-subdifferential of Φ at 𝑥 is 𝜕_𝐵Φ(𝑥)=𝐻∣𝐻=lim_{𝑥𝑘→𝑥}∇Φ𝑥_𝑘,𝑥_𝑘∈𝐷_Φ, (9) where 𝐷_Φ denotes the set of points 𝑥 at which Φ is differential.

The Clarke’s generalized Jacobian of Φ at 𝑥 is defined by 𝜕Φ(𝑥)=conv𝜕_𝐵Φ(𝑥). (10)

Definition 1. Let Φ∶𝑅^𝑛→𝑅^𝑛 be a local Lipschitzian function. If for all  ℎ∈𝑅^𝑛, it holds lim_{𝑉∈𝜕Φ(𝑥+𝑡ℎ′),ℎ′→ℎ,𝑡↓0}{𝑉ℎ^′}, then function Φ is called semismooth at 𝑥.

Lemma 2. Suppose 𝑓∶𝑅^𝑛→𝑅^𝑛, and ∇𝑓 is semismooth at 𝑥. Then for all 𝑉∈𝜕(∇𝑓(𝑥+𝑑)), there hold(i) 𝑉𝑑−(∇𝑓)^′(𝑥;𝑑)=𝑜(‖𝑑‖);(ii) ∇𝑓(𝑥+𝑑)=∇𝑓(𝑥)+(∇𝑓)^′(𝑥;𝑑)+𝑜(‖𝑑‖),
where (∇𝑓)^′(𝑥;𝑑)=lim_𝑡↓0((∇𝑓(𝑥+𝑡𝑑)−∇𝑓(𝑥))/𝑡) is called directional derivative of ∇𝑓 at 𝑥 in the direction 𝑑.

Lemma 3 (see [7]). If 𝜙 is an 𝐿𝐶¹ mapping, then 𝜙(𝑥+𝑑)−𝜙(𝑥)−∇𝜙(𝑥)^𝑇𝑑=𝑂‖𝑑‖²𝑓𝑜𝑟𝑑⟶0. (11)

Lemma 4. There exist constants 𝑐₁,𝑐₂>0, such that 𝑐₁^𝑛_𝑖=1min²𝑥_𝑖,𝐹_𝑖(𝑥)≤𝑓(𝑥)≤𝑐₂^𝑛_𝑖=1min²𝑥_𝑖,𝐹_𝑖(𝑥). (12)

Proof. This follows immediately from Lemma 3.1 of Tseng [8].
In smooth case, for solving the nonlinear equation (5), the Levenberg-Marquardt method can be viewed as a method for generating a sequence {𝑥_𝑘} of iterates where the step 𝑑_𝑘 between iterates is a solution to the problem minΦ𝑥_𝑘+∇Φ𝑥_𝑘^𝑇𝑑₂s.t.‖𝑑‖≤Δ_𝑘 (13) for some bound Δ_𝑘>0. The norm ‖⋅‖₂ denotes the 𝑙₂-norm.
But in the nonsmooth case, ∇Φ(𝑥_𝑘) may not exist at some special points. However, in many cases, one may decompose the nonsmooth function Φ(𝑥) into Φ(𝑥)=𝑝(𝑥)+𝑞(𝑥), where 𝑝∶𝑅^𝑛→𝑅^𝑛 is smooth and 𝑞∶𝑅^𝑛→𝑅^𝑛 is nonsmooth, while 𝑞(𝑥) is relatively small compared to function 𝑝(𝑥). We call such a decomposition a smooth plus nonsmooth (SPN) decomposition. In a certain sense, 𝑞 can be regarded as the perturbation to the system. We now use min𝑄_𝑘(𝑑)=12Φ𝑥_𝑘+∇𝑝𝑥_𝑘^𝑇𝑑²s.t.‖𝑑‖≤Δ_𝑘 (14) to replace (13).

Definition 5. We say that Φ=𝑝+𝑞 is a regular SPN decomposition of Φ if and only if 𝑝∶𝑅^𝑛→𝑅^𝑛 is smooth, and for any 𝑥∈𝑅^𝑛, it holds ∇𝑝(𝑥)Φ(𝑥)≠0 (15) as long as Φ(𝑥)≠0.

Remark 6. In fact, for some given 𝜖>0, define function 𝜙^𝜖 by 𝜙^𝜖()=𝜙(),𝑎,𝑏𝑎,𝑏𝑎²+𝑏²𝜙()+≥𝜖,𝑎,𝑏𝑎²+𝑏²−𝜖²2𝜖,otherwise. (16) Let Φ^𝜖(𝑥)=𝜙^𝜖₁,𝜙^𝜖₂,…,𝜙^𝜖_𝑛^𝑇. (17) So, we can see 𝑝(𝑥)=Φ^𝜖(𝑥),𝑞(𝑥)=Φ(𝑥)−Φ^𝜖(𝑥). (18) Then it is easy to see that 𝑝(𝑥) obtained by the previous decomposition is continuously differential, while 𝑞(𝑥) is nondifferential, and it holds 𝑞(𝑥≤1)2√𝑛𝜖∀𝑥∈𝑅^𝑛. (19)

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: ℎ(𝑥)=‖max{0,−𝑥,−𝐹(𝑥)}‖_∞,𝑓(𝑥)=12‖Φ(𝑥)‖². (20)

The trial step should either reduce the value of the constraint violation function ℎ or the objective value of the function 𝑓. To ensure sufficient decrease of at least one of two criteria, we say that a point 𝑥_𝑖 dominates a point 𝑥_𝑗 whenever ℎ_𝑖≤ℎ_𝑗,𝑓_𝑖≤𝑓_𝑗 (21) for 𝑖≠𝑗, where ℎ_𝑖=ℎ(𝑥_𝑖),  𝑓_𝑖=𝑓(𝑥_𝑖). We thus aim to accept a new iterate 𝑥_𝑖 only if it is not dominated by any other iterate in the filter.

A filter set ℱ is a set of points in 𝑅^𝑛, such that no point dominates any other.

In practical computation, a trial point 𝑥_𝑘 is acceptable to the filter if and only if 𝑓𝑥_𝑘≤𝑓𝑥_𝑙−𝛾𝛿𝜓_𝑘,𝜓_𝑙orℎ𝑥_𝑘≤ℎ𝑥_𝑙−𝛾𝛿𝜓_𝑘,𝜓_𝑙 (22) for all  𝑥_𝑙∈ℱ, where 0<𝛾<1 is a small positive constant and 𝜓_𝑘=(ℎ_𝑘,𝑓_𝑘),  𝛿(𝜓_𝑘,𝜓_𝑙)=min{‖𝜓_𝑘‖,‖𝜓_𝑙‖}.

As the algorithm progresses, we may want to add 𝑥_𝑘 to the filter. If an iteration 𝑥_𝑘 is acceptable for filter ℱ, we do this by adding the point 𝑥_𝑘 to the filter and removing those 𝑥_𝑗 satisfying 𝑓_𝑗≥𝑓_𝑘,ℎ_𝑗≥ℎ_𝑘. (23) We also refer to this operation as “adding 𝑥_𝑘 to the filter.”

3. 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.

Algorithm A

Step 1. Choose 𝑥₀∈𝑅^𝑛, the initial decomposition parameter 𝜖₀>0,  𝜖>0,  0<𝜂,  𝛾<1,  Δ₀>0,  and  ℱ=[ℎ₀,𝑓₀]; set 𝑘=0.

Step 2. Decompose Φ(𝑥_𝑘) into Φ(𝑥_𝑘)=𝑝(𝑥_𝑘)+𝑞(𝑥_𝑘). If ‖Φ(𝑥_𝑘)‖≤𝜖, then stop.

Step 3. Solve (14) to get 𝑑_𝑘. Compute 𝑟_𝑘=𝑎red_𝑘/𝑝red_𝑘=(𝑓(𝑥_𝑘)−𝑓(𝑥_𝑘+𝑑_𝑘))/(𝑓(𝑥_𝑘)−𝑄_𝑘(𝑑_𝑘)).

Step 4. If 𝑟_𝑘≥𝜂, go to Step 6. Or else, go to Step 5.

Step 5. If 𝑥^𝑘+𝑑^𝑘 is acceptable for the filter ℱ, add 𝑥^𝑘 to the filter, and go to Step 6. Otherwise, 𝑥_𝑘+1=𝑥_𝑘,  Δ_𝑘+1=Δ_𝑘/2,  𝜖_𝑘+1=‖𝑑_𝑘‖²,  𝑘=𝑘+1, and go to Step 3 (inner loop).

Step 6. 𝑥_𝑘+1=𝑥_𝑘+𝑑_𝑘,  Δ_𝑘+1=2Δ_𝑘,  𝜖_𝑘+1=‖𝑑_𝑘‖²,  𝑘=𝑘+1, and go to Step 2 (outer loop).

Remark 7. Algorithm A is a trust-region-type filter method combined with a decomposition technique for the nonlinear complementarity problem.
Throughout this paper, we always assume that following conditions hold.
Assumptions (A1)  𝐹(𝑥) is an 𝐿𝐶¹ function, and ∇𝐹(𝑥) is local Lipschitzian. (A2)  The sequences {𝑥^𝑘} and {𝑥_𝑘+𝑑_𝑘} remain in a closed, bounded convex subset Ω⊂𝑅^𝑛.

Lemma 8. The inner loop can terminate in finite times.

Proof. Suppose by contradiction that it cannot terminate finitely, then it holds Δ_𝑘→0, consequently, 𝑑_𝑘→0as𝑘→∞.
By the definition of 𝑎red_𝑘 and 𝑝red_𝑘, we have 𝑎red_𝑘−𝑝red_𝑘=𝑓𝑥_𝑘+𝑑_𝑘−𝑄_𝑘𝑑_𝑘=12Φ𝑥_𝑘+𝑑_𝑘²−12Φ𝑥_𝑘+∇𝑝𝑥_𝑘^𝑇𝑑_𝑘²=12Φ𝑥_𝑘+𝑑_𝑘−Φ𝑥_𝑘²+Φ𝑥_𝑘^𝑇×𝑝𝑥_𝑘+𝑑_𝑘−𝑝𝑥_𝑘−∇𝑝𝑥_𝑘^𝑇𝑑_𝑘+Φ𝑥_𝑘^𝑇𝑞𝑥_𝑘+𝑑_𝑘−𝑞𝑥_𝑘−12𝑑^𝑇_𝑘∇𝑝𝑥_𝑘∇𝑃𝑥_𝑘^𝑇𝑑_𝑘=𝑂𝑑_𝑘²,𝑝red_𝑘=𝑓𝑥_𝑘−𝑄_𝑘𝑑_𝑘≥𝑓𝑥_𝑘−𝑄_𝑘−𝛼Δ_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘=12Φ𝑥_𝑘²−12Φ𝑥_𝑘+∇𝑝𝑥_𝑘^𝑇×−𝛼Δ_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘²=𝛼Δ_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘−𝛼²Δ²_𝑘2∇𝑝𝑥_𝑘Φ𝑥_𝑘²×∇𝑝𝑥_𝑘^𝑇∇𝑝𝑥_𝑘Φ𝑥_𝑘²≥𝛼Δ_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘−𝛼²Δ²_𝑘2∇𝑝𝑥_𝑘^𝑇2≥max_0≤𝛼≤1𝛼Δ_𝑘∇𝑝𝑥_𝑘Φ𝑥_𝑘−𝛼²Δ²_𝑘2∇𝑝𝑥_𝑘²≥12∇𝑝𝑥_𝑘Φ𝑥_𝑘×minΔ_𝑘,∇𝑝𝑥_𝑘Φ𝑥_𝑘∇𝑝𝑥_𝑘². (24)(25)
By Δ_𝑘→0 and Assumption (A2), we have Δ_𝑘<‖∇𝑝(𝑥_𝑘)Φ(𝑥_𝑘)‖/‖∇𝑝(𝑥_𝑘)‖² for 𝑘 sufficiently large. Then, it holds 𝑝red_𝑘≥12∇𝑝𝑥_𝑘Φ𝑥_𝑘Δ_𝑘≥12∇𝑝𝑥_𝑘Φ𝑥_𝑘𝑑_𝑘=𝑂𝑑_𝑘. (26) Hence 𝑟_𝑘−1=𝑎red_𝑘𝑝red_𝑘≤𝑂𝑑−1^𝑘2𝑂𝑑_𝑘⟶0as𝑑_𝑘⟶0, (27) which is implied that there exists a constant 𝜂>0, such that 𝑟_𝑘≥𝜂 for 𝑘 sufficiently large. The result follows.

4. 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_𝑘→∞‖𝜓(𝑥_𝑘)‖=0; that is, any accumulation point of {𝑥_𝑘} is a solution to the nonlinear complementarity problem.

Proof. Suppose 𝑘_𝑖 index the subsequence of iteration at which 𝑥_𝑘𝑖 is added to the filter ℱ. Now suppose by contradiction that there exist a constant 𝜖₁>0 and a subsequence {𝑘_𝑗}⊂{𝑘_𝑖}, such that 𝜓𝑥_𝑘𝑗>𝜖₁. (28)
From Assumption (A2), we have lim_𝑗→∞𝜓𝑥_𝑘𝑗=𝜓^∞<∞with𝜓^∞≥𝜖₁. (29)
By the definition of {𝑘_𝑗}, 𝑥_𝑘𝑗 is acceptable for the filter, which implies that 𝑓𝑥_𝑘𝑗≤𝑓𝑥_𝑘𝑗−1−𝛾𝛿𝑥_𝑘𝑗,𝑥_𝑘𝑗−1orℎ𝑥_𝑘𝑗≤ℎ𝑥_𝑘𝑗−1−𝛾𝛿𝑥_𝑘𝑗,𝑥_𝑘𝑗−1. (30)
Together with (29) and the definition of 𝛿(⋅), we deduce that there exists a constant 𝜖₂>0, such that 𝛿(𝑥_𝑘𝑗,𝑥_𝑘𝑗−1)>𝜖₂.
Then by (30), it holds 𝑓𝑥_𝑘𝑗−1−𝑓𝑥_𝑘𝑗≥𝛾𝜖₂orℎ𝑥_𝑘𝑗−1−ℎ𝑥_𝑘𝑗≥𝛾𝜖₂. (31)
Let 𝑗→∞, and it is easy to see that 0≥𝛾𝜖₂>0, (32) which is a contradiction. Hence lim_𝑖→∞𝜓𝑥_𝑘𝑖=0. (33)
Consider now any 𝑙∉{𝑘_𝑖}, and let 𝑘_𝑖(𝑙) be the last iteration before 𝑙, such that 𝑥_{𝑘𝑖(𝑙)} was added to the filter. By the construction of Algorithm A, if 𝑥_𝑙 is not included in the filter, it must result in the decrease of the objective function 𝑓(𝑥). Hence for all 𝑙∉{𝑘_𝑖}, it holds 𝑓_𝑙≤𝑓_{𝑘𝑖(𝑙)}≤𝜓𝑥_{𝑘𝑖(𝑙)}, (34) which follows lim_𝑘→∞𝑓(𝑥_𝑘)=0. Moreover by Lemma 4, we have lim^𝑛_{𝑘→∞𝑖=1}min²𝑥_𝑘𝑖,𝐹_𝑖𝑥_𝑘=0. (35) Therefore lim_𝑘→∞‖𝜓(𝑥_𝑘)‖=0.

Lemma 10. Suppose that there are finite many points entered into the filter ℱ. Then any accumulation point of {𝑥_𝑘} is a solution of the NCP.

Proof. By Assumption (A2), we know that {𝑥_𝑘} has at least one accumulation point 𝑥^∗. Suppose by contradiction that 𝑥^∗ is not the solution to the nonlinear complementarity problem, then there exist 𝜖>0 and 𝑘₁>0 such that ‖Φ(𝑥_𝑘)‖>𝜖 for 𝑘>𝑘₁. Then by Definition 5, we have liminf‖∇𝑝(𝑥_𝑘)Φ(𝑥_𝑘)‖>𝜖 for 𝑘>𝑘₁. If there are finite many points entered into ℱ, then by Lemma 8, it must exist a constant 𝑘₂, such that 𝑟_𝑘≥𝜂 for 𝑘>𝑘₂. Moreover, by the construction of Algorithm A, there also exists a constant Δ>0, such that Δ_𝑘≥Δfor𝑘>𝑘₂. (36)
Together with (25), we have 𝑎red_𝑘=𝑓𝑥_𝑘−𝑓𝑥_𝑘+𝑑_𝑘≥𝜂𝑝red_𝑘≥𝜂2∇𝑝𝑥_𝑘Φ𝑥_𝑘minΔ_𝑘,∇𝑝𝑥_𝑘Φ𝑥_𝑘∇𝑝𝑥_𝑘². (37)
So there exists 𝑘₃=max{𝑘₁,𝑘₂}, such that the sequence 𝑓(𝑥_𝑘) is decreasing monotonically for 𝑘>𝑘₃. In other hand, it is below bounded by Assumption (A2). Hence, we have 𝑓(𝑥_𝑘)−𝑓(𝑥_𝑘+𝑑_𝑘)→0 for 𝑘>𝑘₃. Consequently, it follows Δ_𝑘⟶0for𝑘>𝑘₃, (38) which contradicts to (36). The desired conclusion holds.

Theorem 11. Suppose that Assumptions (A1)-(A2) hold. Let {𝑥_𝑘} be the iterate sequence generated by Algorithm A, then any accumulation point of {𝑥_𝑘} is a solution to the nonlinear complementarity problem.

Proof. It is natural by the previous results.

5. A Numerical Example

In this section, we give a numerical example to test Algorithm A. We use the example as following.

Example 12. One has 𝐹₁(𝑥)=3𝑥²₁+2𝑥₁𝑥₂+2𝑥²₂+𝑥₃+3𝑥₄𝐹−6,₂(𝑥)=2𝑥²₁+𝑥₁+𝑥²₂+10𝑥₃+2𝑥₄𝐹−2,₃(𝑥)=3𝑥²₁+𝑥₁𝑥₂+2𝑥²₂+2𝑥₃+9𝑥₄𝐹−9,₄(𝑥)=𝑥²₁+3𝑥²₂+2𝑥₃+3𝑥₄−3. (39)

This problem has one nondegenerate solution 𝑥^∗=(1,0,3,0)^𝑇 and one degenerate solution 𝑥^∗∗=(√6/2,0,0,1/2)^𝑇.

We choose initial point 𝑥₀=𝑎(1,1,1,1), where 𝑎 is a random number. Also, we choose Δ₀=1,  𝜂=0.25,  𝜖₀=0.01,  and  𝛾=0.25. With different 𝑎, Figure 1 shows the change of objective function 𝑓(𝑥) corresponding to the change of iteration 𝑘 in Algorithm A presented in this paper.

Figure 1 

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).

Table 1 

In Table 1, 𝑘 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 𝑓(𝑥) fluctuates to a certain degree, but there exists 𝑘₁, such that 𝑓(𝑥) is also monotone for 𝑘>𝑘₁. 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.

Acknowledgments

The author would like to thank the anonymous referee, whose constructive comments led to a considerable revision of the original paper. This research is supported by the National Natural Science Foundation of China (no. 11101115), the Natural Science Foundation of Hebei Province (nos. A2010000191, A2011201053).