A new smoothing method for solving nonlinear complementarity problems

Abstract In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.


Introduction
Consider the following nonlinear complementarity problems (denoted by NCP), x ≥ , F(x) ≥ , x T F(x) = , (1.1) where F := (F , F , . . . , Fn) T , and F : n → n is continuously di erentiable function. NCP has been extensively studied due to its various important applications in operations research, engineering design, and economic equilibrium. Many algorithms have been developed for solving the Problem (1.1) [1][2][3][4][5][6][7][8][9]. Recently, smoothing method has attracted much attention because that it is a very e cient algorithm to solve NCP. It's main idea is to use a smoothing function to approximate NCP via a family of parameterized smooth equations and solve the smooth equations approximately at each iteration. By reducing the parameter to zero, we have hoped that a solution of the original problem can be found. It is obvious that smoothing functions play a very important role in smoothing methods. Up to now, a large number of smoothing functions have been proposed, Fischer-Burmeister smoothing function [10] and CHKS smoothing function [11] are the most famous ones. Based on the smoothing functions, scholars proposed a number of smoothing algorithms. In order to solve systems of nonsmooth equations, Chen et al. [7] proposed a smoothing Newton method and proved that the algorithm is globally convergent and locally supperlinearly convergent; Yang et al. [12] proposed a smoothing trust region algorithm by using trust region technique instead of line search strategy, but they had to solve complicated quadratic programming subproblems in its current version. Recently, smoothing Levenberg-Marquardt method has attracted much attention. Based on the trust region technique, a smoothing Levenberg-Marquardt method is proposed for the extended linear complementarity problems in [13]. By employing Fischer-Burmeister smoothing function, a smoothing Levenberg-Marquardt method is proposed for solving nonlinear complementarity problems with P function in [14]. In [14], a smoothing parameter τ as an independent variable was introduced. In order to ensure the strict positivity of the smoothing parameter, a relatively complicated subproblems have to be solved at each iteration. Based on a partially smoothing function, Wan et al. [15] proposed a partially smoothing Jacobian method for solving the nonlinear complementarity problems. Like most Jacobian smoothing methods, they still have to assume the function F is a P -function. Under the condition that the level set of a merit function is bounded, they proved the proposed algorithm is globally convergent and superlinearly convergent.
In this paper, motivated by the above work, we propose an improved smoothing Newton method for solving the Problem (1.1). First, based on a one-parametric class of smoothing function, the Problem (1.1) can be reformulated to a system of smoothing equations, and an improved smoothing method is proposed for solving the smoothing equations. Di erent from the processing in [12,14], we solve a system of linear equations instead of a quadratic programming problem for each inner iteration. Moreover, when the iteration point is close enough to the solution point of NCP, the algorithm always takes the full steps. Without strict complementarity conditions and the assumption of P property, we prove that the proposed smoothing method possess the global and local quadratic convergence properties. Compared with previous smoothing methods, our method has some other good properties. Especially, -Compared with the classical smoothing Newton method, our proposed method needn't nonsingular of the smoothing approximation function; -Without requiring strict complementarity conditions and the assumption of P property, proposed smoothing method is also proved to possess global and local quadratic convergence rate. We introduce the following notations. Throughout this paper, all vectors are column vectors, the superscript T denotes transpose of a matrix and a vector, x stands for the 2-norm of vector x ∈ n . n + ( n ++ ) denotes the nonnegative (positive) orthant in n . For a continuously di erentiable function Φτ(x) : n → n , we denote the Jacobian of Φτ(x) at x ∈ n by Φ τ (x).
The rest of paper is organized as follows. In section 2, we investigate a one-parametric class of smoothing function and discuss its properties, and recall some preliminary result used in the subsequent. The algorithm model is stated in section 3. In section 4, the global and local quadratic convergence of the new algorithm is established. Some numerical test results are reported in section 5, which show that the proposed algorithm is e cient.

Smoothing function and its properties
In this section, we investigate a parameter smoothing function and discuss its properties. Based on this smoothing function, the equivalent smoothing reformulation of NCP is given. Firstly, we recall a class of NCP function, which was de ned in [16], It has the following characterizations φ θ (a, b) = ⇔ a ≥ , b ≥ and ab = .
Using the smoothing function (2.2), a smooth approximation of Φ is de ned by Φτ : and the corresponding merit function can also be de ned by So, to solve Problem (1.1), we only need to solve Φτ(x) = and make τ ↓ . Next, we review the concept of semismooth, which was rst introduced by Mi in [40] for functions and extended to vector-valued functions by Qi and Sun [41].
De nition 2.1. Suppose that F : n → n is locally Lipschitz function. The generalized Jacobian of F at x in the sense of Clark [42] denote by ∂F(x), then, F is said to be semismooth (or strongly semismooth) at x ∈ n , if F is directionally di erentiable at x ∈ n and F(

Lemma 2.1. The function Φτ(x) satis es the inequality
for all x ∈ n and τ , τ ≥ , where κ = √ n. In particular, we have for all x ∈ n and τ ≥ .
Proof. It is obvious to hold for τ = τ = . So, we suppose at least one of the perturbation parameters is positive. By simple calculation, we can have Obviously, for any x ∈ n , Similar to the proof of Lemma 2.8 in [22], we have the following lemma.

Lemma 2.2. Let Φτ(x) be de ned by (2.3). Then,
Similar to the proof of Proposition 3.4 in [8], we have the following lemma.

Lemma 2.3. Let x ∈ n be arbitrary but xed. Assume that x is not a solution of NCP. Let us de ne the constants
Let δ > be given, and de ne ), otherwise.

Proposed Algorithm
In this section, we establish a new smoothing method for Problem(1.1) and prove that well-de niteness of the proposed algorithm. Algorithm 3.1.
Step 0 Given a starting point Step 1 Find a solution d k ∈ n of the linear system Step 2 If the condition is satis ed, then set x k+ := x k + d k (we call this 'fast step') and go to Step 4.
Step 6 Set µ k+ = Φτ k+ (x k+ ) and k := k + . Go to Step 1. Remark Di erently from the algorithm in [7,15], our proposed algorithm is well-de ned without the assumption that smoothing function Φτ k (x k ) is nonsingular. For proving that the algorithm 3.1 is well-de ned and has the global convergence property, we assume that Algorithm 3.1 does not terminate in a nite number of iterations. De ne the index set It follows from Lemma 2.1 and Step 5 of the Algorithm 3.1 that which indicates Φτ k (x k ) ≠ for all k ∈ K. By the de nition τ(·, ·) in Lemma 2.3 and the updating rule (3.5), it is not di cult to nd that holds, for any k ∈ K with k ≥ . The following proposition shows the well-de niteness of the proposed algorithm.
It follows from (3.9) and σ ∈ ( , ) that By (3.10) and (3.11), we can obtain the following inequality So, there exists a nite nonnegative integer m k such that Therefore, Algorithm 3.1 is well-de ned.

Convergence of the proposed algorithm
In this section, we will give the global and local quadratic convergence of the proposed algorithm. Proof. Assume that x * is an arbitrary accumulation point of the sequence {x k }, and {x k } k∈K be a subsequence converging to x * . We rst show that the index set K is in nite. By contradiction, we assume that the set K is nite. Let k be the largest number in K, then for any k ≥ k, wehaveτ k = τ k and β k = β k . Denote and In the following paragraphs, we will show that the following equation holds.
On the other hand, in view of Lemma 2.1 and (4.3), we have Ψ τ (x k ) → Ψ τ (x * ) = , then, there exists k ≥ k such that for all k ∈ L with k ≥ k, In view of (4.1) and (4.2), we deduce that for all k ∈ K with k ≥ k, i.e., Therefore, which in contradiction to (4.1). Hence the set K is in nite. By the updating rule of τ k , we have {τ k } → . Similar to the proof of Proposition 4.2 in [8], we deduce that where r = max{ , η}.
Since the set K is in nite, it follows from (3.8) and (4.16) that As a consequence of Theorem 4.1, we can obtain the following global convergence result. In order to obtain the local convergent result, we introduce the following lemma.
Proof. By the condition, we know that the matrix V is nonsingular. Notice that ∂ C Φ(x) is compact for all x ∈ n . Therefore, there exists Combining with (3.8), we can deduce that for all k ∈ K, By theorem 4.1, we have { Φ(x k ) } → . This together with the compactness and the upper semicontinuity of ∂ C Φ(x * ), we obtain from the above inequality that the matrices Φ τ k (x k ) and Φ τ k (x k ) T Φ τ k (x k ) are nonsingular and there exist M > , M > such that (4.17) holds.
The following lemma can be seen in Ref. [44].

Lemma 4.2. Assume that A, B ∈ n×n and A is nonsingular. If A − B < , then A + B is nonsingular and satis es
From Lemmas 4.1 and 4.2, we can prove that the following lemmas hold. Proof. It follows from Theorem 4.1 that x * is a solution of Φ(x) = . Notice that ∂ B Φ(x * ) ⊆ ∂ C Φ(x * ). From Proposition 2.5 in [45], it follows that there exists a neighbourhood of x * such that x * is the unique solution.

Lemma 4.3. Assume that the conditions of Lemma 4.1 hold. Then, for µ k ≤ M ,
Since the sequence {x k } has an accumulation point x * , there exists a subsequence {x k } k∈K that converges to x * . By Lemma 4.1,there exist M > , M > and k ∈ K, Further, from Lemma 2.1, (3.5) and the Lipschitz continuity of Φ(x), we deduce that Then, it follows from (4.21) and Lemmas 4.2, 4.3 that Therefore, for all k > k and k ∈ K, we have By Proposition 2.3 in [16], we known that Φ(x) is strongly semismooth, combining with Theorem 3.2 in [41], we have where V i k denotes the i-th row of V k . Hence, From (4.19)-(4.23), we can deduce (4.24) The above equation together with (4.21) indicate that It follows from triangular inequality and (3.8) that The above equations (4.25) and (4.26) imply that which means that there exists k ≥ k such that k ∈ K and for any k ≥ k, That is, x k+ = x k + d k for all k ≥ k and k ∈ K. This together with (4.24) implies that {x k } converges to x * quadratically.

Numerical experiments
In this section, we implement Algorithm 3.1 on some typical test problems for two purposes: one is to see the numerical behavior of Algorithm 3.1; and the other is to investigate the behavior of these test problems for di erent θ ∈ [ , ]. All the codes are nished in MATLAB 7.8 and done using a PC with Intel (R) Core (TM) i3-3240 CPU @ 3.40 GHz and RAM of 4 GB.
We use ∇Ψ(x k ) ≤ − as the stop rule.
In the tables of experimental results, ST denotes the starting point; DIM denotes the dimension of the problem; θ denotes the values of θ; IT denotes the number of iteration; Fast denotes the number of 'full step' taken during the iteration; B denotes the number of backtracking steps; τ denotes the value of τ at the nal iteration.
From Tables 1-3, It is easy to see that not all the best numerical results based on the proposed smoothing algorithm occur in the case of θ = (in this case, the smoothing function is the famous CHKS smoothing function) or θ = (in this case, the smoothing function is the famous Fischer-Burmeister smoothing function). On the other hand, from the column Fast, it can be seen that, in general, the number of the iteration in which 'fast step' are accepted occupied almost all the iterations. From Table 2, we can see that θ = seems to be more suitable to nd a non-degenerate solution. We also investigate the behavior of our proposed algorithm for solving large scale linear complementarity problems with θ = . The numerical results are listed in Tables 4 and 5. From Tables 4 and 5, we can see that our proposed method is e cient for solving large scale linear complementarity problems. From all the test results, we can see that our algorithm is promising.
Example 5.1. Mathiesen Problem. This test problem was used by Jiang and Qi [46] with four variables, which was also tested by Pang and Gabriel [47]. Let We test this problem by using di erent starting points. The test results are listed in Table 1.

Example 5.2. Kojima-Shindo
Problem. This problem was tested by Pang and Gabriel in [47], which was also tested by Mangasarian and Solodov [48], and Kanzow [49] with four variables. Let This example has one degenerate solution ( √ , , , ) T and one nondegenerate solution ( , , , ) T . We summarize the results in Table 2 by using di erent starting points.
We summarize the test results in Table 3 by using di erent starting points.  . e-. e-.
. e-. e-. e-. e-Example 5.4. This problem is from Geiger and Kanzow [51], which was also tested by Kanzow [11]: We test this problem by using di erent starting points. The test results are listed in Table 4.
Example 5.5. This example was used by Ahn [52]. The test results for Example 5.5 are listed in Table 5 by using di erent starting points.

Conclusion
In this paper, we have presented a new improved smoothing algorithm for the NCP. Compared with the classical smoothing method, our proposed method needn't nonsingular of the smoothing approximation function. We have established the global convergence and the local quadratic convergence for the developed algorithm without strict complementarity conditions and the assumption of P property. Numerical results have showed that the new algorithm works very well.