The Levenberg-Marquardt-Type Methods for a Kind of Vertical Complementarity Problem

Abstract

Two kinds of the Levenberg-Marquardt-type methods for the solution of vertical complementarity problem are introduced. The methods are based on a nonsmooth equation reformulation of the vertical complementarity problem for its solution. Local and global convergence results and some remarks about the two kinds of the Levenberg-Marquardt-type methods are also given. Finally, numerical experiments are reported.

1. Introduction

In this paper, we consider the following kind of vertical complementarity problem: 𝐹_𝑖(𝑥)Π≥0,𝑖=1,…,𝑚,_𝑚𝑖=1𝐹^𝑗_𝑖(𝑥)=0,𝑗=1,…,𝑛, (1.1) where the functions 𝐹_𝑖(𝑥)∶𝑅^𝑛→𝑅^𝑛 are assumed to be continuously differentiable and 𝐹^𝑗_𝑖(𝑥) denotes the 𝑗th component of the function 𝐹_𝑖(𝑥). The above vertical complementarity problem is of concrete background, for instance, the Karush-Kuhn-Tucker system of nonlinear programs, complementarity problems, and many problems in mechanics and engineering. When 𝑚=2, (1.1) is the generalized complementarity problem, which has been considered in [1]. When 𝑥 satisfied min_𝑗∈𝐽1𝐹¹_𝑗(𝑥⋮)=0,min_{𝑗∈𝐽𝑛}𝐹^𝑛_𝑗(𝑥)=0, (1.2)𝑥 solves (1.1), where “min” denotes the componentwise minimum operator, 𝐹^𝑖_𝑗∶𝑅^𝑛→𝑅, for, 𝑗∈𝐽_𝑖, 𝑖=1,2,…,𝑛, are also continuously differentiable, and 𝐽_𝑖, for 𝑖=1,2,…,𝑛, are finite index sets. Throughout this section, we denote 𝑔_𝑖(𝑥)=min_{𝑗∈𝐽𝑖}𝐹^𝑖_𝑗(𝑥),𝑥∈𝑅^𝑛𝐺(𝑥)=𝑔,𝑖=1,2,…,𝑛,₁(𝑥),…,𝑔_𝑛(𝑥)^𝑇,𝑥∈𝑅^𝑛,𝐽_𝑖=𝑗∈𝐽_𝑖∣𝐹^𝑖_𝑗(𝑥)=𝑔_𝑖(𝑥),𝑥∈𝑅^𝑛,𝑖=1,2,…,𝑛. (1.3)

Thus, (1.2) can be briefly rewritten as 𝐺(𝑥)=𝑔₁(𝑥),…,𝑔_𝑛(𝑥)^𝑇=0, (1.4) which is nonsmooth equation. Nonsmooth equations are much more difficult than smooth equations. Solving the above equations is a classical problem of numerical analysis, see [1–11]. Many existing classical methods for smooth equations cannot be extended to nonsmooth equations directly. Those difficult motivate us to seek higher-quality methods for nonsmooth equations. One of the classical iterative methods for solving (1.4) is the Levenberg-Marquardt-type method, which is based on the generalized Jacobian. The Levenberg-Marquardt method and its variants are of particular importance also because of their locally fast convergent rates. Levenberg-Marquardt method is also a famous method for nonlinear equations, which can be regarded as a modification of Newton method [12–18]. In the smoothing-type methods, some conditions are needed to ensure that the Jacobian matrix is nonsingular; the Levenberg-Marquardt method does not need such conditions [12–19].

By the above analysis, the purpose of this paper is to consider using the Levenberg-Marquardt-type methods for solving (1.1). We reformulate (1.1) as a system of nonsmooth equations and present two kinds of the Levenberg-Marquardt-type methods for solving it. The outline of the paper is as follows. In Section 2, we recall some background concepts and propositions which will play an important role in the subsequent analysis of convergence results. In Section 3, we give the two kinds of Levenberg-Marquardt-type methods and present their local and global convergence results and some remarks. Finally, numerical experiments and some final remarks about the numerical experiments are listed.

Throughout this paper, 𝑅^𝑛 denotes the space of 𝑛-dimensional real column vectors and 𝑇 denotes transpose. For any differentiable function 𝑓∶𝑅^𝑛→𝑅, ∇𝑓(𝑥) denotes the gradient of 𝑓 at 𝑥.

2. Preliminaries

A locally Lipschitz function 𝐻∶𝑅^𝑛→𝑅^𝑛 is said to be semismooth at 𝑥 provided that the following limit lim_{𝑉∈𝜕𝐻𝑥+𝑡ℎ′ℎ′→ℎ,𝑡→0+}𝑉ℎ^′ (2.1)

exists for any ℎ∈𝑅^𝑛. The class of semismooth functions is very broad; it includes smooth functions, convex functions, and piecewise smooth functions. Moreover, the sums, differences, products, and composites of semismooth functions are also semismooth. Maximum of a finite number of smooth functions are semismooth too.

Proposition 2.1. Function 𝐺 in (1.4) is semismooth.

Proposition 2.2. The following statements are equivalent: (i)𝐻(𝑥) is semismooth at 𝑥.(ii)For 𝑉∈𝜕𝐻(𝑥+ℎ), ℎ→0. 𝑉ℎ−𝐻^′(𝑥;ℎ)=𝑜(‖ℎ‖). (2.2)(iii)One has lim_{𝑥+ℎ∈𝐷𝐻,ℎ→0}𝐻^′(𝑥+ℎ;ℎ)−𝐻^′(𝑥;ℎ)‖ℎ‖=0. (2.3) If for any 𝑉∈𝜕𝐻(𝑥+ℎ), ℎ→0, 𝑉ℎ−𝐻^′(𝑥;ℎ)=𝑜‖ℎ‖^1+𝑝, (2.4) where 0<𝑝≤1, then one says 𝐻(𝑥) is 𝑝-order semismooth at 𝑥. Obviously, 𝑝-order semismoothness implies semismoothness. From [8], one knows that if 𝐻(𝑥) is semismooth at 𝑥, then, for any ℎ→0𝐻(𝑥+ℎ)−𝐻(𝑥)−𝐻^′(𝑥;ℎ)=𝑜(‖ℎ‖). (2.5)

In [6], Gao considered a system of equations of max-type functions, which is the system of equations of smooth compositions of max-type functions. For solving the systems of equations, Gao take 𝜕_⋆ as a tool instead of the Clarke generalized Jacobian, 𝐵-differential, and 𝑏-differential. Based on [6], we give the following 𝜕_⋆𝐺(𝑥) for 𝐺 in (1.4): 𝜕_⋆𝐺(𝑥)=∇𝐹¹_𝑗1(𝑥),…,∇𝐹^𝑛_𝑗𝑛(𝑥)^𝑇∣𝑗₁∈𝐽₁(𝑥),…,𝑗_𝑛∈𝐽_𝑛(𝑥),𝑥∈𝑅^𝑛. (2.6) Obviously, 𝜕_⋆𝐺(𝑥) is a finite set of points and can be easily calculated by determining 𝐽_𝑖(𝑥), 𝑖=1,…,𝑛, and ∇𝐹^𝑖_𝑗𝑖(𝑥), 𝑗_𝑖∈𝐽_𝑖(𝑥), 𝑖=1,…,𝑛. In what follows, we use (2.6) as a tool instead of the Clarke generalized Jacobian and 𝑏-differential in the Levenberg-Marquardt method (I).

Proposition 2.3. The 𝜕_⋆𝐺(𝑥) defined as (2.6) is nonempty and compact set for any 𝑥, and the point to set map is also upper semicontinuous.

Proposition 2.4. Suppose that 𝐺(𝑥) and 𝜕_⋆𝐺(𝑥) are defined by (1.4) and by (2.6), and all 𝑉∈𝜕_⋆𝐺(𝑥) are nonsingular. Then there exists a scalar 𝛽>0 such that 𝑉⁻¹≤𝛽,∀𝑉∈𝜕_⋆𝐺(𝑥),‖𝑉‖≤𝛾,∀𝑉∈𝜕_⋆𝐺(𝑥),𝑥∈𝑁(𝑥,𝜖), (2.7) holds for some constants 𝛾>0, 𝜖>0, and 𝑁(𝑥,𝜖) is a neighbor of 𝑥.

By the continuously differentiable property of 𝐺 in (1.1), the above Proposition 2.4 can be easily obtained.

Definition 2.5. One says that 𝐺 is BD-regular at 𝑥 if all elements in 𝜕𝐺(𝑥) are nonsingular.

Definition 2.6. 𝑀∈𝑅^𝑛×𝑛 is a 𝑃₀ matrix if and only if there exists an index 𝑖∈{1,2,…,𝑛} such that 𝑥_𝑖≠0 and 𝑥_𝑖[𝑀𝑥]_𝑖≥0, for all 𝑥∈𝑅^𝑛, 𝑥≠0.

3. The Levenberg-Marquardt-type Methods and Their Convergence Results

In this section, we present two kinds of the Levenberg-Marquardt-type methods for solving vertical complementarity problem (1.1). Firstly, we briefly recall some results on the Levenberg-Marquardt-type method for the solution of nonsmooth equations and their convergence results, see, for example, [7] and also [9, 10]. And we also give the new kinds of the Levenberg-Marquardt methods for solving vertical complementarity problem (1.1) and analyze their convergence results. We are now in the position to consider exact and inexact versions of the Levenberg-Marquardt-type method.

Given a starting vector 𝑥₀∈𝑅^𝑛, let 𝑥_𝑘+1=𝑥_𝑘+𝑑_𝑘, (3.1)

where 𝑑_𝑘 is the solution of the system 𝑉_𝑘^𝑇𝑉_𝑘+𝜎_𝑘𝐼𝑑=−𝑉_𝑘^𝑇𝐺𝑥_𝑘,𝑉_𝑘∈𝜕_⋆𝐺𝑥_𝑘,𝜎_𝑘≥0. (3.2)

In the inexact versions of this method, 𝑑_𝑘 can be given by the solution of the system 𝑉_𝑘^𝑇𝑉_𝑘+𝜎_𝑘𝐼𝑑=−𝑉_𝑘^𝑇𝐺𝑥_𝑘+𝑟_𝑘,𝑉_𝑘∈𝜕_⋆𝐺𝑥_𝑘,𝜎_𝑘≥0, (3.3)

where 𝑟_𝑘 is the vector of residuals, and we can assume ‖𝑟_𝑘‖≤𝛼_𝑘‖(𝑉_𝑘)^𝑇𝐺(𝑥_𝑘)‖ for some 𝛼_𝑘≥0.

We now give a local convergence Levenberg-Marquardt-type method (I) for (1.1) as follows.

3.1. The Levenberg-Marquardt Method (I)

Step 1. Given 𝑥₀, 𝜖>0, 𝜆^𝑘_𝑖∈𝑅^𝑛, 0<|𝜆^𝑘_𝑖|<+∞.

Step 2. Solve the system to get 𝑑_𝑘, 𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘𝑑=−𝑉_𝑘^𝑇𝐺𝑥_𝑘+𝑟_𝑘,𝑉_𝑘∈𝜕_⋆𝐺(𝑥), (3.4) for 𝑖=1,⋯𝑛, and 𝑟_𝑘 is the vector of residuals 𝑟_𝑘≤𝛼_𝑘𝑉_𝑘^𝑇𝐺𝑥_𝑘,𝛼_𝑘≥0. (3.5)

Step 3. Set 𝑥_𝑘+1=𝑥_𝑘+𝑑_𝑘; if ‖𝐺(𝑥_𝑘)‖≤𝜖, terminate. Otherwise, let 𝑘∶=𝑘+1, and go to Step 2.

Based upon the above analysis, we give the following local convergence result. Similar results have also been mentioned in [9].

Theorem 3.1. Suppose that {𝑥_𝑘} is a sequence generated by the above method and there exist constants 𝑎>0, 𝛼_𝑘≤𝑎 for all 𝑘, and there exist constant 𝑀>0, ‖diag(𝜆_𝑖^(𝑘)𝑔_𝑖(𝑥))‖≤𝑀<+∞. Let 𝑥^⋆ be a BD-regular solution of 𝐺(𝑥)=0. Then the sequence {𝑥_𝑘} converge 𝑄-linearly to 𝑥^⋆ for ‖𝑥₀−𝑥^⋆‖≤𝜖.

Theorem 3.2. Suppose that {𝑥_𝑘} is a sequence generated by the above method and there exist constants 𝑎>0, 𝛼_𝑘≤𝑎 for all 𝑘. ‖𝑟_𝑘‖≤𝛼_𝑘‖𝐺(𝑥_𝑘)‖, 𝛼_𝑘≥0. And suppose that there exist constants 𝑀>0, ‖diag(𝜆_𝑖^(𝑘)𝑔_𝑖(𝑥))‖≤𝑀<+∞. Then the sequence {𝑥_𝑘} converge 𝑄-linearly to 𝑥^⋆ for ‖𝑥₀−𝑥^⋆‖≤𝜖.

By Propositions 2.2, 2.3, and 2.4, we can get the proof of Theorem 3.2 similarly to [9, Theorem 3.1], so we omit it.

Remark 3.3. Theorems 3.1 and 3.2 hold with ‖𝑟_𝑘‖=0.

Remark 3.4. In the Levenberg-Marquardt method (I), if 𝑑_𝑘 is computed by (3.3), Theorems 3.1, 3.2, and Remark 3.3 can also be obtained.

When 𝑚=2 in (1.1), the vertical complementarity problem reduces to the well-known generalized complementarity problem(GCP) in [1]. The GCP is to find 𝑥_⋆∈𝑅^𝑛 satisfying 𝐹₁(𝑥)≥0,𝐹₂≥0,𝐹₁(𝑥)^𝑇𝐹₂(𝑥)=0, (3.6)

where 𝐹_𝑖∶𝑅^𝑛→𝑅^𝑛, 𝑖=1,2 and are continuously differentiable functions. If 𝐹₁(𝑥)=𝑥, then the generalized complementarity problem (GCP) is the nonlinear complementarity problem (NCP). In the following, we give the Levenberg-Marquardt-type method (II) for the generalized complementarity problem  (GCP). The different merit function is based on the well-known 𝐹-𝐵 function𝜑(𝑎,𝑏)=𝑎²+𝑏²−𝑎−𝑏. (3.7)

Then 𝐺(𝑥)=𝜑(𝐹₁,𝐹₂), and we denote the corresponding merit function as Ψ(𝑥)=12‖𝐺(𝑥)‖², (3.8)

where Ψ is a continuously differentiable function. Now, we give the following Levenberg-Marquardt-type method (II).

3.2. The Levenberg-Marquardt Method (II)

Step 1. Given a staring vector 𝑥₀∈𝑅^𝑛, 𝜌>0, 𝑝>2, 𝛽∈(0,1/2), 𝜖≥0, 𝜆^𝑘_𝑖∈𝑅^𝑛, 0<|𝜆^𝑘_𝑖|<+∞.

Step 2. If Ψ(𝑥_𝑘)≤𝜖, stop.

Step 3. Select an element 𝑉_𝑘∈𝜕𝐺(𝑥_𝑘), and find an approximate solution 𝑑_𝑘∈𝑅^𝑛 of the system 𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘𝑑=−𝑉_𝑘^𝑇𝐺𝑥_𝑘,𝑉_𝑘∈𝜕𝐺(𝑥), (3.9) where 𝜆_𝑖^(𝑘) are the Levenberg-Marquardt parameters. If the condition ∇Ψ𝑥_𝑘^𝑇𝑑_𝑘≤−𝜌𝑑_𝑘^𝑝 (3.10) is not satisfied, set 𝑑_𝑘=−(𝑉_𝑘)^𝑇𝐺(𝑥_𝑘).

Step 4. Find the smallest 𝑖^𝑘∈{0,1,2,⋯} such that Ψ𝑥_𝑘+2^−𝑖𝑘𝑑_𝑘≤Ψ𝑥_𝑘+𝛽2^−𝑖𝑘∇Ψ𝑥_𝑘^𝑇𝑑_𝑘. (3.11) Set 𝑥_𝑘+1=𝑥_𝑘+2^−𝑖𝑘𝑑_𝑘, let 𝑘∶=𝑘+1, and go to Step 2.

Notice that if 𝑉_𝑘 is nonsingular in (3.9), the choice of 𝜆^𝑘_𝑖=0, at each step, is allowed by the above algorithm. Then (3.9) is equivalent to the generalized Newton equation in [4]. In what follows, as usual in analyzing the behavior of algorithms, we also assume that the above Levenberg-Marquardt method (II) produces an infinite sequence of points. Based upon the above analysis, we can get the following global convergence results about vertical complementarity problem (1.1). The main proof of the following theorem is similar to [7, Theorem 12]. But the system (3.9), which is used for the solution of 𝑑_𝑘, is different from [7, Theorem 12].

Theorem 3.5. Suppose that there exist constants 𝑀>0, ‖diag(𝜆_𝑖^(𝑘)𝑔_𝑖(𝑥))‖≤𝑀<+∞. Then each accumulation point of the sequence {𝑥_𝑘} generated by the above Levenberg-Marquardt method (II) is a stationary point of Ψ.

Proof. Assume that {𝑥_𝑘}_𝐾→𝑥^⋆. If there are infinitely many 𝑘∈𝐾 such that 𝑑_𝑘=−∇Ψ(𝑥_𝑘), then the assertion follows immediately from [5, Proposition 1.16]. Hence we can assume without loss of generality that if {𝑥_𝑘}_𝐾 is a convergent subsequence of {𝑥_𝑘}, then 𝑑_𝑘 is always given by (3.9). We show that for every convergent subsequence {𝑥_𝑘}_𝐾 for which lim_{𝑘∈𝐾,𝑘→∞}∇Ψ𝑥_𝑘≠0, (3.12) there holds limsup_{𝑘∈𝐾,𝑘→∞}𝑑_𝑘<∞,limsup_{𝑘∈𝐾,𝑘→∞}∇Ψ𝑥_𝑘^𝑇𝑑_𝑘>0 (3.13) In the following, we assume that 𝑥_𝑘→𝑥^⋆. Suppose that 𝑥^⋆ is not a stationary point of Ψ. By (3.9), we have ∇Ψ𝑥_𝑘=𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘𝑑_𝑘≤𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘𝑑_𝑘, (3.14) so 𝑑_𝑘≥∇Ψ𝑥_𝑘𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘. (3.15) Note that the denominator in the above inequality is nonzero; otherwise by (3.14), we have ‖∇Ψ(𝑥_𝑘)‖=0. 𝑥_𝑘 would be a stationary point and the algorithm would have stopped. By assumption ‖diag(𝜆_𝑖^(𝑘)𝑔_𝑖(𝑥))‖≤𝑀<+∞ and Proposition 2.4, there exists a constant 𝑘₁>0 such that 𝑉_𝑘^𝑇𝑉_𝑘+diag𝜆_𝑖^(𝑘)𝑔_𝑖𝑥_𝑘≤𝑘₁. (3.16) From the above inequality and (3.13), we get 𝑑_𝑘≥1−𝛼_𝑘𝑘₁∇Ψ𝑥_𝑘≥1−𝑎𝑘₁∇Ψ𝑥_𝑘. (3.17) Formula (3.13) now readily follows from the fact that we are assuming that the direction satisfies (3.10) with 𝑝>2, while the gradient ∇Ψ(𝑥_𝑘) is bounded on the convergent sequence {𝑥_𝑘}. If (12) is not satisfied, there exists a subsequence {𝑥_𝑘}_𝐾′ of {𝑥_𝑘}_𝐾lim_{𝑘∈𝐾′,𝑘→∞}∇Ψ𝑥_𝑘^𝑇𝑑_𝑘=0. (3.18) This implies, by (3.10), that lim_{𝑘∈𝐾′,𝑘→∞}‖𝑑_𝑘‖=0. Together with (3.17) one implies lim_{𝑘∈𝐾′,𝑘→∞}∇Ψ𝑥_𝑘=0 (3.19) contradicting (3.12). The sequence {𝑑_𝑘} is uniformly gradient related to {𝑥_𝑘} according to the definition given in [5], and the assertion of the theorem also follows from [5, Proposition 1.16]. We completed the proof.

Remark 3.6. In the Levenberg-Marquardt method (II), (3.11) can be replaced by the following nonmonotone line search: Ψ𝑥_𝑘+2^−𝑖𝑘𝑑_𝑘−max_{(𝑘)0≤𝑗≤𝑚}Ψ𝑥_𝑘−𝑗≤𝛽2^−𝑖𝑘∇Ψ𝑥_𝑘^𝑇𝑑_𝑘, (3.20) where 𝑚(0)=0, 𝑚(𝑘)=min{𝑚(𝑘−1)+1,𝑀₀}, and 𝑀₀ is a nonnegative integer.

Remark 3.7. Let the assumptions of Theorem 3.5 hold. Equation (3.9) is replaced by (3.3). Then each accumulation point of the sequence {𝑥_𝑘} generated by the above Levenberg-Marquardt method (II) is a stationary point of Ψ.

Remark 3.8. The assumption that there exist constants 𝑀>0, ‖diag(𝜆_𝑖^(𝑘)𝑔_𝑖(𝑥))‖≤𝑀<+∞ in Theorems 3.1, 3.5, and Remark 3.7 can be easily obtained, for the continuously differentiable of 𝐹 in (1.1).

Remark 3.9. Let 𝑥_⋆ be a stationary point of Ψ, 𝐹^′₂(𝑥_⋆) nonsingular, and 𝐹^′₁(𝑥_⋆)𝐹^′₂(𝑥_⋆)⁻¹ a 𝑃₀ matrix. Then 𝑥_⋆ is a solution of GCP(𝐹₁,𝐹₂).

Remark 3.10. We can use 𝜑_𝑀𝑆 in [20] to construct merit function Ψ in the Levenberg-Marquardt method (II).

Remark 3.11. We can use a family of new NCP functions based on the Fischer-Burmeister function to construct merit function in the Levenberg-Marquardt method (II), which is defined by 𝜙_𝑝(𝑎,𝑏)=‖(𝑎,𝑏)‖_𝑝−(𝑎+𝑏), (3.21) where 𝑝 is any fixed real number in the interval (1,+∞) and ‖(𝑎,𝑏)‖_𝑝 denotes the 𝑝-norm of (𝑎,𝑏). Numerical results based on the function 𝜙_𝑝 for the test problems from MCPLIB indicated that the algorithm has better performance in practice [21].

4. Numerical Tests and Final Remarks

In this section, in order to show the performance of the above Levenberg-Marquardt type methods, we present some numerical results for the Levenberg-Marquardt method (I) and the Levenberg-Marquardt method (II). The results indicate that the Levenberg-Marquardt algorithms work quite well in practice. We coded the algorithms in Matlab 7.0. Some remarks are also attached.

Example 4.1. We consider the following vertical complementarity problem: 𝐹_𝑖(𝑥)Π≥0,𝑖=1,…,𝑚,_𝑚𝑖=1𝐹^𝑗_𝑖(𝑥)=0,𝑗=1,…,𝑛, (4.1) where the functions 𝐹₁𝑥₁,𝑥₂=𝑥²₁,𝑥²₁+𝑥²₂^𝑇,𝐹₂𝑥₁,𝑥₂=2𝑥²₁1+6,2𝑥²₁^𝑇. (4.2) Both 𝐹₁ and 𝐹₂ are 𝑅²→𝑅² continuously differentiable functions.

We use the Levenberg-Marquardt method (I) to compute Example 4.1. Results for Example 4.1 with initial point 𝑥₀=(0.1,0.7)^𝑇, 𝛼_𝑘≡1 and 𝜆₁=0.01, 𝜆₂=0.01 are presented in Table 1.

Table 1 

We use the Levenberg-Marquardt method (I) to compute Example 4.1. Results for Example 4.1 with initial point 𝑥₀=(10,10)^𝑇, and 𝑥₀=(100,100)^𝑇, 𝛼_𝑘≡1 and 𝜆₁=0.01, 𝜆₂=0.01 are presented in Table 2.

Table 2 

When we compute 𝑑_𝑘 by (3.3) in the Levenberg-Marquardt method (I) to compute Example 4.1. Results for Example 4.1 with initial point 𝑥₀=(0.1,0.7)^𝑇, 𝑥₀=(10,10)^𝑇, and 𝑥₀=(100,100)^𝑇, 𝛼_𝑘≡1 and 𝜆₁=0.01, 𝜆₂=0.01 are presented in Table 3.

Table 3 

Remark 4.2. From the numerical results for the Levenberg-Marquardt method (I) in Table 1, Table 2, and Table 3, we can see that the modification of (3.4) in the Levenberg-Marquardt method (I) works quite better in practice than (3.3) which has been used in [7].

Example 4.3. We consider the following vertical complementarity problem: 𝐹_𝑖(𝑥)Π≥0,𝑖=1,…,𝑚,_𝑚𝑖=1𝐹^𝑗_𝑖(𝑥)=0,𝑗=1,…,𝑛, (4.3) where the functions 𝐹₁𝑥₁,𝑥₂=𝑥²₁+𝑥²₂,𝑥²₂^𝑇,𝐹₂𝑥₁,𝑥₂=𝑥²₁,𝑥²₁+12𝑥²₂+1^𝑇. (4.4) Both 𝐹₁ and 𝐹₂ are 𝑅²→𝑅² continuously differentiable functions.
Now, we use the Levenberg-Marquardt method (II) to compute Example 4.3. For the special construction of Example 4.3, we can use (2.6) to get the Clarke generalized Jacobian in the Levenberg-Marquardt method (II). Results for Example 4.3 with initial point 𝑥₀=(1,1)^𝑇, 𝑥₀=(10,0.1)^𝑇, and 𝑥₀=(100,10)^𝑇 and 𝜆₁=0.01, 𝜆₂=1 are presented in Table 4.

Table 4 

Then we compute 𝑑_𝑘 by (3.3) in the Levenberg-Marquardt method (II) to compute Example 4.3. Results for Example 4.3 with initial point 𝑥₀=(1,1)^𝑇, 𝑥₀=(10,0.1)^𝑇, and 𝑥₀=(100,10)^𝑇 and 𝜆₁=0.01, 𝜆₂=1 are presented in Table 5.

Table 5 

Remark 4.4. From the numerical results for in the Levenberg-Marquardt method (II) in Table 4 and Table 5, we can see that the modification of (3.9) in the Levenberg-Marquardt method (II) works quite as well as (3.3) which has been used in [7].

Acknowledgments

This work is supported by National Science Foundation of China (10971118, 11171221, 11101231), a project of Shandong province Higher Education Science and Technology Program (J10LA05), Shanghai Municipal Committee of Science and Technology (10550500800), Innovation Program of Shanghai Municipal Education Commission (10YZ99).