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:

()

where the functions F_i(x) : Rⁿ → Rⁿ are assumed to be continuously differentiable and

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

()

x solves (1.1), where “min” denotes the componentwise minimum operator,

, 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

()

Thus, (1.2) can be briefly rewritten as

()

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

2. Preliminaries

A locally Lipschitz function H : Rⁿ → Rⁿ is said to be semismooth at x provided that the following limit

()

exists for any h ∈ Rⁿ. 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.

()

(iii)
One has

()

If for any V ∈ ∂H(x + h), h → 0,

()

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 → 0

()

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):

()

Obviously, ∂_⋆G(x) is a finite set of points and can be easily calculated by determining J_i(x), i = 1, …, n, and

, 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

()

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 ∈ Rⁿ, 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 ∈ Rⁿ, let

()

where d_k is the solution of the system

()

In the inexact versions of this method, d_k can be given by the solution of the system

()

where r_k is the vector of residuals, and we can assume 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, , .

Step 2. Solve the system to get d_k,

()

for i = 1, ⋯n, and r_k is the vector of residuals

()

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, . 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, . 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_⋆ ∈ Rⁿ satisfying

()

where F_i : Rⁿ → Rⁿ, 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 function

()

Then G(x) = φ(F₁, F₂), and we denote the corresponding merit function as

()

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 ∈ Rⁿ, ρ > 0, p > 2, β ∈ (0, 1/2), ϵ ≥ 0, , .

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 ∈ Rⁿ of the system

()

where

are the Levenberg-Marquardt parameters. If the condition

()

is not satisfied, set d_k = −(V_k) ^TG(x_k).

Step 4. Find the smallest i^k ∈ {0,1, 2, ⋯} such that

()

Set

, let k : = k + 1, and go to Step 2.

Notice that if V_k is nonsingular in (3.9), the choice of , 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, . 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

()

there holds

()

In the following, we assume that x_k → x^⋆. Suppose that x^⋆ is not a stationary point of Ψ. By (3.9), we have

()

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

and Proposition 2.4, there exists a constant k₁ > 0 such that

()

From the above inequality and (3.13), we get

()

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

of {x_k} _K

()

This implies, by (3.10), that

. Together with (3.17) one implies

()

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:

()

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, 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, nonsingular, and 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

()

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:

()

where the functions

()

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.

Step	x_k	G(x)
1	(0.1000000,0.7000000)^T	(0.0100000,0.0050000)^T
2	(0.0500998,0.7000000)^T	(0.0025099,0.0012549)^T
3	(0.0250999,0.7000000)^T	1.0e − 003*(0.630004,0.315002)^T
4	(0.0125749,0.7000000)^T	1.0e − 003*(0.158130,0.079065)^T
5	(0.0063000,0.7000000)^T	1.0e − 004*(0.396906,0.198453)^T
6	(0.0031563,0.7000000)^T	1.0e − 005*(0.996230,0.498115)^T
7	(0.0015813,0.7000000)^T	1.0e − 005*(0.250052,0.125026)^T
8	(0.0007922,0.7000000)^T	1.0e − 006*(0.627630,0.313815)^T
9	(0.0003969,0.7000000)^T	1.0e − 006*(0.157534,0.078767)^T
10	(0.0001988,0.7000000)^T	1.0e − 007*(0.395410,0.197705)^T
11	(0.0000996,0.7000000)^T	1.0e − 008*(0.992475,0.496237)^T

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.

Step	x_k	G(x)
x₀ = (10,10) ^T
1	(10.000000,10.0000000)^T	(100.000000,25.000000)^T
2	(5.009980,10.0000000)^T	(25.099900,12.549950)^T
3	(2.509990,10.0000000)^T	(6.300049;3.150024)^T
4	(1.257499,10.0000000)^T	(1.581306,0.7906530)^T
5	(0.630004;10.0000000)^T	(0.396906,0.198453)^T
6	(0.315631,10.0000000)^T	(0.099623,0.049811)^T
7	(0.158130,10.0000000)^T	(0.025005,0.012502)^T
8	(0.079223,10.0000000)^T	(0.006276, 0.003138)T
9	(0.039690,10.0000000)^T	(0.001575,0.000787)^T
10	(0.019884,10.0000000)^T	1.0e − 003*(0.395410,0.197705)^T
11	(0.009962,10.0000000)^T	1.0e − 004*(0.992475,0.496237)^T
12	(0.004991,10.0000000)^T	1.0e − 004*(0.249110,0.124555)^T
13	(0.002500,10.0000000)^T	1.0e − 005*(0.625264,0.312632)^T
14	(0.001252,10.0000000)^T	1.0e − 005*(0.156940,0.078470)^T
15	(0.000627,10.0000000)^T	1.0e − 006*(0.393919,0.196959)^T
16	(0.000314,10.0000000)^T	1.0e − 007*(0.988734,0.494367)^T
17	(0.000157,10.0000000)^T	1.0e − 007*(0.248171,0.124085)^T
18	(0.0000789,10.0000000)^T	1.0e − 008*(0.6229079,0.3114539)^T

x₀ = (100,100) ^T
21	1.0e + 002*(0.0000009,1.0000000)^T	1.0e − 008*(0.985008,0.492504)^T

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.

Step	x_k	G(x)
x₀ = (0.1,0.7) ^T
40	(0.007326,0.700000) ^T	1.0e − 004*(0.536774,0.268387) ^T

x₀ = (10,10) ^T
47	(0.007301,10.000000) ^T	1.0e − 004*(0.533144,0.266572) ^T

x₀ = (100,100) ^T
50	1.0e + 002*(0.000073,1.000000) ^T	1.0e − 004*(0.537728,0.268864) ^T

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:

()

where the functions

()

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.

Step	x_k	G(x)
x₀ = (1,1) ^T
3	(−0.001250,1.000000) ^T	1.0e − 005*(0.156444,0.156444) ^T

x₀ = (10,0.1) ^T
15	(0.099976,0.100000) ^T	(0.009995,0.009995) ^T

x₀ = (100,10) ^T
17	(0.096868,10.000000) ^T	(0.009383,0.009383) ^T

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.

Step	x_k	G(x)
x₀ = (1,1) ^T
3	(−0.001250,1.000000) ^T	1.0e − 005*(0.156444,0.156444) ^T

x₀ = (10,0.1) ^T
15	(0.099989,0.100000) ^T	(0.009997,0.009997) ^T

x₀ = (100,10) ^T
17	(0.097467,10.000000) ^T	(0.009499,0.009499) ^T

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).

References

1 Kanzow C. and Fukushima M., Equivalence of the generalized complementarity problem to differentiable unconstrained minimization, Journal of Optimization Theory and Applications. (1996) 90, no. 3, 581–603, 1402624, https://doi.org/10.1007/BF02189797, ZBL0866.90124.
10.1007/BF02189797
Web of Science® Google Scholar
2 Pang J.-S. and Qi L. Q., Nonsmooth equations: motivation and algorithms, SIAM Journal on Optimization. (1993) 3, no. 3, 443–465, 1230150, https://doi.org/10.1137/0803021, ZBL0784.90082.
10.1137/0803021
Web of Science® Google Scholar
3 Qi L. and Tseng P., On almost smooth functions and piecewise smooth functions, Nonlinear Analysis, Theory, Methods & Applications. (2007) 67, no. 3, 773–794, https://doi.org/10.1016/j.na.2006.06.029, 2319208, ZBL1125.26019.
10.1016/j.na.2006.06.029
Web of Science® Google Scholar
4 Qi L. Q. and Sun J., A nonsmooth version of Newton′s method, Mathematical Programming. (1993) 58, no. 3, 353–367, 1216791, https://doi.org/10.1007/BF01581275, ZBL0780.90090.
10.1007/BF01581275
Web of Science® Google Scholar
5 Bertsekas D. P., Constrained Optimization and Lagrange Multiplier Methods, 1982, Academic Press, New York, NY, USA, Computer Science and Applied Mathematics, 690767.
Google Scholar
6 Gao Y., Newton methods for solving two classes of nonsmooth equations, Applications of Mathematics. (2001) 46, no. 3, 215–229, 1828306, https://doi.org/10.1023/A:1013791923957, ZBL1068.65063.
10.1023/A:1013791923957
Google Scholar
7 Facchinei F. and Kanzow C., A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Mathematical Programming. (1997) 76, no. 3, 493–512, https://doi.org/10.1016/S0025-5610(96)00058-5, 1433968, ZBL0871.90096.
10.1007/BF02614395
Web of Science® Google Scholar
8 Fischer A., Jeyakumar V., and Luc D. T., Solution point characterizations and convergence analysis of a descent algorithm for nonsmooth continuous complementarity problems, Journal of Optimization Theory and Applications. (2001) 110, no. 3, 493–513, 1854013, https://doi.org/10.1023/A:1017580126509, ZBL1064.90048.
10.1023/A:1017580126509
Web of Science® Google Scholar
9 Du S. Q. and Gao Y., A modified Levenberg-Marquardt method for nonsmooth equations with finitely many maximum functions, Mathematical Problems in Engineering. (2008) 2008, 10, 942391, https://doi.org/10.1155/2008/942391, 2476501.
10.1155/2008/942391
Web of Science® Google Scholar
10 Yamashita N. and Fukushima M., Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming. (1997) 76, no. 3, 469–491, https://doi.org/10.1016/S0025-5610(96)00057-3, 1433967, ZBL0872.90102.
10.1007/BF02614394
Web of Science® Google Scholar
11 Śmietański M. J., An approximate Newton method for non-smooth equations with finite max functions, Numerical Algorithms. (2006) 41, no. 3, 219–238, 2222248, https://doi.org/10.1007/s11075-005-9009-z, ZBL1141.65031.
10.1007/s11075-005-9009-z
Web of Science® Google Scholar
12 Ma C. and Tang J., The quadratic convergence of a smoothing Levenberg-Marquardt method for nonlinear complementarity problem, Applied Mathematics and Computation. (2008) 197, no. 2, 566–581, 2400679, https://doi.org/10.1016/j.amc.2007.07.060, ZBL1141.65044.
10.1016/j.amc.2007.07.060
Web of Science® Google Scholar
13 Kanzow C., Yamashita N., and Fukushima M., Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints, Journal of Computational and Applied Mathematics. (2004) 172, no. 2, 375–397, 2095326, https://doi.org/10.1016/j.cam.2004.02.013, ZBL1064.65037.
10.1016/j.cam.2004.02.013
Web of Science® Google Scholar
14 Yamashita N. and Fukushima M., On the rate of convergence of the Levenberg-Marquardt method, Computing. (2001) 15, 227–238, 1874516, https://doi.org/10.1007/978-3-7091-6217-0_18, ZBL1001.65047.
10.1007/978?3?7091?6217?0_18
Google Scholar
15 Fan J. and Pan J., Convergence properties of a self-adaptive Levenberg-Marquardt algorithm under local error bound condition, Computational Optimization and Applications. (2006) 34, no. 1, 47–62, 2224974, https://doi.org/10.1007/s10589-005-3074-z, ZBL1111.90103.
10.1007/s10589-005-3074-z
Web of Science® Google Scholar
16 Ueda K. and [email protected], Yamashita N., [email protected], On a Global Complexity Bound of the Levenberg-Marquardt Method, Journal of Optimization Theory and Applications. (2010) 147, no. 3, 443–453, https://doi.org/10.1007/s10957-010-9731-0.
10.1007/s10957-010-9731-0
Web of Science® Google Scholar
17 Polyak R. A., Regularized Newton method for unconstrained convex optimization, Mathematical Programming. (2009) 120, no. 1, 125–145, https://doi.org/10.1007/s10107-007-0143-3, 2496429, ZBL1189.90121.
10.1007/s10107-007-0143-3
Web of Science® Google Scholar
18 Ueda K. and Yamashita N., Convergence properties of the regularized Newton method for the unconstrained nonconvex optimization, Applied Mathematics and Optimization. (2010) 62, no. 1, 27–46, https://doi.org/10.1007/s00245-009-9094-9, 2653894.
10.1007/s00245-009-9094-9
Web of Science® Google Scholar
19 Yin H., Huang Z.-H., and Qi L., The convergence of a Levenberg-Marquardt method for nonlinear inequalities, Numerical Functional Analysis and Optimization. (2008) 29, no. 5-6, 687–716, 2417950, https://doi.org/10.1080/01630560802099936, ZBL1191.90075.
10.1080/01630560802099936
Web of Science® Google Scholar
20 Mangasarian O. L. and Solodov M. V., Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming. (1993) 62, no. 2, 277–297, https://doi.org/10.1007/BF01585171, 1247618, ZBL0813.90117.
10.1007/BF01585171
Web of Science® Google Scholar
21 Chen J. S. and Pan S., A family of NCP functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications. (2008) 40, no. 3, 389–404, 2411201, https://doi.org/10.1007/s10589-007-9086-0, ZBL1153.90542.
10.1007/s10589-007-9086-0
Web of Science® Google Scholar

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

Abstract

1. Introduction

2. Preliminaries

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

3.1. The Levenberg-Marquardt Method (I)

3.2. The Levenberg-Marquardt Method (II)

4. Numerical Tests and Final Remarks

Acknowledgments

References

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References

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

Abstract

1. Introduction

2. Preliminaries

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

3.1. The Levenberg-Marquardt Method (I)

3.2. The Levenberg-Marquardt Method (II)

4. Numerical Tests and Final Remarks

Acknowledgments

References

Citing Literature

References

Related

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