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

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Shou-qiang Du ◽  
Yan Gao
Keyword(s):  
Global Convergence ◽  
Complementarity Problem ◽  
Numerical Experiments ◽  
Convergence Results ◽  
Nonsmooth Equation ◽  
Levenberg Marquardt ◽  
Local And Global Convergence

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.

scholarly journals Generalized Newton Method for a Kind of Complementarity Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shou-qiang Du
Keyword(s):  
Global Convergence ◽  
Newton Method ◽  
Differentiable Function ◽  
Local Convergence ◽  
Complementarity Problem ◽  
Merit Function ◽  
Nonsmooth Equations ◽  
Generalized Newton Method ◽  
Convergence Results ◽  
Generalized Newton

A generalized Newton method for the solution of a kind of complementarity problem is given. The method is based on a nonsmooth equations reformulation of the problem byF-Bfunction and on a generalized Newton method. The merit function used is a differentiable function. The global convergence and superlinear local convergence results are also given under suitable assumptions. Finally, some numerical results and discussions are presented.

scholarly journals Levenberg-Marquardt Method for the Eigenvalue Complementarity Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yuan-yuan Chen ◽  
Yan Gao
Keyword(s):  
Complementarity Problem ◽  
Numerical Experiments ◽  
Nonlinear Complementarity Problem ◽  
Mathematical Programming Problem ◽  
Complementarity Constraints ◽  
Marquardt Method ◽  
Globally Convergent ◽  
Eigenvalue Complementarity Problem ◽  
Semismooth Newton ◽  
Levenberg Marquardt

The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.

scholarly journals A Modified Nonsmooth Levenberg–Marquardt Algorithm for the General Mixed Complementarity Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Linsen Song ◽  
Yan Gao
Keyword(s):  
Complementarity Problem ◽  
Local Error ◽  
Maximum Function ◽  
Local Error Bound ◽  
Mixed Complementarity Problem ◽  
Nonsmooth Equation ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
Median Function ◽  
Vector Valued

As is well known, the mixed complementarity problem is equivalent to a nonsmooth equation by using a median function. By investigating the generalized Jacobi of a composite vector-valued maximum function, a nonsmooth Levenberg–Marquardt algorithm is proposed in this paper. In the present algorithm, we adopt a new LM parameter form and discuss the local convergence rate under the local error bound condition, which is weaker than nonsingularity. Finally, the numerical experiments and the application for the real-time pricing in smart grid illustrate the effectiveness of the algorithm.

A Non-Monotone Line Search Combination Technique for Unconstrained Optimization

2014 ◽  
Vol 8 (1) ◽  
pp. 218-221 ◽  
Author(s):  
Ping Hu ◽  
Zong-yao Wang
Keyword(s):  
Global Convergence ◽  
Unconstrained Optimization ◽  
Numerical Experiments ◽  
Line Search ◽  
Optimization Problems ◽  
Search Algorithm ◽  
New Combination ◽  
Combination Rule ◽  
Combination Technique ◽  
Unconstrained Optimization Problems

We propose a non-monotone line search combination rule for unconstrained optimization problems, the corresponding non-monotone search algorithm is established and its global convergence can be proved. Finally, we use some numerical experiments to illustrate the new combination of non-monotone search algorithm’s effectiveness.

scholarly journals The Picard-HSS-SOR iteration method for absolute value equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lin Zheng
Keyword(s):  
Iteration Method ◽  
Numerical Experiments ◽  
Absolute Value Equation ◽  
Absolute Value Equations ◽  
Absolute Value ◽  
Convergence Results ◽  
Hss Iteration ◽  
The Absolute ◽  
Hss Iteration Method

AbstractIn this paper, we present the Picard-HSS-SOR iteration method for finding the solution of the absolute value equation (AVE), which is more efficient than the Picard-HSS iteration method for AVE. The convergence results of the Picard-HSS-SOR iteration method are proved under certain assumptions imposed on the involved parameter. Numerical experiments demonstrate that the Picard-HSS-SOR iteration method for solving absolute value equations is feasible and effective.

scholarly journals Strict lower subdifferentiability and applications

1999 ◽  
Vol 40 (3) ◽  
pp. 379-391 ◽  
Author(s):  
H. Xu ◽  
A. M. Rubinov ◽  
B. M. Glover
Keyword(s):  
Global Convergence ◽  
Optimality Conditions ◽  
Convex Subset ◽  
Sufficient Optimality Conditions ◽  
Descent Direction ◽  
Minimization Problems ◽  
Convergence Results ◽  
Lower Subdifferential ◽  
Necessary And Sufficient ◽  
Convex Subdifferential

AbstractWe investigate the strict lower subdifferentiability of a real-valued function on a closed convex subset of Rn. Relations between the strict lower subdifferential, lower subdifferential, and the usual convex subdifferential are established. Furthermore, we present necessary and sufficient optimality conditions for a class of quasiconvex minimization problems in terms of lower and strict lower subdifferentials. Finally, a descent direction method is proposed and global convergence results of the consequent algorithm are obtained.

Verification of Iterative Methods for the Linear Complementarity Problem

2016 ◽  
pp. 545-580 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Linear Complementarity Problem ◽  
Numerical Experiments ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Convergence Results ◽  
Fixed Point Principle ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods

In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.

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