A globally convergent damped Gauss–Newton method for solving the extended linear complementarity problem

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Newton Method ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Optimization Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
System Of Equations ◽  
Extended Linear Complementarity Problem ◽  
Special Cases

AbstractThe extended linear complementarity problem (denoted by XLCP), of which the linear and horizontal linear complementarity problems are two special cases, can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed smoothing Fischer-Burmeister function, the XLCP is approximated by a family of parameterized smoothness optimization problems. Asmoothing damped Gauss-Newton method is designed for solving the XLCP. The proposed algorithm is proved to be convergent globally under suitable assumptions. Some numerical results are reported in the paper

scholarly journals Two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2171-2184
Author(s):  
Lu Jia ◽  
Xiang Wang ◽  
Xuan-Sheng Wang
Keyword(s):  
Theoretical Analysis ◽  
Linear Complementarity Problem ◽  
Iteration Method ◽  
Complementarity Problem ◽  
Numerical Experiments ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Matrix Splitting ◽  
Splitting Iteration Method

The modulus-based matrix splitting iteration has received substantial attention as a momentous tool for complementarity problems. For the purpose of solving the horizontal linear complementarity problem, we introduce the two-step modulus-based matrix splitting iteration method. We also show the theoretical analysis of the convergence. Numerical experiments illustrate the effectiveness of the proposed approach.

A New Searching Direction for Linear Complementarity Problems

2011 ◽  
Vol 219-220 ◽  
pp. 1089-1092
Author(s):  
Li Pu Zhang ◽  
Ying Hong Xu
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Quadratic Convergence ◽  
Complementarity Problems ◽  
Convergence Property ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Simple Function ◽  
Interior Point Algorithm ◽  
Local Quadratic Convergence

In this paper, we investigate the properties of a simple function. As an application, we present a full-step interior-point algorithm for linear complementarity problem. The algorithm uses the simple function to determine the searching direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by this simple function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bounds for linear complementarity problem.

scholarly journals The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Hongchun Sun ◽  
Yiju Wang
Keyword(s):  
Error Bound ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Solution Set ◽  
Global Error Bound ◽  
Mixed Linear Complementarity Problem

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

scholarly journals Extensions of P-property, R0-property and semidefinite linear complementarity problems

2017 ◽  
Vol 27 (2) ◽  
pp. 135-151
Author(s):  
I. Jeyaraman ◽  
Kavita Bisht ◽  
K.C. Sivakumar
Keyword(s):  
Linear Transformation ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Linear Transformations ◽  
Semidefinite Linear Complementarity Problems ◽  
Semidefinite Linear Complementarity Problem

In this manuscript, we present some new results for the semidefinite linear complementarity problem, in the context of three notions for linear transformations, viz., pseudo w-P property, pseudo Jordan w-P property and pseudo SSM property. Interconnections with the P#-property (proposed recently in the literature) is presented. We also study the R#-property of a linear transformation, extending the rather well known notion of an R0-matrix. In particular, results are presented for the Lyapunov, Stein, and the multiplicative transformations

scholarly journals A Preconditioned Multisplitting and Schwarz Method for Linear Complementarity Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Cuiyu Liu ◽  
Chen-liang Li
Keyword(s):  
Convergence Rate ◽  
Iterative Methods ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Numerical Experiments ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Coefficient Matrix ◽  
Schwarz Method

The preconditioner presented by Hadjidimos et al. (2003) can improve on the convergence rate of the classical iterative methods to solve linear systems. In this paper, we extend this preconditioner to solve linear complementarity problems whose coefficient matrix isM-matrix orH-matrix and present a multisplitting and Schwarz method. The convergence theorems are given. The numerical experiments show that the methods are efficient.

SOLVING STRONGLY MONOTONE LINEAR COMPLEMENTARITY PROBLEMS

2013 ◽  
Vol 15 (04) ◽  
pp. 1340035 ◽  
Author(s):  
A. CHANDRASHEKARAN ◽  
T. PARTHASARATHY ◽  
V. VETRIVEL
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Inner Product ◽  
General Linear ◽  
Strongly Monotone ◽  
Semidefinite Linear Complementarity Problems ◽  
Lipschitz Constants

Given a linear transformation L on a finite dimensional real inner product space V to itself and an element q ∈ V we consider the general linear complementarity problem LCP (L, K, q) on a proper cone K ⊆ V. We observe that the iterates generated by any closed algorithmic map will converge to a solution for LCP (L, K, q), whenever L is strongly monotone. Lipschitz constants of L is vital in establishing the above said convergence. Hence we compute the Lipschitz constants for certain classes of Lyapunov, Stein and double-sided multiplicative transformations in the setting of semidefinite linear complementarity problems. We give a numerical illustration of a closed algorithmic map in the setting of a standard linear complementarity problem. On account of the difficulties in numerically implementing such algorithms for general linear complementarity problems, we give an alternative algorithm for computing the solution for a special class of strongly monotone semidefinite linear complementarity problems along with a numerical example.

scholarly journals A new error bound for linear complementarity problems for B-matrices

2016 ◽  
Vol 31 ◽  
pp. 476-484 ◽  
Author(s):  
Chaoqian Li ◽  
Mengting Gan ◽  
Shaorong Yang
Keyword(s):  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Appl Math

A new error bound for the linear complementarity problem is given when the involved matrix is a $B$-matrix. It is shown that this bound improves the corresponding result in [M. Garc\'{i}a-Esnaola and J.M. Pe\~{n}a. Error bounds for linear complementarity problems for $B$-matrices. {\em Appl. Math. Lett.}, 22:1071--1075, 2009.] in some cases, and that it is sharper than that in [C.Q. Li and Y.T. Li. Note on error bounds for linear complementarity problems for $B$-matrices. {\em Appl. Math. Lett.}, 57:108--113, 2016.].

scholarly journals New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 1599-1614
Author(s):  
Zhiwu Hou ◽  
Xia Jing ◽  
Lei Gao
Keyword(s):  
Linear Algebra ◽  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Numerical Examples ◽  
Numer Algor ◽  
Better Than

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

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