scholarly journals A Smoothing Method with Appropriate Parameter Control Based on Fischer-Burmeister Function for Second-Order Cone Complementarity Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Yasushi Narushima ◽  
Hideho Ogasawara ◽  
Shunsuke Hayashi
Keyword(s):  
Complementarity Problem ◽  
Quadratic Convergence ◽  
Complementarity Problems ◽  
Optimization Problems ◽  
Second Order ◽  
Smoothing Newton Method ◽  
System Of Equations ◽  
Important Class ◽  
Nonsmooth System ◽  
Second Order Cone

We deal with complementarity problems over second-order cones. The complementarity problem is an important class of problems in the real world and involves many optimization problems. The complementarity problem can be reformulated as a nonsmooth system of equations. Based on the smoothed Fischer-Burmeister function, we construct a smoothing Newton method for solving such a nonsmooth system. The proposed method controls a smoothing parameter appropriately. We show the global and quadratic convergence of the method. Finally, some numerical results are given.

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 The Monotone Extended Second-Order Cone and Mixed Complementarity Problems

2021 ◽  
Author(s):  
Yingchao Gao ◽  
Sándor Zoltán Németh ◽  
Roman Sznajder
Keyword(s):  
Complementarity Problem ◽  
Complementarity Problems ◽  
Nonlinear Complementarity Problem ◽  
Second Order ◽  
Mixed Complementarity Problem ◽  
Second Order Cone ◽  
Mixed Complementarity Problems ◽  
Lorentz Cone ◽  
Lyapunov Rank ◽  
Extended Second Order Cone

AbstractIn this paper, we study a new generalization of the Lorentz cone $$\mathcal{L}^n_+$$ L + n , called the monotone extended second-order cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a nonlinear complementarity problem on a cylinder, which is equivalent to a suitable mixed complementarity problem, and provide a computational example illustrating applicability of MESOC.

scholarly journals A smoothing Newton method for the second-order cone complementarity problem

2013 ◽  
Vol 58 (2) ◽  
pp. 223-247 ◽  
Author(s):  
Jingyong Tang ◽  
Guoping He ◽  
Li Dong ◽  
Liang Fang ◽  
Jinchuan Zhou
Keyword(s):  
Newton Method ◽  
Complementarity Problem ◽  
Second Order ◽  
Smoothing Newton Method ◽  
Second Order Cone

scholarly journals Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wenli Liu ◽  
Xiaoni Chi ◽  
Qili Yang ◽  
Ranran Cui
Keyword(s):  
Complementarity Problem ◽  
Complementarity Problems ◽  
Second Order ◽  
Smoothing Function ◽  
Rapid Convergence ◽  
Smoothing Methods ◽  
Complementarity Function ◽  
Second Order Cone ◽  
Computable Formula

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

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