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.

scholarly journals The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoni Chi ◽  
Zhongping Wan ◽  
Zijun Hao
Keyword(s):  
Directional Derivative ◽  
Second Order ◽  
Smoothing Function ◽  
Smoothing Methods ◽  
Second Order Cone ◽  
Complementarity Functions ◽  
Computable Formula ◽  
Smoothing Functions ◽  
The One ◽  
Parametric Class

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.

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.

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.

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