Generalized Quasi-Newton Method for Variational Inequality Problems

2013 ◽  
Vol 8 (7) ◽  
pp. 1085-1092
Author(s):  
Zhang Bo ◽  
Hua Qiang ◽  
Wang Xizhao
Keyword(s):  
Variational Inequality ◽  
Newton Method ◽  
Variational Inequality Problems ◽  
Quasi Newton

A Hybrid Newton Method for Stochastic Variational Inequality Problems and Application to Traffic Equilibrium

2020 ◽  
pp. 2050036
Author(s):  
Yan-Chao Liang ◽  
Qiao-Na Fan ◽  
Pei-Ping Shen
Keyword(s):  
Variational Inequality ◽  
Newton Method ◽  
Sample Average Approximation ◽  
Gap Function ◽  
Variational Inequality Problems ◽  
Traffic Equilibrium ◽  
Stochastic Variational Inequality ◽  
Sample Average ◽  
Optimization Reformulation ◽  
Average Approximation

In this paper, we consider a class of stochastic variational inequality problems (SVIPs). Different from the classical variational inequality problems, the SVIP contains a mathematical expectation, which may not be evaluated in an explicit form in general. We combine a hybrid Newton method for deterministic cases with an unconstrained optimization reformulation based on the well-known D-gap function and sample average approximation (SAA) techniques to present an SAA-based hybrid Newton method for solving the SVIP. We show that the level sets of the approximation D-gap function are bounded. Furthermore, we prove that the sequence generated by the hybrid Newton method converges to a solution of the SVIP under appropriate conditions, and some numerical experiments are presented to prove the effectiveness and competitiveness of the hybrid Newton method. Finally, we apply this method to solve two specific traffic equilibrium problems.

scholarly journals Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
B. Borsos ◽  
János Karátson
Keyword(s):  
Finite Element ◽  
Newton Method ◽  
Newton's Method ◽  
Numerical Experiments ◽  
Elliptic Problems ◽  
Iterative Solvers ◽  
Finite Element Solution ◽  
Element Solution ◽  
Nonlinear Beam ◽  
Quasi Newton

Abstract The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.

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