A New Unconstrained Differentiable Merit Function for Box Constrained Variational Inequality Problems and a Damped Gauss--Newton Method

1999 ◽  
Vol 9 (2) ◽  
pp. 388-413 ◽  
Author(s):  
Defeng Sun ◽  
Robert S. Womersley
Keyword(s):  
Variational Inequality ◽  
Newton Method ◽  
Merit Function ◽  
Variational Inequality Problems

scholarly journals Monte Carlo Sampling Method for a Class of Box-Constrained Stochastic Variational Inequality Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Pei-Yu Li
Keyword(s):  
Monte Carlo ◽  
Variational Inequality ◽  
Optimization Problem ◽  
Sampling Method ◽  
Monte Carlo Sampling ◽  
Merit Function ◽  
Variational Inequality Problems ◽  
Sufficient Condition ◽  
Existence Of A Solution ◽  
Stochastic Variational Inequality

This paper uses a merit function derived from the Fishcher–Burmeister function and formulates box-constrained stochastic variational inequality problems as an optimization problem that minimizes this merit function. A sufficient condition for the existence of a solution to the optimization problem is suggested. Finally, this paper proposes a Monte Carlo sampling method for solving the problem. Under some moderate conditions, comprehensive convergence analysis is included as well.

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.

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