scholarly journals Stochastic Variational Inequality Problems: Applications, Analysis, and Algorithms

2013 ◽  
pp. 71-107 ◽  
Author(s):  
Uday V. Shanbhag
Keyword(s):  
Variational Inequality ◽  
Variational Inequality Problems ◽  
Stochastic Variational Inequality

SAMPLE AVERAGE APPROXIMATION METHOD FOR SOLVING A DETERMINISTIC FORMULATION FOR BOX CONSTRAINED STOCHASTIC VARIATIONAL INEQUALITY PROBLEMS

2012 ◽  
Vol 29 (02) ◽  
pp. 1250014
Author(s):  
MEI-JU LUO ◽  
GUI-HUA LIN
Keyword(s):  
Variational Inequality ◽  
Approximation Method ◽  
Deterministic Model ◽  
Random Parameter ◽  
Sample Average Approximation ◽  
Variational Inequality Problems ◽  
Stochastic Variational Inequality ◽  
Sample Average ◽  
Sample Average Approximation Method ◽  
Average Approximation

In this paper, we discuss the Expected Residual Minimization (ERM) method, which is to minimize the expected residue of some merit function for box constrained stochastic variational inequality problems (BSVIPs). This method provides a deterministic model, which formulates BSVIPs as an optimization problem. We first study the conditions under which the level sets of the ERM problem are bounded. Then, we show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in BSVIPs. Since the integrality involved in the ERM problem is difficult to compute generally, we then employ sample average approximation method to solve it. Finally, we show that the global optimal solutions and generalized KKT points of the approximate problems converge to their counterparts of the ERM problem. On the other hand, as an application, we consider the model of European natural gas market under price uncertainty. Preliminary numerical experiments indicate that the proposed approach is applicable.

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.

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