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.

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.

SAMPLE AVERAGE APPROXIMATION METHODS FOR A CLASS OF STOCHASTIC VARIATIONAL INEQUALITY PROBLEMS

2010 ◽  
Vol 27 (01) ◽  
pp. 103-119 ◽  
Author(s):  
HUIFU XU
Keyword(s):  
Variational Inequality ◽  
Complementarity Problems ◽  
Sample Average Approximation ◽  
Variational Inequality Problems ◽  
Stochastic Programs ◽  
Stochastic Variational Inequality ◽  
Convergence Results ◽  
Sample Average ◽  
Stochastic Constraints ◽  
Average Approximation

In this paper we apply the well known sample average approximation (SAA) method to solve a class of stochastic variational inequality problems (SVIPs). We investigate the existence and convergence of a solution to the sample average approximated SVIP. Under some moderate conditions, we show that the sample average approximated SVIP has a solution with probability one and with probability approaching one exponentially fast with the increase of sample size, the solution converges to its true counterpart. Finally, we apply the existence and convergence results to SAA method for solving a class of stochastic nonlinear complementarity problems and stochastic programs with stochastic constraints.

scholarly journals A New Gap Function for Vector Variational Inequalities with an Application

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hui-qiang Ma ◽  
Nan-jing Huang ◽  
Meng Wu ◽  
Donal O'Regan
Keyword(s):  
Variational Inequality ◽  
Optimization Problem ◽  
Dimensional Space ◽  
Sample Average Approximation ◽  
Gap Function ◽  
Vector Variational Inequality ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Sample Average ◽  
Stochastic Vector

We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.

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