Sample average approximation method for a class of stochastic variational inequality problems

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.

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A sample average approximation method based on a D-gap function for stochastic variational inequality problems

Journal of Industrial & Management Optimization ◽

10.3934/jimo.2014.10.977 ◽

2014 ◽

Vol 10 (3) ◽

pp. 977-987 ◽

Cited By ~ 3

Author(s):

Suxiang He ◽

◽

Pan Zhang ◽

Xiao Hu ◽

Rong Hu

Keyword(s):

Variational Inequality ◽

Approximation Method ◽

Sample Average Approximation ◽

Gap Function ◽

Variational Inequality Problems ◽

Stochastic Variational Inequality ◽

Sample Average ◽

Sample Average Approximation Method ◽

Average Approximation

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

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595920500360 ◽

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

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595910002569 ◽

2010 ◽

Vol 27 (01) ◽

pp. 103-119 ◽

Cited By ~ 53

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.

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