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

Yan-Chao Liang, Qiao-Na Fan () and Pei-Ping Shen ()
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Yan-Chao Liang: College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China
Qiao-Na Fan: College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China
Pei-Ping Shen: School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450045, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2021, vol. 38, issue 01, 1-25

Abstract: 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.

Keywords: Stochastic variational inequality; D-gap function; sample average approximation; hybrid Newton method; traffic equilibrium (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0217595920500360

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