Dynamic Stochastic Variational Inequalities and Convergence of Discrete Approximation
Links to Fileshttps://epubs.siam.org/doi/abs/10.1137/21M145536X
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Type of Work29 pages
Citation of Original PublicationCHEN, XIAOJUN and JINGLAI SHEN. “Dynamic Stochastic Variational Inequalities and Convergence of Discrete Approximation,” SIAM J. OPTIM 32 4 (November, 2022): 2909 – 37. https://doi.org/10.1137/21M145536X
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This paper studies dynamic stochastic variational inequalities (DSVIs) to deal with uncertainties in dynamic variational inequalities (DVIs). We show the existence and uniqueness of a solution for a class of DSVIs in C¹× γ, where C¹ is the space of continuously differentiable functions and γ is the space of measurable functions, and discuss non-Zeno behavior. We use the sample aver-age approximation (SAA) and time-stepping schemes as discrete approximation for the uncertainty and dynamics of the DSVIs. We then show the uniform convergence and an exponential convergence rate of the SAA of the DSVI. A time-stepping EDIIS (energy direct inversion on the iterative subspace) method is proposed to solve the DVI arising from the SAA of DSVI; its convergence is established. Our results are illustrated by a point-queue model for an instantaneous dynamic user equilibrium in traffic assignment problems.