Moderate deviations for stochastic variational inequalities

Abstract

Stochastic variational inequalities (SVIs) have been used widely in modelling various optimization and equilibrium problems subject to data uncertainty. The sample average approximation (SAA) solution is an asymptotically consistent point estimator for the true solution to a stochastic variational inequality. Some central limit results and large deviation estimates for the SAA solution have been obtained. The purpose of this paper is to study the convergences in regimes of moderate deviations for the SAA solution. Using the delta method and the exponential approximation, we establish some results on moderate deviations. We apply the results to the hypotheses testing for the SVIs, and prove that the rejection region constructed by the central limit theorem has the probability of the type II error with exponential decay speed. We also give some simulations and numerical results for the tail probabilities.

Keywords:

• Moderate deviation
• sample average approximation
• stochastic optimization
• stochastic variational inequality

Acknowledgments

The first author is supported by the Natural Science Foundation of Guangdong Province [grant number 2021A1515010368], and the National Natural Science Foundation of China [grant numbers 11801184 and 12171168]. The second author is supported by he Research Grants Council of Hong Hong (PolyU 15223419), the Hong Kong Polytechnic University (G-UAHF, 1-WZ0E) and Research Centre for itative Finance (1-CE03). The authors want to thank the editor and two anonymous referees for helpful comments and suggestions, which greatly improves the quality of the paper.

Table 1. n = 1000.

Table 2. r = 0.3.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported in part by the Natural Science Foundation of Guangdong Province [grant number 2021A1515010368], and in part by the National Natural Science Foundation of China [grant numbers 11801184 and 12171168], and in part by the Research Grants Council of Hong Hong (PolyU 15223419), the Hong Kong Polytechnic University (G-UAHF, 1-WZ0E) and Research Centre for Quantitative Finance (1-CE03).