Stochastic variational inequality (SVI) is a mathematical model and has been extensively studied over the years. The model appears in diverse fields such as transportation engineering, game theory, and physics and plays an essential role in formulating an equilibrium state in the real world. As is known in the literature on optimization, an expected residual minimization (ERM) method is one of the approaches to solve SVI. However, the conventional ERM assumes that the probability distribution of random variables involving in the model is sufficiently close to the true distribution (which is impossible to know in practice); in the absence of the assumption, such as the lack of observation data to identify the distribution, the ERM method may output a solution which may occur a significant loss in a real world. To tackle the issue, we utilize a distributionally robust optimization approach to obtain more robust solutions of SVI, referred to as a distributionally robust ERM (DRERM). In general, the (DR)ERM requires a numerical integral, which is computationally much expensive, but we demonstrate that the DRERM can be approximately solved via semidefinite programming. We also show that under a suitable assumption the optimal solution to the approximated problem coincides with the original DRERM.
A. Hori, Y. Yamakawa, and N. Yamashita, Distributionally Robust Expected Residual Minimization for Stochastic Variational Inequality Problems. Optimization Methods and Software, to appear, 2023.