Distributionally robust expected residual minimization for stochastic variational inequality problems

Abstract

The stochastic variational inequality problem (SVIP) is an equilibrium model that includes random variables and has been widely applied in various fields such as economics and engineering. Expected residual minimization (ERM) is an established model for obtaining a reasonable solution for the SVIP, and its objective function is an expected value of a suitable merit function for the SVIP. However, the ERM is restricted to the case where the distribution is known in advance. We extend the ERM to ensure the attainment of robust solutions for the SVIP under the uncertainty distribution (the extended ERM is referred to as distributionally robust expected residual minimization (DRERM), where the worst-case distribution is derived from the set of probability measures in which the expected value and variance take the same sample mean and variance, respectively). Under suitable assumptions, we demonstrate that the DRERM can be reformulated as a deterministic convex nonlinear semidefinite programming to avoid numerical integration.

Keywords:

• Stochastic variational inequality
• expected residual minimization
• distributionally robust optimization

AMS CLASSIFICATIONS:

• 90C33
• 90C15
• 65K15

Acknowledgements

The authors are grateful to two anonymous reviewers for careful reading of the manuscript and insightful comments to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Only when the complementarity measure is evaluated by $\Vert x\circ F(x,\xi ){\Vert }_{\mathrm{\infty }}$, the mapping F of (4(4) $\begin{array}{rl}\underset{x\in {\mathrm{\Re }}^n}{min}& \underset{P\in \mathcal{P}}{sup}\{{\mathbb{E}}_P[\mathrm{\Psi }(x,\xi )]\mid P(\{F(x,\xi )\ge 0\}\cap \mathrm{\Xi })\ge 1-\epsilon \}\\ \mathrm{s}.\mathrm{t}.& x\ge 0,\end{array}$(4) ) is allowed up to second-order with respect to ξ.

Additional information

Funding

This work was supported by the JSPS KAKENHI under Grant JP17K00032.

Notes on contributors

Atsushi Hori

Atsushi Hori was born in Aichi, Japan. He received a master's degree in Mathematical Science from Nanzan University, Aichi, Japan, in 2018. Now, he is a Ph.D. candidate at the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University. His research interests include variational inequality and complementarity problems, game theory, bilevel optimization, distributionally robust optimization.

Yuya Yamakawa

Yuya Yamakawa is an assistant professor at the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University. His current research interests include nonlinear semidefinite programming problems, Riemannian optimization problems, optimal control problems, and their optimization methods, such as primal dual interior point methods, sequential quadratic programming methods, and augmented Lagrangian methods.

Nobuo Yamashita

Nobuo Yamashita is a full professor at the Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University. His research interests include nonlinear optimization and equilibrium problems, and their applications.