CAREER: Stochastic and Robust Variational Inequality Problems: Analysis, Computation and Applications to Power Markets

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The research objective of this Faculty Early Career Development (CAREER) project is to develop an analytical and algorithmic framework for addressing variational inequality (VI) problems under uncertainty. Variational inequality problems can accommodate an expansive range of problems, including optimization problems, Nash games and equilibrium problems. Yet, there is limited understanding of how to incorporate uncertainty in such problems. The proposed research intends to fill this gap by considering two extensions: (1) stochastic variational inequality (SVI) problems, which generalize VIs by replacing the mapping with its expected-value counterpart; and (2) robust variational inequality (RVI) problems, where robust solutions to VIs are obtained by parameterizing uncertainty in the feasible solution set and the mapping. In the context of each problem, the proposed research will be aggregated around two thrusts: (i) analysis and (ii) computation. As part of (i), tractable integration-free characterization statements will be developed, including those pertaining to the existence, uniqueness and stability of the associated solutions. Additionally, extensions accommodating nonconvexity will also be investigated. In the context of (ii), the proposed research will investigate the development of adaptive step-size stochastic approximation schemes implementable over possibly evolving networks, as well as globally convergent and scalable decomposition schemes.If successful, this project will lead to new and enhanced tools for the design and operation of networked systems, complicated by uncertainty, nonlinearity, nonsmoothness and competition, as arising in transportation, telecommunications and energy sectors. More specifically, this research will lead to robust and reliable power markets, effected through ongoing interactions with the independent system operator in New England (ISO-NE). The project incorporates a comprehensive education plan aggregated around high-school discovery courses, undergraduate research projects and graduate-level seminars and will be accompanied by efforts toward increasing diversity through student advising and mentoring.