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

本教师早期职业发展（Career）项目的研究目标是开发一个分析和算法框架，以解决不确定条件下的变分不等式（VI）问题。变分不等式问题可以容纳广泛的问题范围，包括优化问题，纳什博弈和平衡问题。然而，人们对如何将不确定性纳入这类问题的理解有限。本研究旨在通过考虑两种扩展来填补这一空白：(1)随机变分不等式（SVI）问题，该问题通过用其期望值对应物替换映射来推广VIs；(2)鲁棒变分不等式（RVI）问题，通过参数化可行解集和映射中的不确定性得到鲁棒变分不等式的鲁棒解。在每个问题的背景下，拟议的研究将围绕两个重点进行：(i)分析和（ii）计算。作为(i)的一部分，将开发可处理的无集成特性陈述，包括与相关解决方案的存在性、唯一性和稳定性有关的陈述。此外，还将研究可容纳非凸性的扩展。在（ii）的背景下，提出的研究将研究可在可能不断发展的网络上实现的自适应步长随机逼近方案的发展，以及全球收敛和可扩展的分解方案。如果成功，该项目将为网络系统的设计和操作带来新的和增强的工具，这些系统因运输、电信和能源部门出现的不确定性、非线性、非平滑和竞争而复杂化。更具体地说，这项研究将通过与新英格兰独立系统运营商（ISO-NE）的持续互动来实现强大可靠的电力市场。该项目包含了一个综合的教育计划，包括高中探索课程、本科研究项目和研究生水平的研讨会，并将通过学生咨询和指导来增加多样性。

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.