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

这个教师早期职业发展（CAREER）项目的研究目标是开发一个分析和算法框架，用于解决不确定性下的变分不等式（VI）问题。变分不等式问题可以容纳广泛的问题，包括优化问题，纳什博弈和均衡问题。 然而，人们对如何将不确定性纳入此类问题的理解有限。所提出的研究旨在通过考虑两个扩展来填补这一空白：（1）随机变分不等式（SVI）问题，通过将映射替换为其期望值对应物来推广维斯;（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来获得维斯的鲁棒解。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，将开发易于处理的无积分特征陈述，包括与相关解的存在性，唯一性和稳定性有关的陈述。此外，扩展容纳非凸性也将被调查。 在（ii）的背景下，拟议的研究将调查可在可能演变的网络上实现的自适应步长随机逼近方案的发展，以及全局收敛和可扩展的分解方案。如果成功，该项目将导致新的和增强的工具，用于网络系统的设计和操作，复杂的不确定性，非线性，非平滑性和竞争，如运输，电信和能源部门。更具体地说，这项研究将导致强大和可靠的电力市场，通过持续的互动与独立的系统运营商在新英格兰（ISO-NE）的影响。该项目包括一个综合教育计划，围绕高中发现课程，本科生研究项目和研究生级别的研讨会，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.