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CRII: AF: Variational Inequality and Saddle Point Problems with Complex Constraints

CRII：AF：具有复杂约束的变分不等式和鞍点问题

基本信息

批准号：
2245705
负责人：
Digvijay Boob
金额：
$ 17.49万
依托单位：
Southern Methodist University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-03-01 至 2025-02-28
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2245705&HistoricalAwards=false
关键词：
CRII AF Variational Inequality Saddle

项目摘要

The variational inequality (VI) problem is a general tool for modeling various optimization and equilibrium problems. Such problems have applications in traffic networks, power market pricing, signal processing, risk-averse/robust optimization, and adversarial learning. Many of these application problems impose complicated constraints that must be satisfied. These constraints can arise due to engineering, ethical or legal concerns for the context of the respective application, and often have a functional form. In some cases, the constraint function is data-driven with unknown data distribution - making it impractical to evaluate its value even at a single point. This project aims to develop novel first-order algorithms that can work for VI problems with complex constraints in functional form.Existing algorithms for VI problems assume that one can efficiently project onto the constraint sets. This assumption is not satisfied when constraints have a general functional form, even more so when the function is data-driven. Hence, the projection requirement severely restricts the applicability of current algorithms to real-world problems of consequence. The algorithms developed as part of this project will solve VI problems without requiring any projection onto complex constraints in functional form. It includes (1) the deterministic problems algorithm evaluates the VI objective and constraint function exactly; (2) data-driven problems where an exact evaluation of the VI objective or the constraint function may not be possible due to large data/unknown underlying distribution; and (3) extending these methods for min-max saddle point problems - an important specific case of the VI problem. The successful completion of this project will yield state-of-the-art algorithms for the VI problem with these complex constraint satisfaction requirements. The proposed algorithms will be equipped with provable convergence guarantees for each of the three subcases above. The project will include a comprehensive numerical validation of the proposed schemes for an equilibrium problem arising from wireless communication. The intellectual property generated as a part of this project, e.g., computer codes, data sets, research articles, and conference proceedings, will be shared with the public through open-source online repositories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

变分不等式(VI)问题是模拟各种优化和均衡问题的通用工具。这类问题在交通网络、电力市场定价、信号处理、风险厌恶/稳健优化和对抗性学习中都有应用。这些应用问题中的许多都施加了必须满足的复杂约束。这些限制可能是由于各自应用环境的工程、伦理或法律方面的考虑而产生的，并且通常具有功能形式。在某些情况下，约束函数是由数据驱动的，具有未知的数据分布，这使得即使在单个点上评估其值也是不切实际的。该项目旨在开发一种新的一阶算法，能够处理函数形式的复杂约束的虚拟仪器问题，现有的虚拟仪器问题的算法假设一个人可以有效地投影到约束集上。当约束具有一般的函数形式时，这一假设不被满足，当函数是数据驱动的时更是如此。因此，投影要求严重限制了当前算法对现实世界中重要问题的适用性。作为该项目的一部分开发的算法将解决VI问题，而不需要对函数形式的复杂约束进行任何投影。它包括(1)确定性问题算法准确地评估VI目标和约束函数；(2)数据驱动的问题，其中由于大数据/未知的潜在分布，可能无法准确评估VI目标或约束函数；以及(3)将这些方法扩展到最小-最大鞍点问题--VI问题的一个重要的具体情况。该项目的成功完成将为具有这些复杂约束满足要求的VI问题产生最先进的算法。所提出的算法将为上述三种子情形中的每一种配备可证明的收敛保证。该项目将包括对提议的无线通信平衡问题的方案进行全面的数值验证。作为该项目的一部分产生的知识产权，例如计算机代码、数据集、研究文章和会议记录，将通过开源在线资源库与公众共享。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。