Collaborative Research: Computationally Efficient Algorithms for Large-scale Bilevel Optimization Problems

机器学习和具有分层决策结构的电力系统的最新进展要求开发有效的方案来解决双层优化问题。双层优化问题是一个分层决策过程，也是一类重要的数学模型，其中找到最优决策（上层问题）依赖于预测另一个决策问题（下层问题）。尽管在研究双层优化方面取得了进展，但大多数现有方法在应用于大规模、不确定或分布式环境时可能很慢或效率低下。该项目旨在通过研究双层优化的新公式和开发用于解决分层决策问题的计算效率算法来解决这些挑战。该项目的成果将对储能系统、电力系统的投资和运营规划、推荐平台以及语音和图像识别软件产生革命性的影响。在教育方面，此计划将提供一个刺激和创新的研究环境，让代表性不足的学生和少数族裔学生参与研究；它还将包括为pi机构的课程编制课程材料。该项目列出了一个详细的议程，用于探索双层优化重新制定和开发高效和可扩展的方案，以解决最先进的双层优化框架在面对机器学习和电力系统中最近出现的范式挑战时的主要局限性。研究包括三个不同的重点：(I)检验非凸双层优化的重新表述，并为如何以寻找局部最优为目标重新表述双层优化问题提供新的见解。（二）利用随机优化和在线学习的工具，研究不确定条件下双层优化问题的快速收敛保证的高效计算方法。（三）研究分散制度下的双层优化问题，目标是开发和分析具有局部计算和通信的分布式算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The recent advancements in machine learning and power systems with hierarchical decision-making structure necessitate developing efficient schemes to solve bilevel optimization problems. A bilevel optimization problem is a hierarchical decision-making process and an important class of mathematical models in which finding the optimal decision (the upper-level problem) depends on anticipating another decision-making problem (the lower-level problem). Despite the progress in studying bilevel optimization, most existing methods could be slow or inefficient when applied in large-scale, uncertain, or distributed settings. This project aims to address these challenges by examining novel reformulations of bilevel optimization and developing computationally efficient algorithms for solving hierarchical decision-making problems. The outcomes of this project will be transformational for energy storage systems, investment and operation planning in power systems, recommendation platforms, and speech and image recognition software. On the education front, this project will provide a stimulating and innovative research environment to include under-representative and minority students in the project research; it will also incorporate the development of curricular material for courses in the PIs’ institutions. This project lays out a detailed agenda for exploring bilevel optimization reformulations and developing efficient and scalable schemes to address major limitations of state-of-the-art bilevel optimization frameworks when confronted with the challenges of recently emerged paradigms in machine learning and power systems. The research encompasses three different thrusts: (I) Examining reformulations of nonconvex bilevel optimization and offering new insights on how to reformulate a bilevel optimization problem with the goal of finding a local optimum. (II) Developing computationally efficient methods with fast convergence guarantees for bilevel optimization problems under uncertainty by leveraging tools from stochastic optimization and online learning. (III) Investigating bilevel optimization problems in a decentralized regime with the goal of developing and analyzing distributed algorithms with local computations and communications.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.