CAREER: Efficient computational methods for nonlinear optimization and machine learning problems with applications to power systems

职业：非线性优化和机器学习问题的有效计算方法及其在电力系统中的应用

基本信息

批准号：
2045829
负责人：
Somayeh Sojoudi
金额：
$ 50万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-01-15 至 2025-12-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2045829&HistoricalAwards=false
关键词：
CAREER Efficient computational methods nonlinear

项目摘要

Optimization is an important tool for the design, analysis, control and operation of real-world systems, such as power systems. It also plays a central role in machine learning and artificial intelligence, particularly in deep learning, reinforcement leaning, and statistical learning. The mathematical foundation of optimization has heavily relied on the notion of convexity since convex optimization problems can be solved using fast algorithms. Nevertheless, many optimization problems in real-world applications are non-convex, and therefore it is extremely difficult to solve those problems reliably and efficiently using the existing methods. As an example, this issue is one of the main bottlenecks in the upgrade of the legacy power grids and has been incurring billions of dollars annually in the United States. This CAREER project aims to develop a set of computational tools for solving complex optimization and learning problems using efficient computational methods. This project has a significant impact on many societal problems through the development of a rich mathematical foundation for non-convex optimization, and its outcomes can be exploited in a variety of fields. The developed techniques enable solving large-scale computational problems for improving the efficiency, reliability, resiliency and sustainability of power grids, which has major societal, economical, and environmental impacts. Moreover, these tools significantly extend the application of artificial intelligence to safety-critical systems. This project has a wide range of outreach plans for K-12 and underrepresented students, and it also has several educational activities at both undergraduate and graduate levels. The state-of-the-art techniques for solving non-convex problems are based on various approximation and relaxation methods, whose practical use remains limited due to their scalability issues for real-world systems. On the other hand, the staggering advances made in artificial intelligence in the last 5 years (e.g., in deep learning) are due, in part, to handling computationally-intensive machine learning problems directly as non-convex optimization without relying on convex optimization. Motivated by the resounding success of local search methods for artificial intelligence, this CAREER project aims to design low-complexity computational methods for non-convex optimization problems. To this end, it studies the notion of spurious solutions, which are those solutions of an optimization problem that satisfy the local optimality conditions but are not globally optimal. The main property of convex optimization is the absence of spurious solutions. This project introduces the class of global functions which is far broader than the class of convex functions but benefits from the same spurious-solution-free property. Using the notions of global functions and kernel structure property, four objectives will be addressed: (i) analysis of the spurious solutions of key non-convex problems in machine learning and studying how the amount of data and the structural properties of each problem affects the inexistence of such solutions, (ii) analysis of the spurious solutions of an arbitrary polynomial optimization problem via its conversion to a machine learning problem and then discovering what structural properties guarantee the inexistence of spurious solutions, (iii) approximation of an arbitrary polynomial optimization problem having a spurious solution with a sequence of spurious-minima-free non-convex problems in a higher-dimensional space, (iv) software development and performing case studies on key problems for power systems and machine learning. This project is interdisciplinary and contributes to the areas of optimization theory, machine learning, control theory, and energy.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.

优化是现实系统（例如电力系统）设计、分析、控制和运行的重要工具。它还在机器学习和人工智能中发挥着核心作用，特别是在深度学习、强化学习和统计学习中。优化的数学基础在很大程度上依赖于凸性的概念，因为凸优化问题可以使用快速算法来解决。然而，现实应用中的许多优化问题都是非凸的，因此使用现有方法可靠有效地解决这些问题极其困难。例如，这个问题是传统电网升级的主要瓶颈之一，在美国每年造成数十亿美元的损失。该职业项目旨在开发一套计算工具，使用高效的计算方法解决复杂的优化和学习问题。该项目通过为非凸优化奠定了丰富的数学基础，对许多社会问题产生了重大影响，其成果可在各个领域得到利用。所开发的技术能够解决大规模计算问题，从而提高电网的效率、可靠性、弹性和可持续性，从而产生重大的社会、经济和环境影响。此外，这些工具显着地将人工智能的应用扩展到安全关键系统。该项目为 K-12 和代表性不足的学生制定了广泛的外展计划，并且还在本科生和研究生层面开展了多项教育活动。解决非凸问题的最先进技术基于各种近似和松弛方法，但由于现实系统的可扩展性问题，其实际使用仍然受到限制。另一方面，过去 5 年人工智能（例如深度学习）取得的惊人进步部分归因于将计算密集型机器学习问题直接作为非凸优化处理，而不依赖于凸优化。受人工智能局部搜索方法取得巨大成功的推动，该职业项目旨在为非凸优化问题设计低复杂度的计算方法。为此，它研究了伪解的概念，即满足局部最优条件但不是全局最优的优化问题的解。凸优化的主要特性是不存在虚假解。该项目引入了全局函数类，它比凸函数类要广泛得多，但受益于相同的无伪解属性。使用全局函数和核结构属性的概念，将解决四个目标：（i）分析机器学习中关键非凸问题的伪解，并研究每个问题的数据量和结构属性如何影响此类解不存在，(ii) 通过将任意多项式优化问题转换为机器学习问题来分析其伪解，然后发现哪些结构属性保证伪解不存在，(iii) 任意多项式优化的近似在高维空间中具有一系列无虚假极小值非凸问题的虚假解决方案的问题，(iv) 软件开发并对电力系统和机器学习的关键问题进行案例研究。该项目是跨学科的，对优化理论、机器学习、控制理论和能源领域做出了贡献。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。