Second-Order Variational Properties of Composite Optimization and Applications

复合材料优化的二阶变分性质及其应用

• 批准号: 2108546
• 负责人: Ebrahim Sarabi
• 金额: $ 19.5万
• 依托单位: Miami University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108546&HistoricalAwards=false
• 关键词: Second; Order; Variational; Properties; Composite

This project will study the behavior of solutions to optimization problems, which appear in applications such as regression models, sparse approximation of signals, image processing, and sensor location problems. In addition, the principal investigator will design numerical algorithms that can solve the optimization problems efficiently. The investigator will exploit various tools and techniques of variational analysis for these optimization problems with data that may not be differentiable in the usual way, and study applications in numerical algorithms. Graduate students and a postdoctoral researcher will participate in this project. The project investigates second-order variational properties of important classes of composite optimization problems, including piecewise linear-quadratic composite problems and different classes of matrix optimization problems. The proposal has three main objectives. First, the investigator will study parabolic regularity and twice epi-differentiability of the aforementioned classes of optimization problems. In particular, the investigator pays special attention to the augmented Lagrangians associated with composite optimization problems, studies their twice epi-differentiability, and characterizes the quadratic growth condition for this class of functions via the second-order sufficient condition. Second, the investigator will study important stability properties of composite problems, including strong metric regularity, strong metric subregularity, and non-criticality of their Lagrange multipliers. Finally, the investigator will conduct local and global convergence analysis of the augmented Lagrangian method for important classes of composite optimization problems with special emphasis on those optimization problems whose Lagrange multipliers are not unique. In doing so, the investigator mainly relies on the concept of the second subderivative and its recent developments.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.

该项目将研究最优化问题的解的行为，这些问题出现在回归模型、信号的稀疏逼近、图像处理和传感器定位问题等应用中。此外，首席研究员将设计能够有效解决优化问题的数值算法。研究者将利用变分分析的各种工具和技术来解决这些优化问题，这些问题的数据可能无法以通常的方式微分，并研究在数值算法中的应用。本项目将有研究生和一名博士后参与。本课题研究了一类重要的复合优化问题的二阶变分性质，包括分段线性二次复合问题和不同类型的矩阵优化问题。该提案有三个主要目标。首先，研究者将研究上述优化问题的抛物正则性和二次外可微性。特别地，研究者特别关注与复合优化问题相关的增广拉格朗日，研究了它们的二次外微性，并通过二阶充分条件刻画了这类函数的二次增长条件。其次，研究者将研究复合问题的重要稳定性性质，包括强度量正则性、强度量子正则性和它们的拉格朗日乘子的非临界性。最后，研究者将对重要类别的复合优化问题进行增广拉格朗日方法的局部和全局收敛性分析，并特别强调那些拉格朗日乘子不是唯一的优化问题。在这样做时，研究者主要依赖于二阶导数的概念及其最近的发展。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。