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.

该项目将研究优化问题解决方案的行为，这些解决方案出现在应用程序中，例如回归模型，信号的稀疏近似，图像处理和传感器位置问题。此外，主要研究者将设计可以有效解决优化问题的数值算法。研究人员将利用各种分析的工具和技术来针对这些优化问题的数据，这些问题可能不会以通常的方式不同，并在数值算法中研究了应用。研究生和博士后研究人员将参加该项目。该项目研究了重要类别优化问题的重要类别的二阶变分特性，包括分段线性 - 季度复合问题和不同类别的矩阵优化问题。该提案有三个主要目标。首先，研究人员将研究上述优化问题类别的抛物面规律性和两倍的Epi差异性。特别是，研究人员特别注意与复合优化问题相关的增强拉格朗日人，研究其两倍的表面分化性，并通过二阶足够条件来表征此类功能的二次生长条件。其次，研究者将研究复合问题的重要稳定性，包括强度的指标规则性，强度较高的次级性和其Lagrange乘数的非关键性。最后，研究者将对增强的拉格朗日方法进行本地和全球融合分析，以特别强调那些lagrange乘数并非唯一的优化问题，以特别强调那些优化问题。在这样做的过程中，研究人员主要依靠第二个亚衍生物的概念及其最近的发展。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。