Second-Order Variational Properties of Composite Optimization and Applications

这个项目将研究最优化问题的解的行为，这些问题出现在诸如回归模型、信号的稀疏近似、图像处理和传感器位置问题等应用中。此外，首席研究员将设计能够有效地解决优化问题的数值算法。研究人员将利用变分分析的各种工具和技术来处理这些具有通常不可微的数据的优化问题，并研究其在数值算法中的应用。研究生和一名博士后研究员将参与这个项目。该项目研究了一类重要的组合优化问题的二阶变分性质，包括分段线性-二次组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，研究上述几类优化问题的抛物线正则性和二次表可微性。特别地，研究了与复合优化问题相关的增广拉格朗日函数，研究了它们的二次表可微性，并利用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、强度量次正则性和拉格朗日乘子的非临界性。最后，对一类重要的复合优化问题的增广拉格朗日方法进行了局部和全局收敛分析，重点分析了拉格朗日乘子不唯一的优化问题。在这样做的过程中，研究人员主要依靠二次衍生的概念及其最新发展。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.