Design and Analysis of Algorithms for Structured Optimization

该项目旨在开发用于解决结构化优化问题的分析算法的先进工具，这些工具在各种科学和工程领域（包括海量数据分析、机器学习、信号处理和图像重建）中对模型起着重要作用。虽然有一些实用和成功的算法来优化这些框架，但它们的基本收敛理论尚未完全理解。该项目旨在开发新的工具，以便更好地理解模型和算法的核心特征，设计更有效的算法，并解决更具挑战性的应用。这个项目的结果将有助于更好地理解如何在改进的经典迭代中实现快速收敛，这将提高它们的效率。该项目将把研究结果整合到研究生课程中，并吸引博士生参与与项目主题相关的研究。本研究项目将专注于设计和分析新的高效投影/近端型方案，以解决（非）凸和（非）光滑复合优化和可行性问题。本研究将探讨非光滑和非凸的不规则现象如何影响算法的性能，并将研究通过利用问题的特殊结构来提高算法收敛复杂性的可能性。分割迭代频繁出现锯齿状，影响了算法的收敛速度。本研究将改进和改进环心反射方法，以提高分割算法的性能和复杂性，以解决更一般的结构化问题。在缺乏经典假设的情况下，该项目还将研究用于解决复合问题的FISTA算法的变化，以及用于解决广义投影方程和互补问题的半光滑牛顿迭代。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。

This project aims to develop advanced tools for analyzing algorithms that solve structured optimization problems, which play an important role for models in various scientific and engineering fields, including massive data analysis, machine learning, signal processing, and image reconstruction. While there are several practical and successful algorithms for optimizing these frameworks, their fundamental convergence theory is not yet fully understood. This project seeks to develop new tools that will enable a better understanding of the core features of both the models and algorithms, design more effective algorithms, and tackle more challenging applications. The outcomes of this project will contribute to a better understanding of how to achieve fast convergence in modified classical iterations, which will improve their efficiency. The project will integrate its findings into graduate-level courses and engage Ph.D. students in research related to the project's topics.This research project will focus on designing and analyzing novel efficient projection/proximal-type schemes for solving (non)convex and (non)smooth composite optimization and feasibility problems. The research will investigate how the irregular phenomena of nonsmoothness and nonconvexity affect algorithmic performance and will study the possibility of improving the convergence complexity of the algorithms by exploiting the particular structure of the problem. Splitting iterations frequently show signs of zigzagging, affecting those schemes' convergence speed. The proposed research will advance and adapt the Circumcentered-Reflection Method to enhance the performance and complexity of splitting algorithms for solving more general structured problems. In the absence of classical assumptions, the project will also investigate variations of the FISTA algorithm for solving composite problems and semismooth Newtonian iterations for solving generalized projection equations and complementarity problems.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.