Optimization Methods for Nonconvex Structured Optimization

该项目将推进基础算法理论和软件工具，以解决在科学、工程和工业中广泛应用的优化问题。具体来说，该项目将在结构化非凸非线性优化领域，这是许多现代应用的关键组成部分，从信号/图像处理，实时最优控制到随机学习。该项目旨在为优化和计算数据科学社区的研究人员开发算法，重点关注以下特征：速度，问题依赖性和易用性。学生将参与并有机会进行跨学科研究。将开发软件。该项目将开发理论强大和数值有效的算法以及解决非凸结构优化的软件。这些算法能够在保证全局收敛的情况下不精确地求解子问题，并且当问题具有凸性结构时具有最优的计算复杂度。这些算法将基于最近的研究成果，包括用于结构复合最小化的近端和随机梯度方法、用于可分离凸/非凸优化的不精确交替方向乘子方法（ADMM）和用于多面体约束优化的活动集方法。此外，还将探讨加速收敛的二阶技术。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 will advance fundamental algorithmic theory and software tools for solving optimization problems with wide applications in science, engineering and industry. Specifically, the project will be in the area of structured nonconvex nonlinear optimization, a critical component in many modern applications ranging from signal/image processing, real-time optimal control to stochastic learning. The project aims to develop algorithms with focus on the following features: speed, problem dependence, and ease of use for researchers in both optimization and computational data science community. Students will be involved and will have opportunities for interdisciplinary research. Software will be developed.This project will develop theoretically strong and numerically efficient algorithms as well as the software for solving nonconvex structured optimization. These algorithms will solve the subproblems inexactly with guaranteed global convergence as well as feature an optimal computational complexity when the problem features convexity structure. The algorithms will be based on recent work on proximal and stochastic gradient methods for structured composite minimization, inexact alternating direction multiplier methods (ADMM) for separable convex/nonconvex optimization and active set methods for polyhedral constrained optimization. In addition, second-order techniques for accelerating the convergence will be also explored.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.