Nonsmooth, Nonconvex Optimization: Algorithms, Theory, and Applications

最优化问题出现在科学和工程的各个分支中。在这些问题中，通常需要选择参数以最小化诸如能源或成本之类的目标。“非光滑”是指目标可能不会随着参数的变化而平滑变化，“非凸”是指问题可能有多个目标值不同的极小值，这使得问题的求解变得特别困难。拟议工作的智力价值涉及所有三个方面的细节，这三个方面都是相互关联的。算法是指定计算方法的正式方式。我们特别关注两种算法：一种是专门为非光滑问题设计的简单但计算密集的新方法，另一种是较老的、众所周知的、高效的光滑优化方法，到目前为止，它还不被认为是非光滑情况下的可行选择。我们打算为这些方法的收敛发展一个理论基础。我们还对其他理论问题感兴趣。最后，非光滑、非凸优化在工程和应用科学中的应用非常广泛，特别是在控制工程中。要在这些应用领域产生影响需要付出很多努力：首先，了解感兴趣的问题及其理论方面，然后将我们的算法应用于问题并分析结果。拟议工作的更广泛影响有许多组成部分。其一是它将带来公开可用的软件，工程师和其他用户可以下载。健壮、易用的便携软件的重要性及其对科学基础设施的影响怎么强调都不为过。非光滑、非凸优化有许多潜在的应用，希望本项目开发的算法对实践中出现的具有挑战性的问题的成功应用将激励其他研究人员将它们直接应用于他们的问题。最后，拟议的工作包括让首席调查员有许多机会继续向学生提供咨询和指导，包括代表人数不足的群体的成员。

Optimization problems arise in every branch of science and engineering. In these problems, typically one needs to choose parameters in order to minimize an objective such as energy or cost. "Nonsmooth" means that the objective may not vary smoothly with the parameters, and "nonconvex" means that the problems may have multiple minimizers with different objective values, making them especially difficult to solve.The project focuses on three aspects of these problems: algorithms, theory, and applications. The intellectual merit of the proposed work concerns the details of all three aspects, which are all interrelated. Algorithms are formal ways to specify computational methods. We focus on two algorithms in particular: a new easily-stated but computationally intensive method designed specifically for nonsmooth problems, and an older, well known, highly effective method for smooth optimization which up until the present has not been considered a viable option in the nonsmooth case. We intend to develop a theoretical foundation for the convergence of these methods. We are also interested in other theoretical issues. Finally, applications of nonsmooth, nonconvex optimization abound in engineering and applied science, particularly in control engineering. Making an impact in these application areas requires much effort: first of all, understanding the problems of interest and their theoretical aspects, and following this by applying our algorithms to the problems and analyzing the results.The broader impact of the proposed work has many components. One is that it will lead to publicly available software that can be downloaded by engineers and other users. The importance of robust, easy-to-use portable software and its impact on the scientific infrastructure cannot be overstated. Nonsmooth, nonconvex optimization has many potential applications, and it is hoped that the successful application of the algorithms developed in this project to challenging problems arising in practice will inspire other researchers to apply them directly to their problems. Finally, the proposed work includes many opportunities for the principal investigator to continue advising and mentoring students, including members of underrepresented groups.