Nonsmooth, Nonconvex Optimization: Algorithms, Theory, and Applications

优化问题出现在科学和工程的每一个分支中。 在这些问题中，通常需要选择参数，以最小化目标，如能源或成本。 “非光滑”意味着目标可能不会随参数平滑变化，“非凸”意味着问题可能有多个具有不同目标值的最小化器，使得它们特别难以解决。该项目重点关注这些问题的三个方面：算法、理论和应用。 拟议工作的知识价值涉及所有三个方面的细节，这些方面都是相互关联的。 算法是指定计算方法的形式化方法。 我们特别关注两种算法：一种专门针对非光滑问题设计的易于描述但计算密集型的新方法，以及一种较旧的、众所周知的、高效的光滑优化方法，直到目前为止，该方法还没有被认为是一种可行的选择在非光滑情况下。 我们打算开发这些方法的收敛的理论基础。 我们对其他理论问题也很感兴趣。 最后，非光滑、非凸优化在工程和应用科学，特别是在控制工程中有着广泛的应用。 在这些应用领域产生影响需要付出很多努力：首先，理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。 一个是它将导致工程师和其他用户可以下载的公开软件。 强大、易于使用的便携式软件的重要性及其对科学基础设施的影响怎么强调都不过分。 非光滑，非凸优化有许多潜在的应用，并希望在这个项目中开发的算法，在实践中出现的挑战性问题的成功应用将激励其他研究人员直接应用到他们的问题。 最后，拟议的工作包括许多机会，为主要研究者继续建议和指导学生，包括代表性不足的群体的成员。

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.