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.
