Nonconvex Optimization Methods with Applications to Portfolio Selection and Hybrid Systems

Abstract

This thesis focuses on formulating selection problems using continuous optimization, and solving them by specialized algorithms. Problems involving selection, i.e., selecting ``best" candidate(s) out of a given set, occur frequently in various applications, and can be formulated as nonconvex optimization problems. We focus on two applications, portfolio optimization in finance and hybrid systems inference in control theory. We show that using techniques including recently developed relaxations for nonconvex functions, we are able to formulate these problems as structured nonconvex problems and develop efficient algorithms with standard convergence guarantees. The effectiveness of the algorithms is demonstrated in detail for both applications with numerical experiments.