Local convergence of a sequential quadratic programming method for a class of nonsmooth nonconvex objectives

Abstract: A sequential quadratic programming (SQP) algorithm is designed for nonsmooth optimization problems with upper-C^2 objective functions. Upper-C^2 functions are locally equivalent to difference-of-convex (DC) functions with smooth convex parts. They arise naturally in many applications such as certain classes of solutions to parametric optimization problems, e.g., recourse of stochastic programming, and projection onto closed sets. The proposed algorithm conducts line search and adopts an exact penalty merit function. The potential inconsistency due to the linearization of constraints are addressed through relaxation, similar to that of Sl_1QP. We show that the algorithm is globally convergent under reasonable assumptions. Moreover, we study the local convergence behavior of the algorithm under additional assumptions of Kurdyka-{\L}ojasiewicz (KL) properties, which have been applied to many nonsmooth optimization problems. Due to the nonconvex nature of the problems, a special potential function is used to analyze local convergence. We show that under acceptable assumptions, upper bounds on local convergence can be proven. Additionally, we show that for a large number of optimization problems with upper-C^2 objectives, their corresponding potential functions are indeed KL functions. Numerical experiment is performed with a power grid optimization problem that is consistent with the assumptions and analysis in this paper.