A sequential quadratic programming method for nonsmooth stochastic optimization with upper-C^2 objective

Abstract

We propose a sequential quadratic programming (SQP) method that can incorporate adaptive sampling for stochastic nonsmooth nonconvex optimization problems with upper-C^2 objectives. Upper-$\Ctwo$ functions can be viewed as difference-of-convex (DC) functions with smooth convex parts. They are common among certain classes of solutions to parametric optimization problems, e.g., recourse of stochastic programming and closest-point projection onto closed sets. Our proposed algorithm is a stochastic SQP with line search and bounded algorithmic parameters and is shown to achieve subsequential convergence in expectation for nonsmooth problems with upper-C^2 objectives. We discuss various sampling strategies, including an adaptive sampling one, that can potentially improve algorithm efficiency. The capabilities of our algorithm are demonstrated by solving a joint production, pricing and shipment problem, as well as a realistic optimal power flow problem as used in current power grid industry practice.