A successive difference-of-convex method for a class of two-stage nonconvex nonsmooth stochastic conic program via SVI

TL;DR

This paper introduces a novel successive difference-of-convex (SDC) method utilizing the Moreau envelope to solve complex two-stage nonconvex nonsmooth stochastic conic programs, with proven convergence and practical effectiveness demonstrated through an extended Markowitz model.

Contribution

It develops a new SDC algorithm for nonconvex nonsmooth stochastic conic programs, transforming the problem into a variational inequality and solving it with convergence guarantees.

Findings

The proposed method converges under mild conditions.

Numerical results validate the effectiveness of the approach.

Application to an extended Markowitz model shows practical utility.

Abstract

We consider a class of two-stage nonconvex nonsmooth stochastic conic program, where the objective functions in both stages can contain nonsmooth terms that are functions with easily computed proximal mappings, further composed with affine mappings. This kind of problem is capable of modeling various applications. Solving these problems, however, can be challenging due to the two-stage structure with possibly large number of scenarios, the nonconvex, nonsmooth and even non-Lipschitz discontinuous terms, as well as the conic constraints. In this paper, we define a KKT point of the problem, show that it is a necessary optimality condition under mild conditions, and transform it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI). We then propose a successive difference-of-convex (SDC) method by making use of Moreau envelope to solve it, the subproblems of which are approximately solved by the progressive hedging method for solving maximal monotone two-stage SVI. We show the rigorous convergence of our method under suitable assumptions. An extension of Markowitz's mean-variance model is provided as an application and numerical results on it demonstrate the effectiveness of the model and the SDC method.