MoSSP: A Momentum-Based Single-Loop Stochastic Penalty Method for Nonconvex Constrained DC Optimization

In this paper, we study a general class of nonconvex constrained stochastic problems with difference-of-convex (DC) regularization, where the feasible set is possibly nonconvex, and the concave part of the DC regularizer is allowed to be nonsmooth. The fundamental challenge lies in maintaining feasibility for nonconvex constraints while achieving favorable oracle complexity. Although single-loop algorithms are efficient in solving unconstrained DC optimization problems, their potential for constrained optimization with DC structure remains largely unexplored. To address this gap, we develop **MoSSP**, a **Mo**mentum-based **S**ingle-loop **S**tochastic **P**enalty method for such problems with provable complexity guarantee. The key idea is to perform a single stochastic proximal-gradient update that approximates the gradient of the Moreau envelope of the composite term, which consists of the penalty function and the convex component of the DC regularizer. Simultaneously, the proximal mapping of its concave component is computed in parallel. We derive two algorithm variants: a Polyak-momentum version with $\mathcal{O}(\varepsilon^{-4})$ oracle complexity for finding stochastic $\varepsilon$-KKT points, and an improved $\mathcal{O}(\varepsilon^{-3})$ version incorporating recursive momentum. Experiment results demonstrate the effectiveness of our proposed algorithms.