RA-DCA Algorithm for Directional Stationarity in Max-Structured DC Programs

The RA-DCA algorithm tackles the issue of directional stationarity in nonsmooth difference-of-convex (DC) programs, specifically when the convex term is derived from a finite maximum of smooth convex functions. Traditional DCA iterations risk converging to critical points lacking directional stationarity, and precise active-vertex screening can be costly. RA-DCA employs a vertex-first randomized active-set approach, projecting active gradients onto selected directions, verifying a sampled vertex residual, and utilizing a compact linear program as a low-residual convex-combination alternative. This method maintains the descent structure of DCA while simplifying the randomized screening to matrix multiplications. Under certain regularity, numerical active-set consistency, and random-embedding conditions, the safeguarded approach guarantees that each accumulation point is directionally stationary with probability one. MATLAB tests validate the theorem on degenerate cases. The paper can be found on arXiv under reference 2605.23550.