RA-DCA: A Randomized Active-Set DCA for Directional Stationarity in Max-Structured DC Programs

2026-05-22

AI Summary

This work addresses the limitations of the standard Difference-of-Convex Algorithm (DCA) in maximizing structured difference-of-convex (DC) functions, where it may converge to non-directionally stationary points and exact active-set identification incurs high computational cost. To overcome these issues, the authors propose RA-DCA, which introduces a vertex-prioritized randomized active-set mechanism. By randomly sampling directional projected gradients and evaluating vertex residuals, RA-DCA invokes a small-scale linear program only when residuals are sufficiently low, thereby reducing active-set identification to efficient matrix operations. Theoretical analysis and experiments demonstrate that RA-DCA guarantees, with probability one, that its accumulation points are directionally stationary. It effectively avoids non-stationary critical points on degenerate max-affine, max-quadratic, and sparse support function models, maintains strong performance in combinatorial settings such as block top-k, and clarifies the applicability boundaries of active-set strategies.

Abstract

We study nonsmooth difference-of-convex programs whose subtracted convex term is a finite maximum of smooth convex functions. In this setting, standard DCA iterations may converge to critical points that are not directionally stationary, whereas exact active-vertex screening can be expensive when active sets are large or combinatorial. We propose RA-DCA, a vertex-first randomized active-set DCA that projects active gradients onto sampled directions, checks a sampled vertex residual, and uses a small linear program only as a low-residual convex-combination fallback. The method preserves the descent structure of DCA and reduces the randomized screening layer to matrix multiplications. Under the stated regularity, numerical active-set consistency, and random-embedding assumptions, every accumulation point generated by the safeguarded method is directionally stationary with probability one. MATLAB experiments first test the theorem on degenerate max-affine, max-quadratic, and sparse support-function models, where the safeguard avoids nonstationary critical points and closely tracks a full active-vertex scan. Block top-k tests then show that the same screening idea remains useful when exact aggregate enumeration is combinatorial. Trimmed-regression, complementarity, and QUBO diagnostics separate cases where active-set selection helps from cases dominated by multistart search, the DC split, or other problem-specific features.

Problem

• difference-of-convex programs
• directional stationarity
• active-set
• nonsmooth optimization
• max-structured convex functions

Innovation

• Randomized Active-Set
• Directional Stationarity
• DC Programming
• Vertex Screening
• Matrix Multiplication