On the efficient computation of proximal operators of affine-constrained nonconvex functions

TL;DR

This work develops a unified dual-representability framework for analyzing and computing affine-constrained proximal mappings, and introduces a multiplier inclusion formulation that connects the primal affine-constrained proximal problem to an unconstrained convex dual problem.

Abstract

Proximal operators with affine constraints arise in numerous models in nonconvex projection, composite optimization, and structured regularization. However, their efficient computation remains challenging due to the simultaneous presence of affine constraints and nonsmooth, possibly nonconvex objectives. In this work, we develop a unified dual-representability framework for analyzing and computing affine-constrained proximal mappings. Specifically, we introduce a multiplier inclusion formulation that connects the primal affine-constrained proximal problem to an unconstrained convex dual problem. Based on this formulation, we prove that, whenever the associated dual inclusion problem admits a solution, strong duality holds. For convex functions and a broad class of prox-regular nonconvex functions, we establish that dual representability holds under a simple subdifferential sum rule, and further develop a hierarchy of verifiable regularity conditions that guarantee this sum rule. In addition, we analyze the smoothness and strong convexity properties of the dual objective, providing a rigorous foundation that guarantees fast local convergence rates for efficient first- and second-order methods. Numerical experiments demonstrate that the proposed dual reformulation enables the reliable computation of globally optimal solutions for a range of large-scale nonconvex proximal and projection problems using existing convex optimization solvers.

Figures (3)

• Figure 1: Epigraphical intersection in Example \ref{['ex:epi-CHIP-R']}: replacing $\mathbb{R}_{+}$ by $\mathbb{R}$ restores a well-defined normal cone at $(0,-1)$ and ensures SSR.
• Figure 2: Convex hierarchy of regularity conditions.
• Figure 3: Illustration of convergence rate in terms of $\mathrm{R}_{\text{res}}$ for affine-constrained low-rank EDM projection problem with $n=1000$.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Definition 3: Prox-boundedness
• Proposition 1
• proof
• Remark 1
• Definition 4: Dual-representable
• Proposition 2
• proof
• Remark 2