LEAF: A Learning-Enabled ADMM Framework for Accelerated Convex Optimization

2026-06-07

AI Summary

This work addresses convex optimization problems involving both smooth and nonsmooth components by proposing the LEAF framework, which introduces scalar-valued Moreau envelope learning into the Alternating Direction Method of Multipliers (ADMM) for the first time. The approach explicitly models the Moreau envelope of the objective function using input-convex neural networks (ICNNs), yielding the MEL-ADMM algorithm and its splitting variant, sMEL-ADMM. These methods substantially reduce model complexity while rigorously preserving convexity and theoretical convergence guarantees. Experimental results demonstrate that the proposed algorithms achieve up to an order-of-magnitude speedup over existing solvers while maintaining a low optimality gap, and exhibit convergence rates comparable to those of classical ADMM.

Abstract

We propose LEAF, a learning-enabled ADMM framework for accelerated convex optimization. The key idea is to approximate the Moreau envelope of the objective function using an Input Convex Neural Network (ICNN), resulting in a learned model that preserves convexity and smoothness. This leads to the proposed Moreau Envelope Learning ADMM (MEL-ADMM) and its splitting variant sMEL-ADMM. Unlike existing approaches that learn high-dimensional operators directly, LEAF learns a scalar-valued Moreau envelope, significantly reducing model complexity and improving data efficiency. The framework accommodates a broad class of convex problems with smooth and non-smooth objectives. By embedding convexity explicitly through the ICNN architecture, the proposed approach maintains high approximation accuracy while preserving key structural properties of the optimization problem. Both MEL-ADMM and sMEL-ADMM are developed with theoretical guarantees of convergence and feasibility under the learned model. Rigorous analysis shows that the proposed methods achieve convergence rates comparable to classical ADMM while reducing per-iteration computational cost. Numerical experiments demonstrate up to an order-of-magnitude speedup over state-of-the-art solvers while maintaining low optimality gaps

Problem

• convex optimization
• accelerated optimization
• ADMM
• Moreau envelope
• large-scale optimization

Innovation

• Moreau Envelope Learning
• Input Convex Neural Network
• ADMM Acceleration
• Convex Optimization
• Model-Based Learning