Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions

2025-02-27

AI Summary

This paper studies smooth nonconvex optimization problems subject to simple polyhedral and convex inequality constraints. Under a local error bound condition—rather than the stronger Mangasarian–Fromovitz or linear independence constraint qualification—it proposes the inexact Moreau–envelope Lagrangian (iMELa) algorithm. At each main iteration, iMELa solves a convex subproblem incorporating a proximal term, synergistically combining Moreau envelope smoothing, Lagrangian relaxation, and proximal gradient techniques. We establish, for the first time under a local error bound, that iMELa converges to an ε-KKT point with optimal gradient complexity Õ(ε⁻²), thereby overcoming theoretical limitations of classical penalty and augmented Lagrangian methods. Crucially, this result holds under significantly weaker constraint regularity assumptions, enhancing both the theoretical efficiency and practical applicability of algorithms for nonconvex constrained optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

In this paper, we study the inexact Moreau envelope Lagrangian (iMELa) method for solving smooth non-convex optimization problems over a simple polytope with additional convex inequality constraints. By incorporating a proximal term into the traditional Lagrangian function, the iMELa method approximately solves a convex optimization subproblem over the polyhedral set at each main iteration. Under the assumption of a local error bound condition for subsets of the feasible set defined by subsets of the constraints, we establish that the iMELa method can find an $epsilon$-Karush-Kuhn-Tucker point with $ ilde O(epsilon^{-2})$ gradient oracle complexity.

Problem

• Non-convex optimization with convex constraints
• Inexact Moreau Envelope Lagrangian method
• Local error bound conditions on constraints

Innovation

• Inexact Moreau Envelope Lagrangian
• Non-convex optimization solution
• Local error bound condition