Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions
Yankun Huang, Qihang Lin, Yangyang Xu
TL;DR
The paper develops the inexact Moreau envelope Lagrangian (iMELa) method for smooth nonconvex constrained optimization over a compact polytope with convex inequality constraints. By embedding a proximal term and solving a convex subproblem at each outer iteration with controlled inexactness, the algorithm achieves an $\epsilon$-KKT point under Slater's condition and a uniform local error bound on active constraint subsets, with gradient oracle complexity $\tilde{O}(\epsilon^{-2})$. This matches the best-known unconstrained rates up to a logarithmic factor and extends proximal ALM analyses to weaker regularity assumptions, broadening applicability. Numerical experiments on fairness-constrained classification corroborate the theory, showing competitive performance and improved feasibility trajectories compared to relevant baselines.
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 $ε$-Karush-Kuhn-Tucker point with $\tilde O(ε^{-2})$ gradient oracle complexity.