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

TL;DR

This paper introduces an inexact Moreau envelope Lagrangian method for non-convex constrained optimization, demonstrating that under weaker local error bound conditions, it achieves near-optimal complexity in finding approximate KKT points.

Contribution

It establishes complexity bounds for the method under local error bound conditions, extending guarantees to broader non-convex constrained problems with weaker assumptions.

Findings

Achieves $ ilde O( ext{epsilon}^{-2d})$ gradient complexity under local error bound with exponent d

Matches best-known complexity when d=1, up to logarithmic factors

Extends complexity guarantees beyond linear independence constraint qualification

Abstract

In this paper, we investigate how structural properties of the constraint system impact the oracle complexity of smooth non-convex optimization problems with convex inequality constraints over a simple polytope. In particular, we show that, under a local error bound condition with exponent $d\in[1,2]$ on constraint functions, an inexact Moreau envelope Lagrangian method can attain an $\epsilon$-Karush--Kuhn--Tucker point with $\tilde O(\epsilon^{-2d})$ gradient oracle complexity. When $d=1$, this result matches the best-known complexity in literature up to logarithmic factors. Importantly, the assumed error bound condition with any $d\in[1,2]$ is strictly weaker than the local linear independence constraint qualification that is required to achieve the best-known complexity. Our results clarify the interplay between error bound conditions of constraints and algorithmic complexity, and extend complexity guarantees to a broader class of constrained non-convex problems.