Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions
This work addresses optimization problems for researchers in mathematical optimization, offering a theoretical improvement but is incremental as it builds on existing Lagrangian methods.
The paper tackles non-convex constrained optimization by proposing the inexact Moreau envelope Lagrangian method, which achieves an $ε$-Karush-Kuhn-Tucker point with $ ilde O(ε^{-2})$ gradient oracle complexity under local error bound conditions.
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.