Penalty-based Methods for Simple Bilevel Optimization under Hölderian Error Bounds

This paper investigates simple bilevel optimization problems where we minimize a convex upper-level objective over the optimal solution set of a convex lower-level objective. Existing methods for such problems either only guarantee asymptotic convergence, have slow sublinear rates, or require strong assumptions. To address these challenges, we propose a penalization framework that delineates the relationship between approximate solutions of the original problem and its reformulated counterparts.