Methodology and first-order algorithms for solving nonsmooth and non-strongly convex bilevel optimization problems

Simple bilevel problems are optimization in which we want to find an optimal solution inner problem that minimizes outer objective function. Such appear many machine learning and signal processing applications as a way eliminate undesirable solutions. In our work, suggest new approach is designed for with simple functions, such the \(l_1\) norm, not required be either smooth or strongly convex. ITerative Approximation Level-set EXpansion (ITALEX) approach, alternate between expanding level-set of function approximately optimizing over this level-set. We show through first-order methods proximal gradient generalized conditional results feasibility convergence rate O(1/k), up now was only achieved by algorithms convex functions. Moreover, prove \(O(1/\sqrt{k})\) function, contrary existing methods, provide asymptotic guarantees. demonstrate performance numerical experiments.