Trilevel and Multilevel Optimization using Monotone Operator Theory

We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized, subject to constraints to optima of a nested convex optimization problem. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term. Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.