Trilevel and Multilevel Optimization using Monotone Operator Theory
This work addresses optimization challenges in nested convex problems, which is incremental as it builds on existing fixed-point theory for multi-level settings.
The paper tackles the problem of solving a general class of multi-level optimization problems, particularly trilevel cases with smooth and non-smooth terms, by presenting a first-order algorithm based on fixed-point theory and analyzing its convergence rates in various parameter regimes.
We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. 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.