Trilevel and Multilevel Optimization using Monotone Operator Theory

TL;DR

This paper introduces a unified approach to solve complex multi-level convex optimization problems using monotone operator theory, providing a new first-order algorithm with proven convergence properties.

Contribution

It develops a novel first-order algorithm for general multi-level convex optimization problems, including trilevel cases, with convergence analysis based on fixed-point theory.

Findings

The proposed algorithm converges under various parameter regimes.

Convergence rates are established for different problem settings.

The method applies to problems with smooth and non-smooth components.

Abstract

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.