Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Trilevel and Multilevel Optimization using Monotone Operator Theory

Allahkaram Shafiei, Vyacheslav Kungurtsev, Jakub Marecek

TL;DR

Based on fixed-point theory and related arguments, a natural first-order algorithm is presented and its convergence and rates of convergence in several regimes of parameters are analyzed.

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.

Trilevel and Multilevel Optimization using Monotone Operator Theory

TL;DR

Based on fixed-point theory and related arguments, a natural first-order algorithm is presented and its convergence and rates of convergence in several regimes of parameters are analyzed.

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.

Paper Structure

This paper contains 15 sections, 24 theorems, 162 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Problem
  3. Related Work
  4. Examples
  5. Preliminaries
  6. Convergence Analysis for Trilevel Optimization Problems
  7. Assumptions and the Algorithm
  8. Properties of the limit points
  9. Convergence analysis under assumptions on step-sizes $\alpha_k$ and $\beta_k$
  10. Convergence analysis under an error-bound condition
  11. Convergence Rate Analysis for Trilevel Optimization Problems
  12. Technical lemmas
  13. The Main Result
  14. An Extension to Multilevel Optimization Problems
  15. Conclusion

Key Result

Lemma 5

xu2002iterative Assume that $a_k$ be a sequence of non-negative real numbers such that where $\gamma_k$ is a sequences in $(0,1)$ and $\delta_k$ is a sequence in $\mathbb{R}$, such that (1) $\sum\limits_{k = 1}^\infty {{\gamma _k} = } \infty$, (2) either $\limsup\limits_{k\to\infty}\frac{\delta_k}{\gamma_k}\le 0$ or $\sum\limits_{k= 1}^\infty {\lvert{\delta _k}\rvert < } \infty$. Th

Theorems & Definitions (56)

  • Lemma 5
  • Lemma 6
  • proof
  • Remark 7
  • Example 1
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • proof
  • ...and 46 more