Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regularization methods for solving hierarchical variational inequalities with complexity guarantees

Daniel Cortild, Meggie Marschner, Mathias Staudigl

TL;DR

The paper addresses solving hierarchical variational inequalities where the lower level is defined by a monotone inclusion, by introducing a double-loop diagonal equilibrium tracking method (DANTE) that combines $T$ikhonov regularization with proximal penalties. The inner loop employs a flexible fixed-point encoding with an inertial Krasnoselskii–Mann iteration to approximate a temporal solution, while the outer loop restarts to update anchors and steer convergence. The authors establish convergence and rate guarantees for the averaged iterates in terms gap functions $\operatorname{Gap}_{\operatorname{opt}}$ and $\operatorname{Gap}_{\operatorname{feas}}$, with explicit complexity bounds and performance across several splitting schemes (two-operator and three-operator). Numerical experiments on equilibrium selection, least-norm least-squares, and image inpainting validate the method’s versatility and show competitive convergence behavior across different problem structures. Overall, the work provides a broad, theory-backed framework for solving hierarchical VI problems with concrete rates and practical splitting strategies that can accommodate infinite-dimensional Hilbert spaces and a wide range of operator structures.

Abstract

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples.

Regularization methods for solving hierarchical variational inequalities with complexity guarantees

TL;DR

The paper addresses solving hierarchical variational inequalities where the lower level is defined by a monotone inclusion, by introducing a double-loop diagonal equilibrium tracking method (DANTE) that combines ikhonov regularization with proximal penalties. The inner loop employs a flexible fixed-point encoding with an inertial Krasnoselskii–Mann iteration to approximate a temporal solution, while the outer loop restarts to update anchors and steer convergence. The authors establish convergence and rate guarantees for the averaged iterates in terms gap functions and , with explicit complexity bounds and performance across several splitting schemes (two-operator and three-operator). Numerical experiments on equilibrium selection, least-norm least-squares, and image inpainting validate the method’s versatility and show competitive convergence behavior across different problem structures. Overall, the work provides a broad, theory-backed framework for solving hierarchical VI problems with concrete rates and practical splitting strategies that can accommodate infinite-dimensional Hilbert spaces and a wide range of operator structures.

Abstract

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples.
Paper Structure (31 sections, 15 theorems, 89 equations, 10 figures, 1 table, 2 algorithms)

This paper contains 31 sections, 15 theorems, 89 equations, 10 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization of the paper.
  4. Notation and Definitions
  5. Outline of the Algorithm
  6. Inner Loop
  7. Restarting Procedure
  8. Convergence Analysis
  9. Convergence Analysis of Inner Loop
  10. Parameters Analysis
  11. Convergence Analysis of Restarting Procedure
  12. Total Complexity Analysis
  13. Relevant Splitting Methods
  14. Two-Operator Splitting Schemes
  15. Forward-backward splitting.
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $(\tau_k)_k\subset [0, 1], (\theta_k)_k\subset [0, \overline\theta]$ for $\overline\theta\in (0, 1)$, $\varepsilon>0$, and let $(\mathsf{T}_k)_k$ be a sequence of operators satisfying Assumption ass:contraction. Denote by $p$ the common fixed-point of $(\mathsf{T}_k)_k$. If the stopping time $\m

Figures (10)

  • Figure 1: Largest acceleration parameter $\overline\tau$ as a function of the relaxation and the contraction parameters $\theta$ and $q$.
  • Figure 2: Average iterates for various initial points and proximal parameters.
  • Figure 3: Optimality gap $\operatorname{Gap}_{\operatorname{opt}}(\overline w_n)$ for various initial points and proximal parameters.
  • Figure 4: Feasibility gap $\operatorname{Gap}_{\operatorname{feas}}(\overline w_n)$ for various initial points and proximal parameters.
  • Figure 5: Optimality gap $\operatorname{Gap}_{\operatorname{opt}}(\overline w_n)$ for various auxiliary problem fixed-point encodings.
  • ...and 5 more figures

Theorems & Definitions (41)

  • Example 1.1: Hierarchical Variational Inequality
  • Example 1.2: Equilibrium Selection
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Remark 3.1
  • ...and 31 more