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Extragradient methods with complexity guarantees for hierarchical variational inequalities

Pavel Dvurechensky, Meggie Marschner, Shimrit Shtern, Mathias Staudigl

TL;DR

The paper addresses solving very general hierarchical variational inequalities in real Hilbert spaces, incorporating lower- and upper-level data via monotone operators ${\mathsf{F}}_2, {\mathsf{F}}_1$ and convex (possibly non-smooth) terms $g_2,g_1$. It introduces an optimistic extragradient algorithm with a decreasing Tikhonov-like regularization sequence $\sigma_k$, achieving explicit sublinear rates for both the lower-level feasibility gap and the overall optimality gap, under an Attouch-Czarnecki-type summability condition and a weak-sharpness geometry on the lower level. The results extend to unbounded domains, general $g_i$ (not just indicator functions), and yield improved rates under strong monotonicity of the upper-level operator, with extensions to Tseng splitting. Numerical experiments on hierarchical Nash equilibria and constrained min-max problems illustrate the method’s practicality and confirm the predicted convergence behavior. Overall, the work advances the theory and practice of hierarchical HVIs by providing general complexity guarantees, broader modeling flexibility, and computationally efficient iterative schemes.

Abstract

In the framework of a real Hilbert space we consider the problem of approaching solutions to a class of hierarchical variational inequality problems, subsuming several other problem classes including certain mathematical programs under equilibrium constraints, constrained min-max problems, hierarchical game problems, optimal control under VI constraints, and simple bilevel optimization problems. For this general problem formulation, we establish rates of convergence in terms of suitably constructed gap functions, measuring feasibility gaps and optimality gaps. We present worst-case iteration complexity results on both levels of the variational problem, as well as weak convergence under a geometric weak sharpness condition on the lower level solution set. Our results match and improve the state of the art in terms of their iteration complexity and the generality of the problem formulation.

Extragradient methods with complexity guarantees for hierarchical variational inequalities

TL;DR

The paper addresses solving very general hierarchical variational inequalities in real Hilbert spaces, incorporating lower- and upper-level data via monotone operators and convex (possibly non-smooth) terms . It introduces an optimistic extragradient algorithm with a decreasing Tikhonov-like regularization sequence , achieving explicit sublinear rates for both the lower-level feasibility gap and the overall optimality gap, under an Attouch-Czarnecki-type summability condition and a weak-sharpness geometry on the lower level. The results extend to unbounded domains, general (not just indicator functions), and yield improved rates under strong monotonicity of the upper-level operator, with extensions to Tseng splitting. Numerical experiments on hierarchical Nash equilibria and constrained min-max problems illustrate the method’s practicality and confirm the predicted convergence behavior. Overall, the work advances the theory and practice of hierarchical HVIs by providing general complexity guarantees, broader modeling flexibility, and computationally efficient iterative schemes.

Abstract

In the framework of a real Hilbert space we consider the problem of approaching solutions to a class of hierarchical variational inequality problems, subsuming several other problem classes including certain mathematical programs under equilibrium constraints, constrained min-max problems, hierarchical game problems, optimal control under VI constraints, and simple bilevel optimization problems. For this general problem formulation, we establish rates of convergence in terms of suitably constructed gap functions, measuring feasibility gaps and optimality gaps. We present worst-case iteration complexity results on both levels of the variational problem, as well as weak convergence under a geometric weak sharpness condition on the lower level solution set. Our results match and improve the state of the art in terms of their iteration complexity and the generality of the problem formulation.
Paper Structure (35 sections, 10 theorems, 115 equations, 2 figures, 1 algorithm)

This paper contains 35 sections, 10 theorems, 115 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivating Examples
  3. Simple bilevel optimization
  4. Hierarchical variational inequalities
  5. Hierarchical jointly convex generalized Nash equilibrium problem
  6. The lower-level Nash Equilibrium Problem (NEP)
  7. The upper level GNEP
  8. Related works
  9. Contributions
  10. Facts on variational inequalities
  11. Preliminaries
  12. Hemi-variational inequalities (HVI's)
  13. Sharpness and error-bound property
  14. Constraint qualifications
  15. Gap functions for hierarchical HVI's
  16. ...and 20 more sections

Key Result

Lemma 2.1

Let $\mathcal{C}\subset\mathop{\mathrm{dom}}\nolimits(g)$ be a nonempty compact convex set. Consider problem $\mathop{\mathrm{HVI}}\nolimits({\mathsf{F}},g)$ with ${\mathsf{F}}:\mathcal{Z}\to\mathcal{Z}$ monotone and Lipschitz continuous. The function $x\mapsto \Theta(x\vert{\mathsf{F}},g,\mathcal{C

Figures (2)

  • Figure 1: Evolution of the strategy profile and error plot for the hierarchical NEP.
  • Figure 2: Evolution of the approximate solution $u^k$ after completion of the $k$-th iteration of the Algorithm (left), and the evolution of the residual relative to the true solution (right).

Theorems & Definitions (24)

  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4: Weak Sharpness
  • Remark 2.1
  • Definition 2.5: Feasibility gap
  • Definition 2.6: Optimality gap
  • Lemma 2.7
  • Remark 3.1
  • Remark 3.2
  • ...and 14 more