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A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities

Manoel Jardim, Claudia Sagastizábal, Mikhail Solodov

TL;DR

This paper introduces a Dantzig-Wolfe decomposition tailored to quasi-variational inequalities, splitting a QVI into a master QVI handling the hard, $x$-dependent constraints $K_g(x)$ and an easily solvable VI on the fixed set $K_h$. The subproblem uses a strongly monotone, regularly regularized operator $F^{k}$ built from approximations of $F$, gradient terms $\Gamma^{k}(y)\mu^{k}$, and a proximal term $Q^{k}(y-x^{k})$, with a gap function guiding convergence. The authors prove global convergence under natural conditions (single-valued continuous or maximally monotone $F$; continuous gradients of $g$; uniform strong monotonicity; bounded multipliers and iterates) and establish finite termination criteria via the gap. They also introduce block-separable and Jacobi-type approximations to enable parallelizable, scalable VI subproblems while preserving convergence, and demonstrate the method on Walrasian equilibrium problems and moving-set examples, showing favorable performance and scalability compared with direct solvers. Overall, the work provides a scalable framework for solving large-scale QVIs arising in economics and equilibrium modeling, with practical effectiveness demonstrated in large generalized Nash settings.

Abstract

We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming that the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-valued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we apply our algorithm to an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. Our numerical assessment demonstrates good performance and usefullness of the approach for the large-scale cases.

A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities

TL;DR

This paper introduces a Dantzig-Wolfe decomposition tailored to quasi-variational inequalities, splitting a QVI into a master QVI handling the hard, -dependent constraints and an easily solvable VI on the fixed set . The subproblem uses a strongly monotone, regularly regularized operator built from approximations of , gradient terms , and a proximal term , with a gap function guiding convergence. The authors prove global convergence under natural conditions (single-valued continuous or maximally monotone ; continuous gradients of ; uniform strong monotonicity; bounded multipliers and iterates) and establish finite termination criteria via the gap. They also introduce block-separable and Jacobi-type approximations to enable parallelizable, scalable VI subproblems while preserving convergence, and demonstrate the method on Walrasian equilibrium problems and moving-set examples, showing favorable performance and scalability compared with direct solvers. Overall, the work provides a scalable framework for solving large-scale QVIs arising in economics and equilibrium modeling, with practical effectiveness demonstrated in large generalized Nash settings.

Abstract

We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming that the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-valued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we apply our algorithm to an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. Our numerical assessment demonstrates good performance and usefullness of the approach for the large-scale cases.
Paper Structure (13 sections, 5 theorems, 63 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 5 theorems, 63 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dantzig-Wolfe decomposition for QVI
  3. General organization and master QVI definition
  4. Subproblem definition
  5. Algorithm statement and its convergence
  6. Gap function and finite termination
  7. Asymptotic convergence of Algorithm \ref{['algo1']}
  8. Block-Separable Approximations
  9. Numerical Results
  10. Walrasian Equilibrium Problems
  11. Problem formulation and QVI blockwise structure
  12. Solvers, data generation, and results
  13. Moving Set Problems

Key Result

Theorem 1

Under any suitable constraint qualification, if the pair $\left(x^k,z^k_m\in F(x^k)\right)$ solves the master QVI master at iteration $k$, then there exists a Lagrange multiplier $\mu^{k} \in \mathbb{R}^m$ such that

Figures (2)

  • Figure 1: Some elements of the DW decomposition for \ref{['qVI']}-\ref{['Kx']}. On the left, the master problem solves a QVI that outputs $x^k$ and a multiplier $\mu^k$ associated to the $g$-constraints. This primal-dual pair is used by the subproblem on the right to define the operator $F^k$, and return a solution $y^{k+1}$ to the master problem. The output of the subproblem is used by the master in the next iteration, to define the set $K_h^{k+1}$.
  • Figure 2: Time statistics (median and 25%-75% quantiles) for direct and dw, separating the three different ratios between goods and consumers in Table \ref{['tab2']}. For better illustration, three different economy configurations are plotted separately, in a semilogarithmic scale and barring the last two lines in the table.

Theorems & Definitions (10)

  • Remark 1: On strong monotonicity of subproblem operators
  • Theorem 1: KKT conditions for master QVI
  • Lemma 2: Gap properties
  • proof
  • Corollary 3: Finite termination
  • proof
  • Theorem 4
  • proof
  • Proposition 5: Monotonicity properties of the Jacobi approximation
  • proof