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Asymptotic Convergence of the Frank-Wolfe Algorithm for Monotone Variational Inequalities

Matthew Hough

Abstract

We consider the Frank-Wolfe algorithm for solving variational inequalities over compact, convex sets under a monotone $C^1$ operator and vanishing, nonsummable step sizes. We introduce a continuous-time interpolation of the discrete iteration and use tools from dynamical systems theory to analyze its asymptotic behavior. This allows us to derive convergence results for the original discrete algorithm. Consequently, every cluster point of the iterates is a solution of the underlying variational inequality, the distance from the iterates to the solution set converges to zero, and the Frank-Wolfe gap vanishes asymptotically. In the strongly monotone case, the solution is unique and the iterates converge to it. In particular, this proves Hammond's conjecture on the convergence of generalized fictitious play.

Asymptotic Convergence of the Frank-Wolfe Algorithm for Monotone Variational Inequalities

Abstract

We consider the Frank-Wolfe algorithm for solving variational inequalities over compact, convex sets under a monotone operator and vanishing, nonsummable step sizes. We introduce a continuous-time interpolation of the discrete iteration and use tools from dynamical systems theory to analyze its asymptotic behavior. This allows us to derive convergence results for the original discrete algorithm. Consequently, every cluster point of the iterates is a solution of the underlying variational inequality, the distance from the iterates to the solution set converges to zero, and the Frank-Wolfe gap vanishes asymptotically. In the strongly monotone case, the solution is unique and the iterates converge to it. In particular, this proves Hammond's conjecture on the convergence of generalized fictitious play.
Paper Structure (6 sections, 10 theorems, 59 equations)

This paper contains 6 sections, 10 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Contributions.
  4. The general monotone case
  5. The strongly monotone case
  6. Conclusion

Key Result

Lemma 2.1

$\mathop{\rm SOL}(\mathcal{C},F)$ is nonempty.

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Lemma 2.4
  • proof
  • ...and 12 more