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A Frank-Wolfe Algorithm for Strongly Monotone Variational Inequalities

Reza Rahimi Baghbadorani, Peyman Mohajerin Esfahani, Sergio Grammatico

TL;DR

This work tackles strongly monotone variational inequalities on a compact convex set by introducing a projection-free, accelerated Frank-Wolfe-based algorithm that uses a FW-minimax oracle to solve inner minimax subproblems. The authors establish non-asymptotic convergence results, showing a fast decay of the gap: $f(\tilde{y}_k) \le \gamma^2 f(x_0) \exp(-k/(\gamma+1))$ with $\gamma = L/\mu$, and under additional conditions achieve $f(\tilde{y}_k)=O(1/k)$; the outer iteration complexity scales as $T \ge (\gamma+1) \log\left( \frac{\gamma^2 f(x_0)}{\varepsilon} \right)$, with overall cost scenarios of $\mathcal{O}(\log^2(1/\varepsilon))$ or $\mathcal{O}\left( \dfrac{\log(1/\varepsilon)}{\varepsilon} \right)$ depending on the inner-step strategy. The method is validated on a Traffic Assignment Problem, illustrating substantial reductions in projection complexity for large-scale network settings. Overall, the paper provides a theoretically grounded, practical projection-free approach for solving VIPs that can scale to problems where projection is expensive, with concrete convergence guarantees and a roadmap for future enhancements including adaptive stepsizes and higher-order variants.

Abstract

We propose an accelerated algorithm with a Frank-Wolfe method as an oracle for solving strongly monotone variational inequality problems. While standard solution approaches, such as projected gradient descent (aka value iteration), involve projecting onto the desired set at each iteration, a distinctive feature of our proposed method is the use of a linear minimization oracle in each iteration. This difference potentially reduces the projection cost, a factor that can become significant for certain sets or in high-dimensional problems. We validate the performance of the proposed algorithm on the traffic assignment problem, motivated by the fact that the projection complexity per iteration increases exponentially with respect to the number of links.

A Frank-Wolfe Algorithm for Strongly Monotone Variational Inequalities

TL;DR

This work tackles strongly monotone variational inequalities on a compact convex set by introducing a projection-free, accelerated Frank-Wolfe-based algorithm that uses a FW-minimax oracle to solve inner minimax subproblems. The authors establish non-asymptotic convergence results, showing a fast decay of the gap: with , and under additional conditions achieve ; the outer iteration complexity scales as , with overall cost scenarios of or depending on the inner-step strategy. The method is validated on a Traffic Assignment Problem, illustrating substantial reductions in projection complexity for large-scale network settings. Overall, the paper provides a theoretically grounded, practical projection-free approach for solving VIPs that can scale to problems where projection is expensive, with concrete convergence guarantees and a roadmap for future enhancements including adaptive stepsizes and higher-order variants.

Abstract

We propose an accelerated algorithm with a Frank-Wolfe method as an oracle for solving strongly monotone variational inequality problems. While standard solution approaches, such as projected gradient descent (aka value iteration), involve projecting onto the desired set at each iteration, a distinctive feature of our proposed method is the use of a linear minimization oracle in each iteration. This difference potentially reduces the projection cost, a factor that can become significant for certain sets or in high-dimensional problems. We validate the performance of the proposed algorithm on the traffic assignment problem, motivated by the fact that the projection complexity per iteration increases exponentially with respect to the number of links.

Paper Structure

This paper contains 5 sections, 5 theorems, 31 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Assumptions and Technical preliminaries
  3. Proposed Frank-Wolfe Algorithm and Convergence Analysis
  4. Numerical Results
  5. Further Discussion, Limitations, and Future Directions

Key Result

Proposition 2.2

Let $F(x,y)$ be $L_0$-smooth continuous and $(\mu_{\mathcal{X}}, \mu_{\mathcal{Y}})$ strongly convex-concave on a convex, compact set $\mathcal{X}\times\mathcal{Y}$ and $(x_*, y_*)$, the solution of p: min-max, is in the interior of $\mathcal{X}\times\mathcal{Y}$. Consider applying the Frank-Wolfe a where the constants are Then, the convergence rate for the error $h_k$gap-minimax is $\mathcal{O}\

Figures (2)

  • Figure 1: TAP model (Figure 2 in bertsekas2009projection).
  • Figure 2: Optimality gap of the VIP in the TAP.

Theorems & Definitions (9)

  • Proposition 2.2: FW-minimax oracle convergence gidel17a
  • Remark 2.3: FW-variants
  • Lemma 2.4: Gap function properties, nesterov2006solving
  • Lemma 2.5: Gap function upper bound, nesterov2006solving
  • Proposition 3.1: Upper bound of $\Delta_k$
  • proof
  • Theorem 3.2: VIP-convergence via FW-minimax oracle
  • Remark 3.3: Overall complexity of VIP solution
  • proof : Proof of Theorem \ref{['th: vi-conv']}