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Modified Bregman Golden Ratio Algorithm for Mixed Variational Inequality Problems

Gourav Kumar, V. Vetrivel

TL;DR

The paper addresses solving a mixed variational inequality (MVIP) without requiring the global Lipschitz constant by introducing a Modified Bregman Golden Ratio Algorithm (B-GRAAL) with an adaptive, increasing step-size rule. The method uses Bregman distances and a proximal update $w_{k+1}=\mathrm{prox}^h_{\lambda_k g}(\bar w_k)$, and it eliminates the need for backtracking or prior $L$-knowledge. Under standard monotonicity and strong convexity assumptions, the authors prove global convergence and an $R$-linear rate of convergence to the solution; they also establish linear convergence under a strong monotonicity condition in the Bregman sense. Numerical experiments on matrix games and sparse logistic regression demonstrate improved performance compared to existing Bregman-based schemes, highlighting the practical appeal of Lipschitz-free MVIP solvers.

Abstract

In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require prior knowledge of the global Lipschitz constant of the cost operator. Under suitable assumptions, we establish the global iterate convergence as well as the R-linear rate of convergence of the modified algorithm. The numerical performance of the proposed approach is validated for the matrix game problem and the sparse logistic regression problem in machine learning.

Modified Bregman Golden Ratio Algorithm for Mixed Variational Inequality Problems

TL;DR

The paper addresses solving a mixed variational inequality (MVIP) without requiring the global Lipschitz constant by introducing a Modified Bregman Golden Ratio Algorithm (B-GRAAL) with an adaptive, increasing step-size rule. The method uses Bregman distances and a proximal update , and it eliminates the need for backtracking or prior -knowledge. Under standard monotonicity and strong convexity assumptions, the authors prove global convergence and an -linear rate of convergence to the solution; they also establish linear convergence under a strong monotonicity condition in the Bregman sense. Numerical experiments on matrix games and sparse logistic regression demonstrate improved performance compared to existing Bregman-based schemes, highlighting the practical appeal of Lipschitz-free MVIP solvers.

Abstract

In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require prior knowledge of the global Lipschitz constant of the cost operator. Under suitable assumptions, we establish the global iterate convergence as well as the R-linear rate of convergence of the modified algorithm. The numerical performance of the proposed approach is validated for the matrix game problem and the sparse logistic regression problem in machine learning.

Paper Structure

This paper contains 8 sections, 8 theorems, 51 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Modified Bregman-Golden Ratio Algorithm
  4. Linear Convergence of Modified B-GRAAL
  5. Numerical Results
  6. Matrix Games
  7. Sparse Logistic Regression
  8. Conclusion

Key Result

Lemma 2.1

hoai2024new Let $(x_k)$ and $(y_k)$ be two sequences of nonnegative numbers satisfying: Then, $(x_k)$ is convergent and $\sum_{k=0}^{+\infty} y_k<+\infty.$

Figures (6)

  • Figure 1: Matrix game results for $k=10$
  • Figure 2: Matrix game results for $k=20$
  • Figure 3: Matrix game results for $k=100$
  • Figure 4: Result of the data ijcnn1.bz2 ($M=35000, n=22$)
  • Figure 5: Result of the data a9a ($M=32561, n=123$)
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • Remark 1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Remark 2
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Theorem 3.5