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A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds

Shikher Sharma, Simeon Reich

TL;DR

This work addresses solving monotone equilibrium problems on Hadamard manifolds by developing a Bregman regularized proximal point method. It defines a regularized bifunction $\\tilde{F}$ using a Bregman distance $D_{\\phi}$ and proves convergence under a weakened coercivity assumption, together with a convexity condition on the regularization-induced set. The proposed algorithm iteratively solves regularized equilibrium problems, yielding nonincreasing Bregman distances and convergence to a solution of $EP(F,C)$. Numerical experiments on illustrative Hadamard manifolds demonstrate practical effectiveness and reveal how the choice of Bregman distance impacts convergence behavior.

Abstract

In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method.

A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds

TL;DR

This work addresses solving monotone equilibrium problems on Hadamard manifolds by developing a Bregman regularized proximal point method. It defines a regularized bifunction using a Bregman distance and proves convergence under a weakened coercivity assumption, together with a convexity condition on the regularization-induced set. The proposed algorithm iteratively solves regularized equilibrium problems, yielding nonincreasing Bregman distances and convergence to a solution of . Numerical experiments on illustrative Hadamard manifolds demonstrate practical effectiveness and reveal how the choice of Bregman distance impacts convergence behavior.

Abstract

In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method.
Paper Structure (11 sections, 10 theorems, 79 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 10 theorems, 79 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equilibrium Problems
  4. Bregman distances in Hadamard manifolds
  5. A Bregman Regularization for Equilibrium Problems
  6. A Bregman Regularized Proximal Point Method
  7. Numerical Experiments
  8. Conclusion
  9. Appendix
  10. Geodesically convex set $K_x$
  11. Geodesically non convex set $K_x$

Key Result

Lemma 1

Let $\{x_n\}$ be a sequence in an Hadamard manifold $\mathcal{M}$ such that $x_n \to x\in \mathcal{M}$. Then, for any $y \in \mathcal{M}$, we have Moreover, for any $u\in \mathcal{T}_{x_0}\mathcal{M}$, the function $V \colon \mathcal{M} \to \mathcal{T}\mathcal{M}$ defined by $V(x)=\underset{x\leftarrow x_0}{\mathrm{P}}(u)$ for all $x\in \mathcal{M}$ is continuous.

Figures (1)

  • Figure 1: Convergence behavior of Algorithm \ref{['BregAlg']} for $\lambda=0.3$, $\lambda=0.6$ and $\lambda=0.3$

Theorems & Definitions (37)

  • Definition 1: Li2009JLM
  • Remark 1: Li2009JLM
  • Definition 2: Li2009JLM
  • Definition 3: Li2009JLM
  • Lemma 1: Li2009JLM
  • Remark 2
  • Remark 3
  • Definition 4: Ferreira2002
  • Definition 5: Udriste
  • Definition 6: Jost1995
  • ...and 27 more