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Generalized Bregman Projection Algorithms for Solving Nonlinear Split Feasibility Problems in Infinite-Dimensional Spaces

Saeed Hashemi Sababe, Ehsan Lotfali Ghasab

TL;DR

This work tackles nonlinear split feasibility problems ($SFP$s) in infinite-dimensional Hilbert spaces by developing generalized Bregman projection algorithms that fuse Bregman distances, proximal gradient steps, and adaptive inertial terms with Armijo-style step-size control. The methods extend classical $SFP$ techniques to nonlinear operators and broader classes of Bregman functions, and they guarantee strong convergence under mild assumptions. They establish a Bregman-distance decrease mechanism, residual summability, and iteration-complexity bounds, with $N=\mathcal{O}(1/\epsilon^2)$ for Algorithm 1 and $N=\mathcal{O}(1/\epsilon)$ for Algorithm 2. Numerical experiments on linear, nonlinear, and composite problems validate robustness and efficiency, demonstrating improvements over traditional $\mathcal{C}\mathcal{Q}$-type methods and highlighting the benefits of the hybrid proximal-Bregman approach.

Abstract

This paper introduces generalized Bregman projection algorithms for solving nonlinear split feasibility problems (SF P s) in infinitedimensional Hilbert spaces. The methods integrate Bregman projections, proximal gradient steps, and adaptive inertial terms to enhance convergence. Strong convergence is established under mild assumptions, and numerical experiments demonstrate the efficiency and robustness of the proposed algorithms in comparison to classical methods. These results contribute to advancing optimization techniques for nonlinear and high-dimensional problems.

Generalized Bregman Projection Algorithms for Solving Nonlinear Split Feasibility Problems in Infinite-Dimensional Spaces

TL;DR

This work tackles nonlinear split feasibility problems (s) in infinite-dimensional Hilbert spaces by developing generalized Bregman projection algorithms that fuse Bregman distances, proximal gradient steps, and adaptive inertial terms with Armijo-style step-size control. The methods extend classical techniques to nonlinear operators and broader classes of Bregman functions, and they guarantee strong convergence under mild assumptions. They establish a Bregman-distance decrease mechanism, residual summability, and iteration-complexity bounds, with for Algorithm 1 and for Algorithm 2. Numerical experiments on linear, nonlinear, and composite problems validate robustness and efficiency, demonstrating improvements over traditional -type methods and highlighting the benefits of the hybrid proximal-Bregman approach.

Abstract

This paper introduces generalized Bregman projection algorithms for solving nonlinear split feasibility problems (SF P s) in infinitedimensional Hilbert spaces. The methods integrate Bregman projections, proximal gradient steps, and adaptive inertial terms to enhance convergence. Strong convergence is established under mild assumptions, and numerical experiments demonstrate the efficiency and robustness of the proposed algorithms in comparison to classical methods. These results contribute to advancing optimization techniques for nonlinear and high-dimensional problems.
Paper Structure (6 sections, 9 theorems, 76 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 6 sections, 9 theorems, 76 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed Algorithms
  4. Convergence Analysis
  5. Numerical Experiments
  6. Conclusion and Future Directions

Key Result

Lemma 2.1

Let $\varpi: H \to \mathbb{R}$ be $\delta$-strongly convex, Fréchet differentiable, and bounded on bounded subsets of $H$. If $\{\zeta_n\}$ and $\{\varsigma_n\}$ are sequences in $H$ such that: then:

Figures (1)

  • Figure 1: Convergence behavior of the residual norm $\|\zeta_{n+1} - \zeta_n\|$ for Example 2. The plot illustrates the faster convergence of the proposed algorithms compared to traditional methods.

Theorems & Definitions (18)

  • Lemma 2.1: Weak Convergence of Bregman Distance Reich2016
  • Lemma 2.2: Nonexpansiveness of Bregman Projections Eckstein1993
  • Lemma 2.3: Strong Convergence Reich2016
  • Lemma 4.1: Well-Definedness of the Armijo Line Search
  • proof
  • Lemma 4.2: Bregman Distance Decrease
  • proof
  • Lemma 4.3: Summability of Residuals
  • proof
  • Theorem 4.4: Strong Convergence
  • ...and 8 more