A Relaxed Randomized Averaging Block Extended Bregman-Kaczmarz Method for Combined Optimization Problems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912756954824704 |
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| author | Dong, Zeyu Xiao, Aqin Yin, Guojian Yin, Junfeng |
| author_facet | Dong, Zeyu Xiao, Aqin Yin, Guojian Yin, Junfeng |
| contents | Randomized Kaczmarz-type methods are widely used for their simplicity and efficiency in solving large-scale linear systems and optimization problems. However, their applicability is limited when dealing with inconsistent systems or incorporating structural information such as sparsity. In this work, we propose a \emph{relaxed randomized averaging block extended Bregman-Kaczmarz} (rRABEBK) method for solving a broad class of combined optimization problems. The proposed method integrates an averaging block strategy with two relaxation parameters to accelerate convergence and enhance numerical stability. We establish a rigorous convergence theory showing that rRABEBK achieves linear convergence in expectation, with explicit constants that quantify the effect of the relaxation mechanism, and a provably faster rate than the classical randomized extended Bregman-Kaczmarz method. Our method can be readily adapted to sparse least-squares problems and extended to both consistent and inconsistent systems without modification. Complementary numerical experiments corroborate the theoretical findings and demonstrate that rRABEBK significantly outperforms the existing Kaczmarz-type algorithms in terms of both iteration complexity and computational efficiency, highlighting both its practical and theoretical advantages. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09825 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Relaxed Randomized Averaging Block Extended Bregman-Kaczmarz Method for Combined Optimization Problems Dong, Zeyu Xiao, Aqin Yin, Guojian Yin, Junfeng Numerical Analysis 65F10, 65F22 Randomized Kaczmarz-type methods are widely used for their simplicity and efficiency in solving large-scale linear systems and optimization problems. However, their applicability is limited when dealing with inconsistent systems or incorporating structural information such as sparsity. In this work, we propose a \emph{relaxed randomized averaging block extended Bregman-Kaczmarz} (rRABEBK) method for solving a broad class of combined optimization problems. The proposed method integrates an averaging block strategy with two relaxation parameters to accelerate convergence and enhance numerical stability. We establish a rigorous convergence theory showing that rRABEBK achieves linear convergence in expectation, with explicit constants that quantify the effect of the relaxation mechanism, and a provably faster rate than the classical randomized extended Bregman-Kaczmarz method. Our method can be readily adapted to sparse least-squares problems and extended to both consistent and inconsistent systems without modification. Complementary numerical experiments corroborate the theoretical findings and demonstrate that rRABEBK significantly outperforms the existing Kaczmarz-type algorithms in terms of both iteration complexity and computational efficiency, highlighting both its practical and theoretical advantages. |
| title | A Relaxed Randomized Averaging Block Extended Bregman-Kaczmarz Method for Combined Optimization Problems |
| topic | Numerical Analysis 65F10, 65F22 |
| url | https://arxiv.org/abs/2512.09825 |