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Adaptive Randomized Extended Bregman-Kaczmarz Method for Combined Optimization Problems

Combined optimization problems that couple data-fidelity and regularization terms arise naturally in a wide range of inverse problems. In this paper, we study an adaptive randomized averaging block extended Bregman-Kaczmarz (aRABEBK) method for solving such problems. The proposed method incorporates iteration-wise relaxation parameters that are automatically adjusted using residual information, allowing for more aggressive step sizes without additional manual tuning. We establish a convergence theory for the proposed framework and derive expected linear convergence rate guarantees. Numerical experiments on both synthetic and real data sets for sparse and minimum-norm least-squares problems demonstrate that our aRABEBK method achieves faster convergence and improved robustness compared with state-of-the-art extended Kaczmarz and Bregman-Kaczmarz-type algorithms, including its nonadaptive counterpart.