The randomized Kaczmarz methods are a popular and effective family of iterative methods for solving large-scale linear systems of equations, which have also been applied to linear feasibility problems. In this work, we propose a new block variant of the randomized Kaczmarz method, B-MRK, for solving linear feasibility problems defined by matrices. We show that B-MRK converges linearly in expectation to the feasible region.Furthermore, we extend the method to solve tensor linear feasibility problems defined under the tensor t-product. A tensor randomized Kaczmarz (TRK) method, TRK-L, is proposed for solving linear feasibility problems that involve mixed equality and inequality constraints. Additionally, we introduce another TRK method, TRK-LB, specifically tailored for cases where the feasible region is defined by linear equality constraints coupled with bound constraints on the variables. We show that both of the TRK methods converge linearly in expectation to the feasible region. Moreover, the effectiveness of our methods is demonstrated through numerical experiments on various Gaussian random data and applications in image deblurring.

翻译：随机Kaczmarz方法是求解大规模线性方程组的一类流行且有效的迭代方法，该方法亦被应用于线性可行性问题。本文针对矩阵定义的线性可行性问题，提出了一种新的随机Kaczmarz块变体方法——B-MRK。我们证明B-MRK在期望意义下线性收敛至可行域。进一步地，我们将该方法扩展至张量t-积框架下定义的张量线性可行性问题求解。针对混合等式与不等式约束的线性可行性问题，提出了张量随机Kaczmarz方法TRK-L。此外，针对变量带边界约束的线性等式约束可行域问题，我们专门设计了另一种TRK方法——TRK-LB。理论分析表明两种TRK方法均能在期望意义下线性收敛至可行域。最后，通过多组高斯随机数据实验及图像去模糊应用，验证了所提方法的有效性。