  Randomized iterative methods, such as the Kaczmarz method and its variants,
have gained growing attention due to their simplicity and efficiency in solving
large-scale linear systems. Meanwhile, absolute value equations (AVE) have
attracted increasing interest due to their connection with the linear
complementarity problem. In this paper, we investigate the application of
randomized iterative methods to generalized AVE (GAVE). Our approach differs
from most existing works in that we tackle GAVE with non-square coefficient
matrices. We establish more comprehensive sufficient and necessary conditions
for characterizing the solvability of GAVE and propose precise error bound
conditions. Furthermore, we introduce a flexible and efficient randomized
iterative algorithmic framework for solving GAVE, which employs sampling
matrices drawn from user-specified distributions. This framework is capable of
encompassing many well-known methods, including the Picard iteration method and
the randomized Kaczmarz method. Leveraging our findings on solvability and
error bounds, we establish both almost sure convergence and linear convergence
rates for this versatile algorithmic framework. Finally, we present numerical
examples to illustrate the advantages of the new algorithms.
