Before revision:  In contrary, the monotonicity-based regularization, where we introduce constraints for the minimization problem via the monotonicity methods, has a rigorously proven theory, i.e., we prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data.
After revision:  In contrary, the monotonicity-based regularization, where we introduce constraints for the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data.