On degenerate preconditioned proximal point methods under restricted monotonicity

Abstract

This work investigates the fundamental properties of the degenerate preconditioned resolvent under restricted monotonicity. We extend key notions of non-expansiveness and demiclosedness to the degenerate case. By deriving an explicit characterization of the solution set of the resolvent, we establish several necessary and sufficient conditions for both the well-posedness of the degenerate resolvent and the weak convergence of the associated degenerate proximal point method -- considered within either the range space of the preconditioner or the entire Hilbert space. These results provide new insights into the behavior of various operator splitting algorithms, particularly within the range space of the preconditioner. Numerical examples extend the augmented Lagrangian method and Douglas--Rachford splitting algorithm to non-convex settings under restricted maximal monotonicity.