On degenerate preconditioned proximal point methods under restricted monotonicity

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.

💡 Research Summary

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This paper investigates the fundamental properties of the degenerate preconditioned resolvent (T=(A+Q)^{-1}Q) when the linear preconditioner (Q) is only positive semidefinite and therefore singular (i.e., (\ker Q\neq{0})). In the classical setting where (Q) is positive definite, the preconditioned proximal point method (PPM) (x_{k+1}=Tx_k) is well‑understood: (T) has full domain, is single‑valued, non‑expansive in the (Q)‑induced norm, and its fixed‑point iteration converges weakly under mild assumptions. When (Q) is degenerate, these properties no longer follow automatically, and existing works either assume an “admissible preconditioner” (essentially (\operatorname{ran}Q\subseteq\operatorname{ran}(A+Q)) and a global injectivity condition) or impose strong monotonicity conditions that are difficult to verify in practice.

The authors introduce restricted monotonicity: the graph of (A) restricted to the subspace (\operatorname{ran}Q) (i.e. (\operatorname{gra}A\cap(H\times\operatorname{ran}Q))) is required to be monotone. This condition is strictly weaker than global monotonicity and directly couples the degeneracy of (Q) with the monotonicity of (A). Lemma 2.1 provides an explicit decomposition of the solution set of the resolvent equation (y\in (A+Q)^{-1}Qx). By writing any vector (x) as (x_r+x_k) with (x_r\in\operatorname{ran}Q) and (x_k\in\ker Q), the authors show that the range component (y_r) is uniquely determined by a transformed operator \