
Journal of Convex Analysis 30 (2023), No. 2, 413440 Copyright Heldermann Verlag 2023 Preservation or Not of the Maximally Monotone Property by GraphConvergence Samir Adly Laboratoire XLIM, Université de Limoges, France samir.adly@unilim.fr Hedy Attouch IMAG, Université de Montpellier, France hedy.attouch@umontpellier.fr Ralph Tyrell Rockafellar Dept. of Mathematics, University of Washington, Seattle, U.S.A. rtr@uw.edu [Abstractpdf] \def\cH{\mathcal H} \def\N{{\mathbb N}} In a general real Hilbert space $\cH$, given a sequence $(A_n)_{n\in\N}$ of maximally monotone operators $A_n: \cH \rightrightarrows \cH$, which graphically converges to an operator $A$ whose domain is nonempty, we analyze if the limit operator $A$ is still maximally monotone. This question is justified by the fact that, as we show on an example in infinite dimension, the graph limit in the sense of Painlev\'eKuratowski of a sequence of maximally monotone operators may not be maximally monotone. Indeed, the answer depends on the type of graph convergence which is considered. In the case of the Painlev\'eKuratowski convergence, we give a positive answer under a local compactness assumption on the graphs of the operators $A_n$. Under this assumption, the sequence $(A_n)_{n\in\N}$ turns out to be convergent for the bounded Hausdorff topology. Inspired by this result, we show that, more generally, when the sequence $(A_n)_{n\in\N}$ of maximally monotone operators converges for the bounded Hausdorff topology to an operator whose domain is nonempty, then the limit is still maximally monotone. The answer to these questions plays a crucial role in the sensitivity analysis of monotone variational inclusions, and makes it possible to understand these questions in a unified way thanks to the concept of protodifferentiability. It also leads to revisit several notions which are based on the convergence of sequences of maximally monotone operators, in particular the notion of variational sum of maximally monotone operators. Keywords: Maximally monotone operator, graph convergence, bounded Hausdorff convergence, protodifferentiability, sensitivity analysis, variational inclusion, variational sum. MSC: 49J53, 49J52, 58C20, 49A50, 47H05, 49K40. [ Fulltextpdf (197 KB)] for subscribers only. 