Preservation or Not of the Maximally Monotone Property by Graph-Convergence

S. Adly ; H. Attouch ; R. T. Rockafellar

Journal of convex analysis, Tome 30 (2023) no. 2, pp. 413-44

Résumé

\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\'e-Kuratowski 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\'e-Kuratowski 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 proto-differentiability. 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.

Détail

Classification : 49J53, 49J52, 58C20, 49A50, 47H05, 49K40
Mots-clés : Maximally monotone operator, graph convergence, bounded Hausdorff convergence, proto-differentiability, sensitivity analysis, variational inclusion, variational sum