[Paper Review] The sum of a maximal monotone operator of type (FPV) and a maximal monotone operator with full domain is maximal monotone

Liangjin Yao|arXiv (Cornell University)|May 13, 2010

Optimization and Variational Analysis24 references3 citations

TL;DR

This paper establishes the maximal monotonicity of the sum of two maximal monotone operators in Banach spaces under relaxed conditions: one operator is of type (FPV), and the other has full domain. Using the Fitzpatrick function and constraint qualifications, the authors prove that $ A + B $ is maximal monotone when $ \operatorname{dom}A \cap \operatorname{int}\operatorname{dom}B \neq \varnothing $, $ A + N_{\overline{\operatorname{dom}B}} $ is of type (FPV), and $ \operatorname{dom}A \cap \overline{\operatorname{dom}B} \subseteq \operatorname{dom}B $.

ABSTRACT

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximal monotone operators such that $\dom A\cap\inte\dom B eq\varnothing$, $A+N_{\overline{\dom B}}$ is of type (FPV), and $\dom A\cap\overline{\dom B}\subseteq\dom B$. The proof utilizes the Fitzpatrick function in an essential way.

Motivation & Objective

• Address the open problem in Monotone Operator Theory concerning the maximal monotonicity of the sum of two maximal monotone operators in nonreflexive Banach spaces.
• Extend the sum theorem beyond reflexive spaces by weakening the standard Rockafellar constraint qualification.
• Provide a general framework that unifies and generalizes prior results on sums involving subdifferentials, normal cones, and linear relations.
• Establish conditions under which $ A + B $ remains maximal monotone even when $ A $ is not necessarily a subdifferential or normal cone operator.
• Use the Fitzpatrick function as a central tool to analyze monotonicity properties in the absence of reflexivity.

Proposed method

• Employ the Fitzpatrick function to characterize maximal monotonicity and analyze monotonicity relations in the sum operator.
• Use the constraint qualification $ \operatorname{dom}A \cap \operatorname{int}\operatorname{dom}B \neq \varnothing $ to ensure the existence of interior points for perturbation arguments.
• Apply the notion of type (FPV) operators, which generalize subdifferentials and normal cones, to control the behavior of $ A $ near the domain of $ B $.
• Use the condition $ \operatorname{dom}A \cap \overline{\operatorname{dom}B} \subseteq \operatorname{dom}B $ to ensure that the domain interaction does not introduce discontinuities.
• Utilize the fact that $ A + N_{\overline{\operatorname{dom}B}} $ is of type (FPV) to extend monotonicity preservation to the sum $ A + B $.
• Leverage known results on subdifferentials and normal cones as special cases to validate the general framework.

Experimental results

Research questions

• RQ1 Under what conditions is the sum $ A + B $ of two maximal monotone operators maximal monotone in nonreflexive Banach spaces?
• RQ2 Can the sum theorem be extended to operators where one is of type (FPV) and the other has full domain?
• RQ3 Does the condition $ \operatorname{dom}A \cap \overline{\operatorname{dom}B} \subseteq \operatorname{dom}B $ ensure the preservation of maximal monotonicity in the sum?
• RQ4 Can the Fitzpatrick function be used effectively to prove maximal monotonicity of sums without reflexivity?
• RQ5 How do known results on subdifferentials, normal cones, and linear relations fit into this generalized framework?

Key findings

• The sum $ A + B $ is maximal monotone if $ A $ is maximal monotone of type (FPV), $ B $ is maximal monotone with full domain, and $ \operatorname{dom}A \cap \operatorname{int}\operatorname{dom}B \neq \varnothing $.
• The condition $ A + N_{\overline{\operatorname{dom}B}} $ being of type (FPV) ensures that the interaction between $ A $ and the closure of $ \operatorname{dom}B $ preserves monotonicity.
• The result generalizes Verona and Verona’s theorem on subdifferentials and full-domain operators, as well as Voisei’s result on type (FPV) operators and normal cones.
• The sum theorem holds for linear relations $ A $ and full-domain maximal monotone operators $ B $, extending Heisler’s result.
• The example in $ L^1[0,1] $ shows that $ A + J $, where $ A $ is a maximal monotone linear relation and $ J $ is the duality mapping, is maximal monotone — a result not deducible from prior theorems.
• The framework includes the case where $ B $ is the subdifferential of a proper, lower semicontinuous, convex function with possibly non-closed domain, generalizing earlier results.