[Paper Review] For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick-Phelps type all coincide with monotonicity of the adjoint

Heinz H. Bauschke, Jonathan M. Borwein+2 more|arXiv (Cornell University)|Mar 31, 2011

Optimization and Variational Analysis43 references7 citations

TL;DR

This paper establishes that for maximally monotone linear relations on a Banach space, the properties of dense type (D), negative-infimum type (NI), and Fitzpatrick-Phelps type (FP) are equivalent and all equivalent to the monotonicity of the adjoint operator. The proof resolves two longstanding open problems in monotone operator theory by showing that monotonicity of the adjoint implies all three types, and vice versa, using advanced tools from convex analysis and duality theory in Banach spaces.

ABSTRACT

It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negative-infimum type, and of Fitzpatrick-Phelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by Simons.

Motivation & Objective

• To resolve a problem posed by Phelps and Simons regarding whether monotonicity of the adjoint implies type (D) for linear, maximally monotone operators.
• To answer Simons’ question on whether type (FP) implies type (NI) for linear, maximally monotone operators.
• To unify three distinct types of maximally monotone operators—(D), (NI), and (FP)—under a single characterization via adjoint monotonicity.
• To extend known results from continuous linear operators to general linear relations in Banach spaces.
• To provide a comprehensive characterization of monotonicity types in the context of linear relations using Fenchel conjugates and duality.

Proposed method

• Utilizes the Fenchel conjugate and duality theory to analyze the structure of the graph of the operator and its adjoint.
• Applies the concept of weak* × strong convergence of nets in the bidual space to characterize type (D) operators.
• Employs the indicator function and epigraphical techniques to handle convexity and closedness properties of the operator's graph.
• Uses the Fenchel-Moreau theorem and subdifferential calculus to derive necessary and sufficient conditions for monotonicity types.
• Applies the Fenchel conjugate of the function $ F(x,x^*) = \langle x,x^*\rangle + \iota_{\operatorname{gra}A}(x,x^*) $ to link the operator’s properties to its adjoint.
• Employs contradiction arguments based on infimal convolutions and continuity of indicator functions to derive key inequalities.

Experimental results

Research questions

• RQ1 Does monotonicity of the adjoint imply that a maximally monotone linear relation is of type (D)?
• RQ2 Does being of type (FP) imply that a maximally monotone linear relation is of type (NI)?
• RQ3 Are the three types—(D), (NI), and (FP)—equivalent for maximally monotone linear relations in a general Banach space?
• RQ4 Can the equivalence of these types be established without assuming continuity of the operator?
• RQ5 What is the precise relationship between the monotonicity of the adjoint and the three types of maximally monotone operators?

Key findings

• For maximally monotone linear relations on a Banach space, the three types—(D), (NI), and (FP)—are equivalent.
• Monotonicity of the adjoint operator is equivalent to all three types of maximally monotone operators.
• The paper provides an affirmative answer to a problem posed by Phelps and Simons regarding the converse of a known implication in monotone operator theory.
• The paper resolves Simons’ Problem 47.6 by showing that type (FP) implies type (NI) for linear, maximally monotone operators.
• The equivalence holds even without assuming continuity of the operator, extending previous results to the general linear relation setting.
• An explicit example is provided where $ A $ on $ L^1[0,1] $ is of type (D), (NI), and (FP), and $ A^* $ is monotone, confirming the theory.