Autoconjugate functions and representations of monotone operators

Penot, Jean-Paul

doi:10.1017/s0004972700033748

Cited by 12 publications

(13 citation statements)

References 20 publications

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“…The kernel average can also be used to represent maximal monotone operators. Representations with attractive duality properties were studied previously by Svaiter (see [23]), by Penot (see [14,15]), by Penot and Zȃlinescu ("autoconjugates"; see [16]), and by Ghoussoub ("selfdual Lagrangians"; see [11]). The works by Svaiter, by Penot, and by Ghoussoub were not explicit in the sense that either Zorn's Lemma or transfinite induction was utilized.…”

Section: Application To Monotone Operatorsmentioning

confidence: 99%

“…A comparison to previous works is in order. The works by Penot, Svaiter, and Ghoussoub (see [14,23,15,11]) are nonconstructive. Ghoussoub also requires that X be separable.…”

Section: Theorem 57 (Extension and Representation) The Function Rmentioning

confidence: 99%

“…Monotone operators play an important role in modern optimization and analysis; see, e.g., [19,20]. The Fitzpatrick function [10] associated with A has recently turned out to be a very effective tool for analyzing A; see, e.g., [1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,21,22,23].…”

Section: The Kernel Average For Two Convex Functions 5961mentioning

confidence: 99%

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The kernel average for two convex functions and its application to the extension and representation of monotone operators

Bauschke

Wang

2009

Trans. Amer. Math. Soc.

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Abstract. We provide and analyze an average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also known as selfdual Lagrangian) which can be used for finding an explicit maximal monotone extension of the given monotone operator. This completely settles one of the open problems posed by Fitzpatrick in the setting of reflexive Banach spaces.

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Section: Application To Monotone Operatorsmentioning

confidence: 99%

“…A comparison to previous works is in order. The works by Penot, Svaiter, and Ghoussoub (see [14,23,15,11]) are nonconstructive. Ghoussoub also requires that X be separable.…”

Section: Theorem 57 (Extension and Representation) The Function Rmentioning

confidence: 99%

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The kernel average for two convex functions and its application to the extension and representation of monotone operators

Bauschke

Wang

2009

Trans. Amer. Math. Soc.

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“…In the following, we present some definitions and properties of abstract monotone operators (see [2,6,10,11,18]). …”

Section: Some Problems On Abstract Convex Functionsmentioning

confidence: 99%

Maximal Abstract Monotonicity and Generalized Fenchel’s Conjugation Formulas

Eberhard

Mohebi

2009

Set-Valued Anal

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In this paper, we present a generalization of a theory of monotone operators in the framework of abstract convexity. We show that how generalized Fenchel's conjugation formulas can be used to obtain some results on maximal abstract monotonicity. We give a necessary and sufficient condition for maximality of abstract monotone operators representable by abstract convex functions by using an additivity constraint qualification.Keywords Generalized Fenchel's duality · Monotone operator · Abstract monotonicity · Abstract convex function · Abstract convexity Mathematics Subject Classifications (2000) 47H05 · 47H04 · 52A01 · 26A51 · 26B25

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“…Now, assume that X is a set and L is a set of real valued abstract linear functions l : X −→ R defined on X, with the coupling function ., . : X × L −→ R defined by x, l := l(x) for all x ∈ X and all l ∈ L. In the following, we present some definitions and properties of abstract monotone operators (see [5,10,15,20]). …”

Section: Introductionmentioning

confidence: 99%

Some criteria for maximal abstract monotonicity

2011

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In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel's duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators. Abstract. In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel's duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators. Some Criteria for Maximal Abstract Monotonicity

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Autoconjugate functions and representations of monotone operators

Abstract: We show the existence of a convex representation of a maximal monotone operator by a convex function which is invariant with respect to the Fenchel conjugacy (up to an interchange of variables). We use the framework of generalised convexity.

Cited by 12 publications

References 20 publications

The kernel average for two convex functions and its application to the extension and representation of monotone operators

The kernel average for two convex functions and its application to the extension and representation of monotone operators

Maximal Abstract Monotonicity and Generalized Fenchel’s Conjugation Formulas

Some criteria for maximal abstract monotonicity

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