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Maximal monotonicity criteria for the composition and the sum under weak interiority conditions
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The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains.Abstract:
The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).read more
Citations
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New properties of forward–backward splitting and a practical proximal-descent algorithm
Yuanyuan Huang,Yunda Dong +1 more
TL;DR: Some new properties of forward–backward splitting are given, which extend the well-known properties of the usual projection and are used to analyze the weak convergence of the proximal-descent algorithm without assuming Lipschitz continuity of the forward operator.
Journal ArticleDOI
Linear Monotone Subspaces of Locally Convex Spaces
TL;DR: In this article, the main focus of the paper is to study multi-valued linear monotone operators in the context of locally convex spaces via the use of their Fitzpatrick and Penot functions, and notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-)representability are provided.
Journal ArticleDOI
The Sum of a Maximally Monotone Linear Relation and the Subdifferential of a Proper Lower Semicontinuous Convex Function is Maximally Monotone
TL;DR: In this paper, the maximal monotonicity of the sum of two maximally monotone operators was shown to be monotonically monotonous provided that Rockafellar's constraint qualification holds.
On Monotone linear relations and the sum problem in Banach spaces
TL;DR: In this article, the authors studied monotone operators in general Banach spaces and provided a sufficient condition for the sum problem, which is the most famous open problem in Monotone Operator Theory.
References
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Book
Convex analysis in general vector spaces
TL;DR: In this article, the authors present preliminary results on functional analysis and convex analysis in Locally Convex Spaces (LCS) and describe some applications of convex analyses in Normed Spaces.
Book
From Hahn-Banach to monotonicity
TL;DR: The Hahn-Banach-Lagrange theorem and some consequences of Fenchel duality have been studied in this article, where the sum problem for general Banach spaces has been studied.