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The main focus of this paper is to study multi-valued linear monotone operators in the contexts of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness,…
The local boundedness of classes of operators is analyzed on different subsets directly related to their Fitzpatrick functions and characterizations of the topological vector spaces for which that local boundedness holds is given in terms…
The paper is devoted to establishing relationships between global and local monotonicity, as well as their maximality versions, for single-valued and set-valued mappings between finite-dimensional and infinite-dimensional spaces. We first…
Several aspects of the interplay between monotone operator theory and convex optimization are presented. The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized.…
In the context of general Banach spaces characterizations for the maximal monotonicity of operators with non-empty domain interior as well as stronger continuity properties for such operators are provided.
Our goal is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their…
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.
The paper is devoted to a systematic study and characterizations of notions of local maximal monotonicity and their strong counterparts for set-valued operators that appear in variational analysis, optimization, and their applications. We…
We study differentiability properties of convex operators defined on a Banach space with values in an $\Lc_p$ space and of their compositions with monotonic convex functionals on this space. We develop new tools for operators enjoying an…
We consider a class of monotone operators which are appropriate for symbolic representation and manipulation within a computer algebra system. Various structural properties of the class (e.g., closure under taking inverses, resolvents) are…
In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the…
We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed…
We study the relations between some geometric properties of maximal monotone operators and generic geometric and analytical properties of the functions on the associate Fitzpatrick family of convex representations. We also investigate under…
In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal…
Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions…
This work deals with a maximal monotone operator $A$ of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value $Ax$ at a given point $x$ via its values at nearby points of $x$. We…
In the first part of the note we prove that a sufficient condition (due to Simons) for the convexity of the closure of the domain/range of a monotone operator is also necessary when the operator has bounded domain and is maximal. Simons'…
Monotone convex operators and time-consistent systems of operators appear naturally in stochastic optimization and mathematical finance in the context of pricing and risk measurement. We study the dual representation of a monotone convex…
The local equicontinuity of an operator $T:X\rightrightarrows X^{*}$ with proper Fitzpatrick function $\varphi_{T}$ and defined in a barreled locally convex space $X$ has been shown to hold on the algebraic interior of…