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The notion of a firmly nonexpansive mapping is central in fixed point theory because of attractive convergence properties for iterates and the correspondence with maximal monotone operators due to Minty. In this paper, we systematically…
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.…
We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed…
Nonexpansive mappings play a central role in modern optimization and monotone operator theory because their fixed points can describe solutions to optimization or critical point problems. It is known that when the mappings are sufficiently…
We provide a novel transcription of monotone operator theory to the non-Euclidean finite-dimensional spaces $\ell_1$ and $\ell_{\infty}$. We first establish properties of mappings which are monotone with respect to the non-Euclidean norms…
While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as…
Maximally monotone operators play important roles in optimization, variational analysis and differential equations. Finding zeros of maximally monotone operators has been a central topic. In a Hilbert space, we show that most resolvents are…
Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood…
The epsilon-enlargement of a maximal monotone operator is a construct similar to the Br{\o}ndsted and Rocakfellar epsilon-subdifferential enlargement of the subdifferential. Like the epsilon-subdifferential, the epsilon-enlargement of a…
We use the method of monotone iterations to obtain fixed point and coupled fixed point results for mixed monotone operators in the setting of partially ordered sets, with no additional assumptions on the partial order and with no…
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 note, we provide a new maximal…
This paper studies the long-time behavior of stochastic differential inclusions driven by maximal monotone operators, motivated by continuous-time models of first-order optimization methods under noisy or approximate operator information.…
The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods.…
The theory of monotone operators plays a major role in modern optimization and many areas of nonlinera analysis. The central classes of monotone operators are matrices with a positive semidefinite symmetric part and subsifferential…
Resolvent compositions were recently introduced as monotonicity-preserving operations that combine a set-valued monotone operator and a bounded linear operator. They generalize in particular the notion of a resolvent average. We analyze the…
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…
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…
Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this…
This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain physical systems. The differential inclusion is described by a time-dependent set-valued mapping…
Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular…