Related papers: On the generalized parallel sum of two maximal mon…
We present a new sufficient condition under which a maximal monotone operator $T:X\tos X^*$ admits a unique maximal monotone extension to the bidual $\widetilde T:X^{**} \rightrightarrows X^*$. For non-linear operators this condition is…
The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some…
A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a…
We give, for general Banach spaces, a characterization of the sequential lower limit of maximal monotone operators of type (D) and prove its representability. As a consequence, using a recent extension of the Moreau-Yosida regularization…
Gossez type (D) operators are defined in non-reflexive Banach spaces and share with the subdifferential a topological related property, characterized by bounded nets. In this work we present new properties and characterizations of these…
We define a family of linear type (D) operators for which the inverse of their maximal monotone extensions to the bidual are not of type (D) and provide an example of an operator in this family.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we…
In a real Banach space, we first prove that the sum of a monotone operator of type (FPV) and maximal monotone operator Rockafellar's constraint qualification is maximal. This prove leads to the solution of most interesting long-time…
Previous constructions of non-type (D) maximal monotone operators were based on the non-type (D) operators introduced by Gossez, and the construction of such operators or the proof that they were non-type (D) were not straightforward. The…
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 paper, we prove the maximal…
In this paper we provide the resolvent computation of the parallel composition of a maximally monotone operator by a linear operator under mild assumptions. Connections with a modification of the warped resolvent are provided. In the…
In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new…
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum…
The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal…
Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued,…
We focus on the linear convergence of generalized proximal point algorithms for solving monotone inclusion problems. Under the assumption that the associated monotone operator is metrically subregular or that the inverse of the monotone…
In his recent Proceedings of the AMS paper "Gossez's skew linear map and its pathological maximally monotone multifunctions", Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the…
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'…
We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of…
The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it…