Related papers: New Demiclosedness Principles for (firmly) nonexpa…
Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and transparent arguments…
In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is…
We extend to $p$-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise $\alpha$-averaged mappings. Our main…
We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary…
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…
This work investigates the fundamental properties of the degenerate preconditioned resolvent under restricted monotonicity. We extend key notions of non-expansiveness and demiclosedness to the degenerate case. By deriving an explicit…
We introduce the class of $\alpha$-firmly nonexpansive and quasi $\alpha$-firmly nonexpansive operators on $r$-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $\alpha$-firmly nonexpansive…
We in this paper study the nonexpansive operators equipped with arbitrary metric and investigate the connections between firm nonexpansiveness, cocoerciveness and averagedness. The convergence of the associated fixed-point iterations is…
Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive operators, such methods suffer many well known downsides, which…
We introduce a new fixed point theorem of Krasnoselskii type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and…
This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock…
We propose a methodology for studying the performance of common splitting methods through semidefinite programming. We prove tightness of the methodology and demonstrate its value by presenting two applications of it. First, we use the…
The Douglas-Rachford algorithm can be represented as the fixed point iteration of a firmly nonexpansive operator. When the operator has no fixed points, the algorithm's iterates diverge, but the difference between consecutive iterates…
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed…
We introduce and analyze an abstract algorithm that aims to find the projection onto a closed convex subset of a Hilbert space. When specialized to the fixed point set of a quasi nonexpansive mapping, the required sufficient condition…
The Douglas--Rachford algorithm is a classic splitting method for finding a zero of the sum of two maximal monotone operators. It has also been applied to settings that involve one weakly and one strongly monotone operator. In this work, we…
The Douglas-Rachford splitting method is a classical and widely used algorithm for solving monotone inclusions involving the sum of two maximally monotone operators. It was recently shown to be the unique frugal, no-lifting…
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…
The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…