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A new characterization of partial metric completenessFeb 08 2019In this article, we present a new characterization of the completeness of a partial metric space--which we call \textit{orbital characterization}-- using fixed point results.
Partial quasi-metric completeness via Kannan-type fixed pointsFeb 08 2019In this short note, we obtain partial quasi-metric versions of Kannan's fixed point theorem for self-mappings. Moreover, we use these fixed points results to characterize a certain type of completeness in partial quasi-metric spaces. We have reported ... More
Chatterjea type fixed point in Partial $b$-metric spacesFeb 08 2019In this paper, we give and prove two Chatterjea type fixed point theorems on partial $b$-metric space. We propose an extension to the Banach contraction principle on partial $b$-metric space which was already presented by Shukla and also study some related ... More
Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learningDec 31 2018Jan 29 2019The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive ... More
On the minimal displacement vector of compositions and convex combinations of nonexpansive mappingsSep 04 2018Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the ... More
On the asymptotic behaviour of the Aragon Artacho-Campoy algorithmMay 28 2018Arag\'on Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone ... More
Linear Convergence of Projection AlgorithmsSep 01 2016Projection algorithms are well known for theirs simplicity and flexibility in solving feasibility problems. They are particularly important in practice since softwares involving projection algorithms require minimal implementation and maintenance. In ... More
Linear Convergence of Projection AlgorithmsSep 01 2016Oct 25 2017Projection algorithms are well known for their simplicity and flexibility in solving feasibility problems. They are particularly important in practice due to minimal requirements for software implementation and maintenance. In this work, we study linear ... More
The forward-backward algorithm and the normal problemAug 07 2016The forward-backward splitting technique is a popular method for solving monotone inclusions that has applications in optimization. In this paper we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula ... More
Position paper: Towards an observer-oriented theory of shape comparisonMar 07 2016In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions ... More
On Slater's condition and finite convergence of the Douglas-Rachford algorithmApr 27 2015The Douglas-Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our ... More
The Douglas-Rachford algorithm for two (not necessarily intersecting) affine subspacesApr 14 2015The Douglas--Rachford algorithm is a classical and very successful splitting method for finding the zeros of the sums of monotone operators. When the underlying operators are normal cone operators, the algorithm solves a convex feasibility problem. In ... More
Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditionsJul 28 2014Aug 05 2016In this paper we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with ... More
On the finite convergence of a projected cutter methodMay 12 2014Aug 14 2014The subgradient projection iteration is a classical method for solving a convex inequality. Motivated by works of Polyak and of Crombez, we present and analyze a more general method for finding a fixed point of a cutter, provided that the fixed point ... More
An abstract inverse problem for boundary triples with an application to the Friedrichs ModelApr 27 2014Dec 05 2014We discuss the detectable subspaces of an operator. We analyse the relation between the M-function (the abstract Dirichlet to Neumann map) and the resolvent bordered by projections onto the detectable subspaces. The abstract results are explored further ... More
Linear Convergence of the Douglas-Rachford Method for Two Closed SetsJan 25 2014Feb 19 2015In this paper, we investigate the Douglas-Rachford method for two closed (possibly nonconvex) sets in Euclidean spaces. We show that under certain regularity conditions, the Douglas-Rachford method converges locally with R-linear rate. In convex settings, ... More
Combining persistent homology and invariance groups for shape comparisonDec 27 2013Jan 28 2016In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While persistent homology ... More
The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the Friedrichs angleSep 18 2013Dec 21 2013The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method ... More
Alternating Projections and Douglas-Rachford for Sparse Affine FeasibilityJul 08 2013Mar 14 2014The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved nonconvex relaxations. ... More
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problemsDec 13 2012Jun 24 2013We consider projection algorithms for solving (nonconvex) feasibility problems in Euclidean spaces. Of special interest are the Method of Alternating Projections (MAP) and the Douglas-Rachford or Averaged Alternating Reflection Algorithm (AAR). In the ... More
Restricted normal cones and sparsity optimization with affine constraintsMay 02 2012The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely-behaved non convex relaxations. ... More
Restricted normal cones and the method of alternating projectionsMay 02 2012The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, ... More
Firmly nonexpansive mappings in classes of geodesic spacesMar 07 2012Nov 23 2012Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard ... More
Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regularDec 21 2011Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex ... More
Attouch-Théra duality revisited: paramonotonicity and operator splittingOct 21 2011The problem of finding the zeros of the sum of two maximally monotone operators is of fundamental importance in optimization and variational analysis. In this paper, we systematically study Attouch-Th\'era duality for this problem. We provide new results ... More
Near equality, near convexity, sums of maximally monotone operators, and averages of firmly nonexpansive mappingsApr 29 2011We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings ... More
Fixed Points of Averages of Resolvents: Geometry and AlgorithmsFeb 08 2011To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century when Carl Friedrich Gauss developed the method of least ... More
Compositions and Averages of Two Resolvents: Relative Geometry of Fixed Points Sets and a Partial Answer to a Question by C. ByrneMar 25 2010We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted considerable attention ... More
$γ$-Radonifying operators -- a surveyNov 19 2009Feb 09 2010We present a survey of the theory of $\gamma$-radonifying operators and their applications to stochastic integration in Banach spaces.