Related papers: Generalized monotone operators and their averaged …
Firmly 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…
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…
Firstly, we invoke the weak convergence (resp. strong convergence) of translated basic methods involving nonexpansive operators to establish the weak convergence (resp. strong convergence) of the associated method with both perturbation and…
We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary.…
Various optimal gradient-based algorithms have been developed for smooth nonconvex optimization. However, many nonconvex machine learning problems do not belong to the class of smooth functions and therefore the existing algorithms are…
This paper seeks to advance the theory of nonexpansive mappings by introducing and exploring a novel class of nonexpansive type mappings, which we aptly designate as perimetric nonexpansive mappings. We establish that the collection of…
We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that---depending upon how parameters are chosen---resemble \emph{alternating projections}, the \emph{Douglas--Rachford method}, \emph{relaxed…
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized…
In this paper, a new optimization framework is defined that includes the optimization framework recently proposed in [1]-[2] as a special case. The convex optimization in [1]-[2] includes centralized optimization and distributed…
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…
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…
This paper is a continuation to the study of generalized quasi contractive operators, essentially due to Akhtar et al. [A multi-step implicit iterative process for common fixed points of generalized C^{q}-operators in convex metric spaces,…
Reduction operators, i.e. the operators of nonclassical (or conditional) symmetry of a class of variable coefficient nonlinear wave equations with power nonlinearities is investigated within the framework of singular reduction operator. A…
We present a generic operator $J$ simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the…
Nonconvex optimization is central to modern machine learning, but the general framework of nonconvex optimization yields weak convergence guarantees that are too pessimistic compared to practice. On the other hand, while convexity enables…
The minimization of a nonconvex composite function can model a variety of imaging tasks. A popular class of algorithms for solving such problems are majorization-minimization techniques which iteratively approximate the composite nonconvex…
This paper initiates a systematic development of a theory of non-commutative optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically…
Motivated by structures that appear in deep neural networks, we investigate nonlinear composite models alternating proximity and affine operators defined on different spaces. We first show that a wide range of activation operators used in…
In this paper we establish a multivariable non-commutative generalization of L\"owner's classical theorem from 1934 characterizing operator monotone functions as real functions admitting analytic continuation mapping the upper complex…
In this paper, subnormal operators, not necessarily bounded, are discussed as generalized observables. In order to describe not only the information about the probability distribution of the output data of their measurement but also a…