Fejér processes in theory and practice: Recent results

Еремин, И. И.; Popov, L. D.

doi:10.3103/s1066369x09010022

Cited by 15 publications

(15 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…arising in hilbertian nonlinear analysis, see for instance [2,5,12,13,14,18,19,30,31,35] and the references therein. In recent years, there have been attempts to generalize standard algorithms such as those discussed in the above references by allowing the underlying metric to vary over the course of the iterations, e.g., [7,10,11,16,26,29].…”

Section: Introductionmentioning

confidence: 99%

Variable metric quasi-Fejér monotonicity

Combettes

Vũ

2013

Nonlinear Analysis: Theory, Methods & Applications

101

View full text Add to dashboard Cite

MSC: 65J05 90C25 47H09 Keywords: Convex feasibility problem Convex optimization Hilbert space Inverse problems Proximal Landweber method Proximal point algorithm Quasi-Fejér sequence Variable metricThe notion of quasi-Fejér monotonicity has proven to be an efficient tool to simplify and unify the convergence analysis of various algorithms arising in applied nonlinear analysis. In this paper, we extend this notion in the context of variable metric algorithms, whereby the underlying norm is allowed to vary at each iteration. Applications to convex optimization and inverse problems are demonstrated.

show abstract

Section: Introductionmentioning

confidence: 99%

Variable metric quasi-Fejér monotonicity

Combettes

Vũ

2013

Nonlinear Analysis: Theory, Methods & Applications

101

View full text Add to dashboard Cite

show abstract

“…The half-spaces (H n ) n∈N in (2.2) are called Fejér approximations to C. The Fejér monotonicity property (2.1) makes it possible to greatly simplify the analysis of the asymptotic behavior of a broad class of algorithms; see [7,9,21,22,30,31] for background, examples, and historical notes. In the following proposition, we consider the problem of constructing a Fejér approximation to the Kuhn-Tucker set (1.7).…”

mentioning

confidence: 99%

Solving Coupled Composite Monotone Inclusions by Successive Fejér Approximations of their Kuhn--Tucker Set

Alotaibi¹,

Combettes²,

Shahzad³

2014

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. We propose a new class of primal-dual Fejér monotone algorithms for solving systems of composite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the model are used to construct a half-space containing the Kuhn-Tucker set associated with the system. The primal-dual update is then obtained via a relaxed projection of the current iterate onto this half-space. An important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators.Key words. duality, Fejér monotonicity, monotone inclusion, monotone operator, primal-dual algorithm, splitting algorithm AMS subject classifications. Primary 47H05; Secondary 65K05, 90C25, 94A081. Introduction. The first monotone operator splitting methods arose in the late 1970s and were motivated by applications in mechanics and partial differential equations [32,35,39]. In recent years, the field of monotone operator splitting algorithms has benefited from a new impetus, fueled by emerging application areas such as signal and image processing, statistics, optimal transport, machine learning, and domain decomposition methods [3,5,24,36,41,43,46]. Three main algorithms dominate the field explicitly or implicitly: the forward-backward method [38], the Douglas-Rachford method [37], and the forward-backward-forward method [47]. These methods were originally designed to solve inclusions of the type 0 ∈ Ax + Bx, where A and B are maximally monotone operators acting on a Hilbert space (via product space reformulations, they can also be extended to problems involving sums of more than 2 operators [9,45]). Until recently, a significant challenge in the field was to design splitting techniques for inclusions involving linearly composed operators, say

show abstract

“…By exploiting the relation between the iterations and a suitable distancelike function, we show in this paper that convergence theorems represent a key ingredient for a wide variety of system-theoretic problems in fixed-point theory, game theory and optimization [1,23,13,24,25]. In many cases, the study of iterative algorithms allows for a systematic analysis that follows from the concept of Féjer monotone sequence.…”

Section: Lyapunov Decrease and Féjer Monotonicitymentioning

confidence: 99%

“…In a sense, the distance used for Féjer sequences can be seen as a specific class of Lyapunov function and Féjer monotonicity shows that it is decreasing along the iterates. The concept was first introduced in 1922 [26], but the term Féjer monotone sequence was first used thirty years later in 1954 [27] and a huge part of the studies on its properties was made in the 60s [25,28,29,30] and still continues [24,31,32,33,34].…”

Section: Lyapunov Decrease and Féjer Monotonicitymentioning

confidence: 99%

Convergence of sequences: a survey

Franci¹,

Grammatico²

2021

Preprint

View full text Add to dashboard Cite

Convergent sequences of real numbers play a fundamental role in many different problems in system theory, e.g., in Lyapunov stability analysis, as well as in optimization theory and computational game theory. In this survey, we provide an overview of the literature on convergence theorems and their connection with Féjer monotonicity in the deterministic and stochastic settings, and we show how to exploit these results.

show abstract

Fejér processes in theory and practice: Recent results

Cited by 15 publications

References 3 publications

Variable metric quasi-Fejér monotonicity

Variable metric quasi-Fejér monotonicity

Solving Coupled Composite Monotone Inclusions by Successive Fejér Approximations of their Kuhn--Tucker Set

Convergence of sequences: a survey

Contact Info

Product

Resources

About