Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

Butnariu, Dan; Iusem, Alfredo N.

doi:10.1007/978-94-011-4066-9

Cited by 272 publications

(234 citation statements)

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“…In this paper, we are concerned with certain analogous classes of operators which are, in some sense, nonexpansive not with respect to the norm, but with respect to Bregman distances [13,18,21]. Since these distances are not symmetric in general, it seems natural to distinguish between left and right Bregman nonexpansive operators.…”

Section: Introductionmentioning

confidence: 99%

Right Bregman nonexpansive operators in Banach spaces

Martín-Márquez¹,

Reich²,

Sabach³

2012

Nonlinear Analysis: Theory, Methods & Applications

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We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

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Section: Introductionmentioning

confidence: 99%

Right Bregman nonexpansive operators in Banach spaces

Martín-Márquez¹,

Reich²,

Sabach³

2012

Nonlinear Analysis: Theory, Methods & Applications

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“…is sequentially weakly-to-weak * continuous on Ω -(see [18,Chapter 3]). Strong convergence may not happen at all even when weak convergence does occur.…”

Section: Convergence and Stability Of A Regularization Methods For Maxmentioning

confidence: 99%

“…The conditions under which the GPPM is known to converge strongly (see [52], [35], [7], [17], [20] and the references therein) are quite restrictive and mostly concern the data of ( ) [in contrast to those ensuring weak convergence which mostly concern the Bregman function whose selection can be done from a relatively large pool of known candidates -cf. [18]]. We are going to prove, by applying Theorem 3.2 and its corollaries, that a regularized version of the GPPM produces sequences which behave better than the sequences…”

Section: Convergence and Stability Of A Regularization Methods For Maxmentioning

confidence: 99%

“…Then, Lemma 1.4 from [45] guarantees that condition ( b ) of Theorem 4.2 is satisfied and we can apply that proposition in this case too. According to [18] applied to k E and k H we deduce that, for any integer the vector exists. Let and observe that, for any , …”

Section: S Pmentioning

confidence: 99%

“…point method in the construction of augmented Lagrangian algorithms (see [53], [18,Chapter 3] and [30]): in this context a better behaved sequence obtained by regularization of the proximal point method may be of use in order to determine better approximations for a solution of the primal problem. It was shown by Butnariu and Iusem [17] that in smooth uniformly convex Banach spaces the generalized proximal point method converges subsequentially weakly, and sometimes weakly, to solutions of the optimization problem to which it is applied.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization

Alber

Butnariu

Kassay

Variational Analysis and Applications

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Abstract:In this paper we study the stability and convergence of a regularization method for solving inclusions f Ax ∈ k , where A is a maximal monotone point-to-set operator from a reflexive smooth Banach space X with the Kadec-Klee property to its dual. We assume that the data A and f involved in the inclusion are given by approximations A and k f converging to A and f, respectively, in the sense of Mosco type topologies. We prove that the sequence 1 ( )f which results from the regularization process converges weakly and, under some conditions, converges strongly to the minimum norm solution of the inclusion f Ax ∈ , provided that the inclusion is consistent.These results lead to a regularization procedure for perturbed convex optimization problems whose objective functions and feasibility sets are given by approximations. In particular, we obtain a strongly convergent version of the generalized proximal point optimization algorithm which is applicable to problems whose feasibility sets are given by Mosco approximations

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An inertial method for a solution of split equality of monotone inclusion and the f$$ f $$‐fixed point problems in Banach spaces

Zegeye

Sangago

et al. 2022

Math Methods in App Sciences

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In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and 𝑓 -fixed point of Bregman relatively 𝑓 -nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we provide some applications of our method and give numerical results to demonstrate the applicability and efficiency of the proposed method.

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Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

Cited by 272 publications

References 0 publications

Right Bregman nonexpansive operators in Banach spaces

Right Bregman nonexpansive operators in Banach spaces

Convergence and Stability of a Regularization Method for Maximal Monotone Inclusions and Its Applications to Convex Optimization

An inertial method for a solution of split equality of monotone inclusion and the f$$ f $$‐fixed point problems in Banach spaces

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