On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

Díaz, Jesús Ildefonso Díaz; Galiano, Gonzalo; Jüngel, Ansgar

doi:10.1016/s0362-546x(00)00102-4

Cited by 26 publications

References 26 publications

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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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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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Some Results About Proximal-Like Methods

Kaplan

Tichatschke

Lecture Notes in Economics and Mathematical Systems

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Summary. We discuss some ideas for improvement, extension and appUcation of proximal point methods and the auxihary problem principle to variational inequalities in Hilbert spaces. These methods are closely related and will be joined in a general framework, which admits a consecutive approximation of the problem data including applications of finite element techniques and the ^-enlargement of monotone operators. With the use of a "reserve of monotonicity" of the operator in the variational inequality, the concepts of weak-and elliptic proximal regularization are developed. Considering Bregman-function-based proximal methods, we analyze their convergence under a relaxed error tolerance criterion in the subproblems. Moreover, the case of variational inequalities with non-paramonotone operators is investigated, and an extension of the auxiliary problem principle with the use of Bregman functions is studied. To emphasize the basic ideas, we renounce all the proofs and sometimes precise descriptions of the convergence results and approximation techniques. Those can be found in the referred papers.

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Asymptotic Limits in Macroscopic Plasma Models

Jüngel

2004

Dispersive Transport Equations and Multiscale Models

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On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

Cited by 26 publications

References 26 publications

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

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

Some Results About Proximal-Like Methods

Asymptotic Limits in Macroscopic Plasma Models

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