Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithm...

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Published in:Rendiconti del Circolo matematico di Palermo Vol. 67; no. 2; pp. 337 - 371
Main Authors: Reem, Daniel, Reich, Simeon
Format: Journal Article
Language:English
Published: Milan Springer Milan 01.08.2018
Springer Nature B.V
Subjects:
Algebra
Algorithms
Analysis
Applications of Mathematics
Continuity (mathematics)
Dependence
Geometry
Iterative methods
Legendre functions
Mathematics
Mathematics and Statistics
Nonlinear analysis
Operators (mathematics)
Optimization
Fully Legendre function
Zero
Inclusion
65K05
Well defined
Maximally monotone operator
90C31
Resolvent
90C30
Algorithmic scheme
Inexact
49M37
47H05
47J25
Protoresolvent
ISSN:0009-725X, 1973-4409
Online Access:Get full text
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Summary:Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.
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ISSN:0009-725X
1973-4409
DOI:10.1007/s12215-017-0318-6