Generalized Solutions for the Sum of Two Maximally Monotone Operators

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this paper,...

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Bibliographic Details
Published in:SIAM journal on control and optimization 2014-01, Vol.52 (2), p.1034-1047
Main Authors: Bauschke, Heinz H, Hare, Warren L, Moursi, Walaa M
Format: Article
Language:English
Subjects:
Applied mathematics
Convex analysis
Engineering research
Least squares method
Linear algebra
Linear equations
Mathematical analysis
Operators
Optimization
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Summary:A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this paper, we introduce a "normal problem" associated with finding a zero of the sum of two maximally monotone operators. If the original problem admits solutions, then the normal problem returns this same set of solutions. The normal problem may yield solutions when the original problem does not admit any; furthermore, it has attractive variational and duality properties. Several examples illustrate our theory. [PUBLICATION ABSTRACT]
ISSN:0363-0129
1095-7138
DOI:10.1137/130924214