Heinz H. Bauschke, Warren L. Hare, Walaa M. Moursi
Published 2013-06-07Version 1
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.
On Douglas-Rachford operators that fail to be proximal mappings
The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings
Locating and Invariance Theorems of Differential Inclusions Governed by Maximally Monotone Operators
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