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Monotone operator theory in convex optimization. (English) Zbl 1471.47033

In this paper, the author provides an overview as well as new insights on several relations between monotone operator theory and convex optimization. Basic and advanced definitions of known concepts in monotone operator theory, such as subdifferentials, maximally monotone operators, proximity operators and resolvents, are explored and the interplay with convex optimization is shown.
Following the great developments of recent years in the area of splitting algorithms for solving complex structured problems, the author illustrates the mutual relations between these methods for convex optimization and monotone inclusion problems. Several open and interesting questions are raised, for example, splitting based on Bregman distances, which is required for extensions to Banach spaces.
In addition, the author presents a new transformation which maps proximity operators to proximity operators, and establishes connections with self-dual classes of firmly nonexpansive operators.

MSC:

47H05 Monotone operators and generalizations
49M27 Decomposition methods
65K05 Numerical mathematical programming methods
90C25 Convex programming
00A27 Lists of open problems

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