Approximate convexity and submonotonicity |
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Authors: | Aris Daniilidis Pando Georgiev |
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Institution: | a CODE, Edifici B, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain b Department of Mathematics and Informatics, University of Sofia “St. Kl. Ohridski”, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria c Laboratory for Advanced Brain Signal Processing, Brain Science Institute, RIKEN, 2-1, Hirosawa, Wako-shi, Saitama, 351-0198, Japan |
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Abstract: | It is shown that a locally Lipschitz function is approximately convex if, and only if, its Clarke subdifferential is a submonotone operator. Consequently, in finite dimensions, the class of locally Lipschitz approximately convex functions coincides with the class of lower-C1 functions. Directional approximate convexity is introduced and shown to be a natural extension of the class of lower-C1 functions in infinite dimensions. The following characterization is established: a multivalued operator is maximal cyclically submonotone if, and only if, it coincides with the Clarke subdifferential of a locally Lipschitz directionally approximately convex function, which is unique up to a constant. Furthermore, it is shown that in Asplund spaces, every regular function is generically approximately convex. |
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Keywords: | Lower-C1 function Approximate convexity Submonotone operator Cyclicity |
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