Semiregularity and generalized subdifferentials with applications to optimization

Semiregularity and generalized subdifferentials with applications to optimization

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Article ID: iaor19941097
Country: United States
Volume: 18
Issue: 4
Start Page Number: 982
End Page Number: 1005
Publication Date: Nov 1993
Journal: Mathematics of Operations Research
Authors: ,
Abstract:

The Michel-Penot subdifferential of a locally Lipschitzian function is the principal part of the Clarke subdifferential. It coincides with the G-derivative at differentiable points. A locally Lipschitzian function can be determined by its Michel-Penot subdifferential uniquely up to an additive constant, though this cannot be done by its Clarke subdifferential if the set of abnormal points is not negligible. A set-valued operator is the Michel-Penot subdifferential of a locally Lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition. Various calculus rules hold for the Michel-Penot subdifferential. Equalities hold for these rules at a point under semiregularity, which is weaker than regularity. For a locally Lipschitzian function in a separable Banach space, semiregularity holds everywhere except for a Haar zero set. Applications in optimization are discussed.

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