Semiregularity and Generalized Subdifferentials with Applications to Optimization

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.