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Hunting for a smaller convex subdifferential. (English) Zbl 0872.90083
Summary: Certain useful basic results of the gradient (in the smooth case), the Clarke subdifferential, the Michel-Penot subdifferential, which is also known as the “small” subdifferential, and the directional derivative (in the nonsmooth case) are stated and discussed. One of the advantages of the Michel-Penot subdifferential is the fact that it is in general “smaller” than the Clarke subdifferential. In this paper, it is shown that there exist subdifferentials which may be smaller than the Michel-Penot subdifferential and which have certain useful calculus. It is further shown that in the case of quasidifferentiability, the Michel-Penot subdifferential enjoys calculus which hold for the Clarke subdifferential only in the regular case.

MSC:
90C30 Nonlinear programming
90C48 Programming in abstract spaces
49J52 Nonsmooth analysis
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