Abstract
Topological properties of the approximate Subdifferential introduced by Mordukhovich are studied. Apart from formulating a sufficient condition for connectedness, it is shown that, up to homeomorphy, each compact subset of ℝp may occur as the approximate subdifferential of some Lipschitz function. Furthermore, even an exact result is possible when considering the partial approximate Subdifferential, which was introduced as a parametric extension by Jourani and Thibault: Given any compact subset of ℝp, there is a locally Lipschitzian function realizing this set as its partial approximate Subdifferential at some predefined point.
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This research is supported by the “Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung” of the Deutsche Forschungsgemeinschaft. The paper is the written version of a lecture given at theMinisymposium on Stochastic Programming which was held at the Humboldt University of Berlin in January 1994.
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Henrion, R. Topological characterization of the approximate subdifferential in the finite-dimensional case. ZOR - Mathematical Methods of Operations Research 41, 161–173 (1995). https://doi.org/10.1007/BF01432653
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DOI: https://doi.org/10.1007/BF01432653