Abstract
This chapter collects some nontrivial results which will be needed in the rest of the book. We try to keep the presentation as self-contained as possible.
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Bibliographical notes
The results on polyhedral multifunctions, collected in Section 2.1, have been taken from Robinson, 1976; Robinson, 1981; Hogan, 1973. As already mentioned, the main source for Sections 2.2 and 2.3 was the monograph Clarke, 1983. Another interesting treatment of this subject can be found in Rockafellar, 1981 and Aubin and Frankowska, 1990. The concept of semismoothness was introduced for real-valued functions in Mifflin, 1977 and proved to be useful in the study of numerical methods for nonsmooth optimization (Schramm and Zowe, 1992). In Qi and Sun, 1993 this concept has been generalized to the form presented here. Proposition 2.26 and Theorem 2.27 stem from Qi and Sun, 1993. The role of semismoothness in nonsmooth Newton techniques will become apparent in the next chapter. The differentiability of the projection map, has been studied in many papers starting with Haraux, 1977 up to recent contributions of Shapiro dealing with the projection onto nonconvex sets. It is well-known that this map need not be directionally differentiable, not even for convex sets in finite dimension (Kruskal, 1969). The presented results concerning polyhedral sets were taken from Robinson, 1984; Pang, 1990a and Robinson, 1991.
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© 1998 Springer Science+Business Media Dordrecht
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Outrata, J., Kočvara, M., Zowe, J. (1998). Auxiliary Results. In: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Nonconvex Optimization and Its Applications, vol 28. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2825-5_2
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DOI: https://doi.org/10.1007/978-1-4757-2825-5_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4804-5
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