Abstract
Recently, new types of tangential cones such as the radial tangent cone, the hypertangent cone, the Clarke's tangent cone, and the set of directions in which a set is epi-Lipschitzian at a point have been introduced by Clarke, Hiriart-Urruty, and Rockafellar. It turns out that in contrast to the classical tangential cones these new approximations are not necessarily isotone with respect to set inclusion.
The present paper is concerned with modifications of the above-mentioned concepts that lead to isotone approximation. A concept of an approximation operator is introduced and a general scheme for construction of isotone approximations in real affine spaces is presented along with a survey of isotone approximations employed in optimization.
Zusammenfassung
Kürzlich wurden neuartige Tangentialkegel von Clarke, Hiriart-Urruty und Rockafellar eingeführt. Es erweist sich, daß diese neuen Approximationen, im Gegensatz zu den klassischen Tangentialkegeln, nicht notwendig isoton bezüglich Inklusion sind. Die vorliegende Arbeit gefaßt sich mit Modifikationen dieser Begriffe, die zu isotonen Approximationen führen. Der Begriff eines Approximationsoperators wird eingeführt, und es wird ein allgemeines Schema zur Erzeugung isotoner Approximationen in reellen affinen Räumen vorgestellt. Die Verwendung isotoner Approximationen in der Optimierung wird zusammenfassend dargestellt.
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Vlach, M. Approximation operators in optimization theory. Zeitschrift für Operations Research 25, 15–23 (1981). https://doi.org/10.1007/BF01917339
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DOI: https://doi.org/10.1007/BF01917339