Directional Derivatives of Marginal Functions

  • Bernd Luderer
  • Leonid Minchenko
  • Tatyana Satsura
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 66)

Abstract

Let X = R n , Y = R m and let U be a compact set in Y. We consider the functions
$$\begin{array}{*{20}{c}} {\varphi \left( x \right) = \inf \left\{ {f\left( {x,y} \right)\left| {y \in U} \right.} \right\},} \\ {\Phi \left( x \right) = \sup \left\{ {f\left( {x,y} \right)\left| {y \in U} \right.} \right\},} \end{array}$$
where f : X × YR is certain function. It is of considerable interest to find out sufficient assumptions to be imposed on the function f that are minimal in a sense and ensure directional differentiability of the functions φ and Φ. Moreover, our aim is to obtain formulas for the calculation of derivatives \(\varphi '\left( {x;\bar x} \right)\) and \(\Phi '\left( {x;\bar x} \right)\).

Keywords

Support Function Multivalued Mapping Closed Convex Cone Lower Semicontinuous Function Marginal Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Bernd Luderer
    • 1
  • Leonid Minchenko
    • 2
  • Tatyana Satsura
    • 2
  1. 1.Faculty of MathematicsChemnitz University of TechnologyChemnitzGermany
  2. 2.Byelorussian State University of Informatics & RadioelectronicsMinskByelorussia

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