Set-Valued and Variational Analysis

, Volume 21, Issue 2, pp 405–430 | Cite as

Variational Analysis of Directional Minimal Time Functions and Applications to Location Problems

Article
First Online:
Received:
Accepted:

Abstract

This paper is devoted to the study of directional minimal time functions that specify the minimal time for a vector to reach an object following its given direction. We provide a careful analysis of general and generalized differentiation properties of this class of functions. The analysis allows us to study a new model of facility location that involves sets.

Keywords

Directional minimal time functions Scalarization functions Generalized differentiation Facility location problems 

Mathematics Subject Classifications (2010)

49J53 49K27 90C46 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bertsekas, D., Nedic, A., Ozdaglar, A.: Convex Analysis and Optimization. Athena Scientific, Boston (2003)MATHGoogle Scholar
  2. 2.
    Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics, vol. 178. Springer, New York (1998)Google Scholar
  3. 3.
    Colombo, G., Wolenski, P.R.: The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Glob. Optim. 28, 269–282 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gerstewitz (Tammer), C., Iwanow, E.: Dualität für Nichtkonvexe Vektor-Ptimierungsprobleme. Wiss. Z. - Tech. Hochsch. Ilmenau 31, 61–81 (1985)MathSciNetGoogle Scholar
  5. 5.
    Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)MATHGoogle Scholar
  6. 6.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vols. 330 and 331. Springer, Berlin (2006)CrossRefGoogle Scholar
  7. 7.
    Mordukhovich, B.S., Nam, N.M.: Subgradients of minimal time functions under minimal assumptions. J. Convex Anal. 18, 915–947 (2011)MathSciNetMATHGoogle Scholar
  8. 8.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)MATHCrossRefGoogle Scholar
  9. 9.
    Tammer, C., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Fariborz Maseeh Department of Mathematics and StatisticsPortland State UniversityPortlandUSA
  2. 2.Faculty of MathematicsUniversity Alexandru Ioan CuzaIasiRomania
  3. 3.Octav Mayer Institute of Mathematics (Romanian Academy)IasiRomania

Personalised recommendations