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

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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.

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  1. Bertsekas, D., Nedic, A., Ozdaglar, A.: Convex Analysis and Optimization. Athena Scientific, Boston (2003)

    MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gerstewitz (Tammer), C., Iwanow, E.: Dualität für Nichtkonvexe Vektor-Ptimierungsprobleme. Wiss. Z. - Tech. Hochsch. Ilmenau 31, 61–81 (1985)

    MathSciNet  Google Scholar 

  5. Göpfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  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)

    Book  Google Scholar 

  7. Mordukhovich, B.S., Nam, N.M.: Subgradients of minimal time functions under minimal assumptions. J. Convex Anal. 18, 915–947 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  9. Tammer, C., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Nguyen Mau Nam.

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The research of Nguyen Mau Nam was partially supported by the Simons Foundation under grant #208785.

The research of C. Zalinescu was supported by the grant PN-II-ID-PCE-2011-3-0084, CNCS-UEFISCDI, Romania.

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Nam, N.M., Zălinescu, C. Variational Analysis of Directional Minimal Time Functions and Applications to Location Problems. Set-Valued Var. Anal 21, 405–430 (2013).

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