Taiwanese Journal of Mathematics

EPIDERIVATIVES WITH RESPECT TO HALF-SPACES

Elvira Hernández, Luis Rodríguez-Marín, and Miguel Sama

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Abstract

In this paper we extend some results given in [L. Rodríguez-Marín, M. Sama, $\tau^{w}$-contingent epiderivatives in reflexive spaces, Nonlinear Analysis, {68} (2008), 3780-3788]. Following the same approach, we associate a set-valued optimization problem with a family of simpler problems by using a decoupling of the ordering cone into half-spaces. In this context we give necessary and sufficient optimality conditions in terms of epiderivatives with respect to half-spaces. Moreover we obtain computation formulas for these conditions in term of derivatives of scalar set-valued maps.

Article information

Source
Taiwanese J. Math., Volume 12, Number 8 (2008), 1965-1978.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405129

Digital Object Identifier
doi:10.11650/twjm/1500405129

Mathematical Reviews number (MathSciNet)
MR2449956

Zentralblatt MATH identifier
1179.26090

Subjects
Primary: 26E25: Set-valued functions [See also 28B20, 49J53, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx} 46G05: Derivatives [See also 46T20, 58C20, 58C25] 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06] 58C20: Differentiation theory (Gateaux, Fréchet, etc.) [See also 26Exx, 46G05]

Keywords
set-valued maps family of epiderivatives contingent derivative calculus rules existence conditions optimization

Citation

Hernández, Elvira; Rodríguez-Marín, Luis; Sama, Miguel. EPIDERIVATIVES WITH RESPECT TO HALF-SPACES. Taiwanese J. Math. 12 (2008), no. 8, 1965--1978. doi:10.11650/twjm/1500405129. https://projecteuclid.org/euclid.twjm/1500405129


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References

  • J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.
  • F. Flores-Bazán, E. Hernández and V. Novo, Characterizing efficiency without linear structure: a unified approach, J. Global Optim., 2007. Published online. DOI 10.1007/s10898-007-9165-x.
  • T. X. D. Ha, D. Kuroiwa and T. Tanaka, On cone convexity of set-valued maps, Nonlinear Anal., 30 (1997), 1487-1496.
  • T. X. D. Ha, Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124 (2005), 187-206.
  • S. Heikkilä, Inclusion problems in ordered topological vector spaces and applications.J. Math. Anal. Appl., 298 (2004), 94-105.
  • E. Hernández and L. Rodr\'\tiny lguez-Mar\'\tiny ln, Nonconvex scalarization in set-optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18.
  • E. Hernández and L. Rodr\'\tiny lguez-Mar\'\tiny ln, Existence theorems for set optimization problems, Nonlinear Analysis, 67 (2007), 1726-1736.
  • E. Hernández and L. Rodr\'\tiny lguez-Mar\'\tiny ln, Duality in set optimization with set-valued maps, Pac. J. Optim., 3 (2007), 245-255.
  • E. Hernández and L. Rodr\'\tiny lguez-Mar\'\tiny ln, Lagrangian duality in set-valued optimization, J. Optim. Theory Appl., 134(1) (2007), 119-134.
  • J. Jahn and R. Rauh, Contingent epiderivatives and set valued optimization, Math. Meth. Oper. Res., 46 (1997) 193-211.
  • J. Jahn and A. A. Khan, Some calculus rules for contingent epiderivatives, Optimization, 52 (2003), 113-125.
  • J. Jahn, Vector optimization. Theory, applications, and extensions. Springer-Verlag, 2004.
  • D. Kuroiwa, On set-valued optimization. Proceedings of the Third World Congress of Nonlinear Analyst. Nonlinear Anal., 47 (2001), 1395-1400.
  • D. Kuroiwa, Existence theorems of set optimization with set-valued maps, J. Inf. Optim. Sci., 24(1) (2003), 73-84.
  • M. A. Krasnoselskii, J. A. Lifshits and A. V. Sobolev, Positive linear systems, Heldermann Verlag, Berlin, 1989.
  • A. Löhne, Optimization with set relations: conjugate duality, Optimization, 54(3) (2005), 265-282.
  • D. T. Luc, Theory of vector optimization, Springer-Verlag, 1989.
  • L. Rodr\'\tiny lguez-Mar\'\tiny ln and M. Sama, About contingent epiderivatives, J. Math. Anal. Appl., 327 (2007), 745-762.
  • L. Rodr\'\tiny lguez-Mar\'\tiny ln and M. Sama, $(\Lambda,C)$-contingent derivatives of set-valued maps, J. Math. Anal. Appl., 335 (2007), 974-989.
  • L. Rodr\'\tiny lguez-Mar\'\tiny ln and M. Sama, Variational characterization of the contingent epiderivative, J. Math. Anal. Appl., 335 (2007), 1374-1382.
  • L. Rodr\'\tiny lguez-Mar\'\tiny ln and M. Sama, $\tau^{w}$-contingent epiderivatives in reflexive spaces, Nonlinear Analysis, 68 (2008), 3780-3788.