On the Approximation of the Global Extremum of a Semi-Lipschitz Function

Article

Abstract

In this paper one obtains a sequential procedure for determining the global extremum of a semi-Lipschitz real-valued function defined on a quasi-metric (asymmetric metric) space.

Mathematics Subject Classification (2000)

Primary 68W25 Secondary 46A22 

Keywords

Spaces with asymmetric metric semi-Lipschitz functions extension and approximation 

References

  1. 1.
    Basso P.: Optimal search for the global maximum of functions with bounded seminorm. SIAM J. Numer. Anal. 22(no. 5), 888–905 (1985)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Borodin P.A.: The Banach-Mazur theorem for spaces with asymetric norm and its applications in convex analysis. Mat. Zametki 69(no. 3), 329–337 (2001)MathSciNetGoogle Scholar
  3. 3.
    Cobzaş S.: Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math. 27(no. 3), 275–296 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Cobzaş S.: Asymmetric locally convex spaces. Int. J. Math. Math. Sci. 16, 2585–2608 (2005)CrossRefGoogle Scholar
  5. 5.
    Cobzaş S., Mustăţa C.: Norm-preserving extyension of convex Lipschitz functions, J. Approx. Theory 24(no. 3), 236–244 (1978)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cobzaş S., Mustăţa C.: Extension of bounded linear functionals and best approximation in spaces with asymmetric norm. Rev. Anal. Numér. Théor. Approx. 32(no. 1), 39–50 (2004)Google Scholar
  7. 7.
    Collins J., Zimmer J.: An asymmetric Arzelà-Ascoli theorem. Topology Appl. 154(no. 11), 2312–2322 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fletcher P., Lindgren W.F.: Quasi-uniform Spaces. Marcel Dekker, New-York (1982)MATHGoogle Scholar
  9. 9.
    Garcia-Ferreira S., Romaguera S., Sanchis M.: Bounded subsets and Grothendieck’s theorem for bispaces. Houston. J. Math. 25(no. 2), 267–283 (1999)MATHMathSciNetGoogle Scholar
  10. 10.
    Garcia-Raffi L.M., Romaguera S., Sánchez-Pérez E.A.: The dual space of an asymmetric normed linear space. Quaest. Math. 26(no. 1), 83–96 (2003)MATHMathSciNetGoogle Scholar
  11. 11.
    Garcia-Raffi L.M., Romaguera S., Sánchez-Pérez E.A.: On Hausdorff asymmetric normed linear spaces. Houston J. Math. 29(no. 3), 717–728 (2003)MATHMathSciNetGoogle Scholar
  12. 12.
    M. G. Krein and A. A. Nudelman, The Markov Moment Problem and Extremum Problems, Nauka, Moscov, 1973 (in Russian), English translation: AMS, Providence, R.I., 1977.Google Scholar
  13. 13.
    H. P. A. Künzi, Nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topology, Handbook of the History of General Topology, ed. by C.E. Aull and R. Lower, vol. 3, Hist. Topol. 3, Kluwer Acad. Publ. (Dordrecht, 2001), 853–968.Google Scholar
  14. 14.
    McShane E.J.: Extension of range of functions. Bull. Amer. Math. Soc. 40, 837–842 (1934)CrossRefMathSciNetGoogle Scholar
  15. 15.
    A. Mennucci, On asymmetric distances, Technical report, Scuola Normale Superiore, Pisa, 2004.Google Scholar
  16. 16.
    Mustăţa C.: Best approximation and unique extension of Lipschitz functions. J. Approx. Theory 19(no. 3), 222–230 (1977)MATHCrossRefGoogle Scholar
  17. 17.
    Mustăţa C. Extension of Hölder Functions and some related problems of best approximation, ”Babeş-Bolyai” University, Faculty of Mathematics, Research Seminars, Seminar onf Mathematical Analysis (1991) 71–86.Google Scholar
  18. 18.
    Mustăţa C.: Extension of semi-Lipschitz functions on quasi-metric spaces. Rev. Anal. Numér. Théor. Approx. 30(no. 1), 61–67 (2001)MATHMathSciNetGoogle Scholar
  19. 19.
    Mustăţa C.: On the extremal semi-Lipschitz functions. Rev. Anal. Numér. Théor. Approx. 31(no. 1), 61–67 (2002)MathSciNetGoogle Scholar
  20. 20.
    Pestov V., Stojmirović A.: Indexing schemes for similarity search: an illustrated paradigm. Fund. Inf. 70(no. 4), 367–385 (2006)MATHGoogle Scholar
  21. 21.
    Romaguera S., Sanchis M.: Semi-Lipschitz functions and best approximation in quasi-metric spaces. J. Approx. Theory 103, 292–301 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Romaguera S., Sanchis M.: Properties of the normed cone of semi-Lipschitz functions. Acta Math. Hungar 108(no. 1-2), 55–70 (2005)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    S. Romaguera, J. M. Sanchez-Álvarez and M. Sanchis, El espacio de funciones semi-Lipschitz, VI Jornadas de Matemática Aplicada, Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, 1-3 septiembrie, 2005.Google Scholar
  24. 24.
    Sánchez-Álvarez J.M.: On semi-Lipschitz functions with values in a quasi-normed linear space. Appl. Gen. Top. 6(no. 2), 217–228 (2005)MATHGoogle Scholar
  25. 25.
    Shubert B.: A sequential method seeking the global maximum of a function. SIAM J. Num. Anal. 9, 379–388 (1972)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Stojmirović A.: Quasi-metric spaces with measures. Proc. 18th Summer Conference on Topology and its Applications, Topology Proc. 28(no. 2), 655–671 (2004)MATHGoogle Scholar
  27. 27.
    Wilson W.A.: On quasi-metric spaces, Amer. J. Math. 53(no. 3), 75–684 (1931)CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.“Tiberiu Popoviciu” Institute of Numerical AnalysisCluj-NapocaRomania

Personalised recommendations