TOP

, Volume 20, Issue 2, pp 503–516 | Cite as

On the infimum of a quasiconvex vector function over an intersection

Original Paper
  • 123 Downloads

Abstract

We give sufficient conditions for the infimum of a quasiconvex vector function f over an intersection \(\bigcap_{i\in I}R_{i}\) to agree with the supremum of the infima of f over the Ri’s.

Keywords

Quasiconvex functions Distance to the intersection 

Mathematics Subject Classification (2000)

06F20 52A41 54E99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aliprantis CD, Border KC (2006) Infinite dimensional analysis, 3rd edn. Springer, Berlin Google Scholar
  2. Aliprantis CD, Tourky R (2007) Cones and duality. American Mathematical Society, Providence Google Scholar
  3. Castillo JMF, Papini PL (2007) Approximation of the limit distance function in Banach spaces. J Math Anal Appl 328:577–589 CrossRefGoogle Scholar
  4. Chacón G, Montesinos V, Octavio A (2004) A note on the intersection of Banach subspaces. Publ Res Inst Math Sci 40:1–6 CrossRefGoogle Scholar
  5. Chen GY, Yang XQ, Yu H (2005) A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J Glob Optim 32:451–466 CrossRefGoogle Scholar
  6. Hoffmann A (1992) The distance to the intersection of two convex sets expressed by the distance to each of them. Math Nachr 157:81–98 CrossRefGoogle Scholar
  7. Kusraev AG (1985) Banach-Kantorovich spaces. Sib Math J 26:254–259 CrossRefGoogle Scholar
  8. Kusraev AG (1993) Kantorovich spaces and the metrization problem. Sib Math J 34:688–694 CrossRefGoogle Scholar
  9. Kusraev AG, Kutateladze SS (2006) Kantorovich spaces and optimization. J Math Sci 133:1449–1455 CrossRefGoogle Scholar
  10. Luc DT (2005) Generalized convexity in vector optimization. In: Hadjisavvas N, Komlósi S, Schaible S (eds) Handbook of generalized convexity and generalized monotonicity. Springer, Berlin, pp 195–236 Google Scholar
  11. Luxemburg WAJ, Zaanen AC (1971) Riesz spaces, vol 1. North-Holland, Amsterdam Google Scholar
  12. Martínez-Legaz J-E, Martinón A (2007) Boundedly connected sets and the distance to the intersection of two sets. J Math Anal Appl 332:400–406 CrossRefGoogle Scholar
  13. Martínez-Legaz J-E, Martinón A (2011) On the infimum of a quasiconvex function on an intersection. Application to the distance function. J Convex Anal 18:687–698 Google Scholar
  14. Martínez-Legaz J-E, Rubinov AM, Singer I (2002) Downward sets and their separation and approximation properties. J Glob Optim 23:111–137 CrossRefGoogle Scholar
  15. Martinón A (2004) Distance to the intersection of two sets. Bull Aust Math Soc 70:329–341 CrossRefGoogle Scholar
  16. Pallaschke D, Urbański R (2002) On the separation and order law of cancellation for bounded sets. Optimization 51:487–496 CrossRefGoogle Scholar
  17. Rubinov AM (2003) Distance to the solution set of an inequality with an increasing function. In: Maugeri A, Danilele P (eds) Equilibrium problems and variational models. Kluwer Academic, Dordrecht, pp 417–431 CrossRefGoogle Scholar
  18. Rubinov AM, Singer I (2000) Best approximation by normal and conormal sets. J Approx Theory 107:212–243 CrossRefGoogle Scholar
  19. Rubinov AM, Singer I (2000) Distance to the intersection of normal sets and applications. Numer Funct Anal Optim 21:521–535 CrossRefGoogle Scholar
  20. Sonmez A, Cakalli H (2010) Cone normed spaces and weighted means. Math Comput Model 52:1660–1666 CrossRefGoogle Scholar
  21. Willard S (1970) General topology. Addison-Wesley, Reading Google Scholar
  22. Zabrejko PP (1997) K-metric and K-normed linear spaces: survey. Collect Math 48:825–859 Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2011

Authors and Affiliations

  • Juan-Enrique Martínez-Legaz
    • 1
  • Antonio Martinón
    • 2
  1. 1.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  2. 2.Departamento de Análisis MatemáticoUniversidad de La LagunaLa Laguna (Tenerife)Spain

Personalised recommendations