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 R i ’s.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis CD, Border KC (2006) Infinite dimensional analysis, 3rd edn. Springer, Berlin
Aliprantis CD, Tourky R (2007) Cones and duality. American Mathematical Society, Providence
Castillo JMF, Papini PL (2007) Approximation of the limit distance function in Banach spaces. J Math Anal Appl 328:577–589
Chacón G, Montesinos V, Octavio A (2004) A note on the intersection of Banach subspaces. Publ Res Inst Math Sci 40:1–6
Chen GY, Yang XQ, Yu H (2005) A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J Glob Optim 32:451–466
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
Kusraev AG (1985) Banach-Kantorovich spaces. Sib Math J 26:254–259
Kusraev AG (1993) Kantorovich spaces and the metrization problem. Sib Math J 34:688–694
Kusraev AG, Kutateladze SS (2006) Kantorovich spaces and optimization. J Math Sci 133:1449–1455
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
Luxemburg WAJ, Zaanen AC (1971) Riesz spaces, vol 1. North-Holland, Amsterdam
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
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
Martínez-Legaz J-E, Rubinov AM, Singer I (2002) Downward sets and their separation and approximation properties. J Glob Optim 23:111–137
Martinón A (2004) Distance to the intersection of two sets. Bull Aust Math Soc 70:329–341
Pallaschke D, Urbański R (2002) On the separation and order law of cancellation for bounded sets. Optimization 51:487–496
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
Rubinov AM, Singer I (2000) Best approximation by normal and conormal sets. J Approx Theory 107:212–243
Rubinov AM, Singer I (2000) Distance to the intersection of normal sets and applications. Numer Funct Anal Optim 21:521–535
Sonmez A, Cakalli H (2010) Cone normed spaces and weighted means. Math Comput Model 52:1660–1666
Willard S (1970) General topology. Addison-Wesley, Reading
Zabrejko PP (1997) K-metric and K-normed linear spaces: survey. Collect Math 48:825–859
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Marco Antonio López Cerdà on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Martínez-Legaz, JE., Martinón, A. On the infimum of a quasiconvex vector function over an intersection. TOP 20, 503–516 (2012). https://doi.org/10.1007/s11750-011-0222-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-011-0222-8