Abstract
This paper considers the problem of minimizing the ordered median function of finitely many rational functions over compact semi-algebraic sets. Ordered median of rational functions are not, in general, neither rational functions nor the supremum of rational functions.We prove that the problem can be transformed into a new problem embedded in a higher dimension space where it admits a convenient representation. This reformulation admits a hierarchy of SDP relaxations that approximates, up to any degree of accuracy, the optimal value of those problems. We apply this general framework to a broad family of continuous location problems solving some difficult problems.
Victor Blanco
Thanks to Grant JCI-2009-03896
Justo Puerto Safae El Haj Ben Ali
Thanks to FQM-5849 (Junta de AndalucÃa\FEDER) and MTM2010-19576-C02-01 (MICINN, Spain)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Blanco V., El-Haj Ben-Ali S. and Puerto J. (2011). A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions. arXiv:1106.2407v2.
Drezner Z. and Nickel S. (2009). Solving the ordered one-median problem in the plane. European J. Oper. Res., 195 no. 1, 46–61.
Lasserre J. B. (2001). Global Optimization with Polynomials and the Problem of Moments. SIAM J. Optim., 11, 796–817.
Lasserre J.B. (2006). Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim., 17, 822–843.
Nickel S. and Puerto J. (2005). Facility Location - A Unified Approach. Springer Verlag. 6. Ogryczak W. and Śliwiński T. (2010). On solving optimization problems with ordered average criteria and constraints. Stud. Fuzziness Soft Comput. Springer, Berlin, 254, 209–230.
Ogryczak W. (2010). Conditional median as a robust solution concept for uncapacitated location problems. TOP, 18 (1), 271–285.
Ogryczak W. and Zawadzki M. (2002). Conditional median: a parametric solution concept for location problems. Ann. Oper. Res., 110, 167–181.
Puerto J. and Tamir A. (2005). Locating tree-shaped facilities using the ordered median objective. Mathematical Programming, 102 (2), 313–338.
PutinarM. (1993). Positive Polynomials on Compact Semi-Algebraic Sets. Ind. Univ. Math. J., 42, 969–984.
Yager R. and Kacprzyk J. (1997). The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer: Norwell, MA.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Blanco, V., Puerto, J., Ali, S.E.H.B. (2012). Optimizing ordered median functions with applications to single facility location. In: Klatte, D., Lüthi, HJ., Schmedders, K. (eds) Operations Research Proceedings 2011. Operations Research Proceedings. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29210-1_53
Download citation
DOI: https://doi.org/10.1007/978-3-642-29210-1_53
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29209-5
Online ISBN: 978-3-642-29210-1
eBook Packages: Business and EconomicsBusiness and Management (R0)