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Revisiting several problems and algorithms in continuous location with \(\ell _\tau \) norms

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Abstract

This paper addresses the general continuous single facility location problems in finite dimension spaces under possibly different \(\ell _\tau \) norms, \(\tau \ge 1\), in the demand points. We analyze the difficulty of this family of problems and revisit convergence properties of some well-known algorithms. The ultimate goal is to provide a common approach to solve the family of continuous \(\ell _\tau \) ordered median location problems Nickel and Puerto (Facility location: a unified approach, 2005) in dimension \(d\) (including of course the \(\ell _\tau \) minisum or Fermat-Weber location problem for any \(\tau \ge 1\)). We prove that this approach has a polynomial worst case complexity for monotone lambda weights and can be also applied to constrained and even non-convex problems.

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Acknowledgments

The authors were partially supported by the Project FQM-5849 (Junta de Andalucía\(\backslash \)FEDER). The first and third author were partially supported by the project MTM2010-19576-C02-01 (MICINN, Spain). The first author was also supported by research group SEJ-534.

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Authors and Affiliations

  1. Departamento de Métodos Cuantitativos para la Economia y la Empresa, Universidad de Granada, 18011 , Granada, Spain

    Victor Blanco

  2. Departamento de Estadística e Investigación Operativa, Universidad de Sevilla, 41012 , Sevilla, Spain

    Justo Puerto & Safae El Haj Ben Ali

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  1. Victor Blanco
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  2. Justo Puerto
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Correspondence to Justo Puerto.

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Blanco, V., Puerto, J. & El Haj Ben Ali, S. Revisiting several problems and algorithms in continuous location with \(\ell _\tau \) norms. Comput Optim Appl 58, 563–595 (2014). https://doi.org/10.1007/s10589-014-9638-z

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