Journal of Global Optimization
, Volume 60, Issue 1, pp 4958
First online:
A hyperbolic smoothing approach to the Multisource Weber problem
 Vinicius L. XavierAffiliated withPESCCOPPE, Universidade Federal do Rio de Janeiro
 , Felipe M. G. FrançaAffiliated withPESCCOPPE, Universidade Federal do Rio de Janeiro
 , Adilson E. XavierAffiliated withPESCCOPPE, Universidade Federal do Rio de Janeiro Email author
 , Priscila M. V. LimaAffiliated withiNCE, Universidade Federal do Rio de Janeiro
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The Multisource Weber problem, also known as the continuous locationallocation problem, or as the FermatWeber problem, is considered here. A particular case of the Multisource Weber problem is the minimum sumofdistances clustering problem, also known as the continuous \(p\)median problem. The mathematical modelling of this problem leads to a \(minsummin\) formulation which, in addition to its intrinsic bilevel nature, is strongly nondifferentiable. Moreover, it has a large number of local minimizers, so it is a typical global optimization problem. In order to overcome the intrinsic difficulties of the problem, the so called Hyperbolic Smoothing methodology, which follows a smoothing strategy using a special \( \, C^{\infty } \, \) differentiable class function, is adopted. The final solution is obtained by solving a sequence of low dimension \( \, C^{\infty } \, \) differentiable unconstrained optimization subproblems which gradually approaches the original problem. For the purpose of illustrating both the reliability and the efficiency of the method, a set of computational experiments making use of traditional test problems described in the literature was performed. Apart from consistently presenting better results when compared to related approaches, the novel technique introduced here was able to deal with instances never tackled before in the context of the Multisource Weber problem.
Keywords
Multisource Weber problem Minsumdistances clustering problem Nondifferentiable programming Title
 A hyperbolic smoothing approach to the Multisource Weber problem
 Journal

Journal of Global Optimization
Volume 60, Issue 1 , pp 4958
 Cover Date
 201409
 DOI
 10.1007/s1089801401608
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Multisource Weber problem
 Minsumdistances clustering problem
 Nondifferentiable programming
 Industry Sectors
 Authors

 Vinicius L. Xavier ^{(1)}
 Felipe M. G. França ^{(1)}
 Adilson E. Xavier ^{(1)}
 Priscila M. V. Lima ^{(2)}
 Author Affiliations

 1. PESCCOPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
 2. iNCE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil