A hyperbolic smoothing approach to the Multisource Weber problem
The Multisource Weber problem, also known as the continuous location-allocation problem, or as the Fermat-Weber problem, is considered here. A particular case of the Multisource Weber problem is the minimum sum-of-distances clustering problem, also known as the continuous <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>p</mi> </math> </EquationSource> </InlineEquation>-median problem. The mathematical modelling of this problem leads to a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$min-sum-min$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>-</mo> <mi>s</mi> <mi>u</mi> <mi>m</mi> <mo>-</mo> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </math> </EquationSource> </InlineEquation> formulation which, in addition to its intrinsic bi-level 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$ \, C^{\infty } \, $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mspace width="0.166667em"/> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mspace width="0.166667em"/> </mrow> </math> </EquationSource> </InlineEquation> differentiable class function, is adopted. The final solution is obtained by solving a sequence of low dimension <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$ \, C^{\infty } \, $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mspace width="0.166667em"/> <msup> <mi>C</mi> <mi>∞</mi> </msup> <mspace width="0.166667em"/> </mrow> </math> </EquationSource> </InlineEquation> differentiable unconstrained optimization sub-problems 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. Copyright Springer Science+Business Media New York 2014
