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A Lagrangian search method for the P-median problem

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Abstract

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.

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Acknowledgments

The research of the first two authors are supported by the National Science Foundation Graduate Research Fellowship under Grant 0946809, the National Science Foundation under Grant CMMI-1413790, Air Force office of Scientific Research under Grant FA-9550-14-1-0059, and AFRL Mathematical Modeling and Optimization Institute. The research of the last author is supported by NSF Award CMMI-1131690. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, the Air Force office of Scientific Research and the AFRL Mathematical Modeling and Optimization Institute.

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Correspondence to Enlu Zhou.

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Hale, J.Q., Zhou, E. & Peng, J. A Lagrangian search method for the P-median problem. J Glob Optim 69, 137–156 (2017). https://doi.org/10.1007/s10898-016-0481-x

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