An aggregation heuristic for large scale p-median problem

Data clustering methods are used extensively in the data mining literature to detect important patterns in large datasets in the form of densely populated regions in a multi-dimensional Euclidean space. Due to the complexity of the problem and the size of the dataset, obtaining quality solutions within reasonable CPU time and memory requirements becomes the central challenge. In this paper, we solve the clustering problem as a large scale p-median model, using a new approach based on the variable neighborhood search (VNS) metaheuristic. Using a highly efficient data structure and local updating procedure taken from the OR literature, our VNS procedure is able to tackle large datasets directly without the need for data reduction or sampling as employed in certain popular methods. Computational results demonstrate that our VNS heuristic outperforms other local search based methods such as CLARA and CLARANS even after upgrading these procedures with the same efficient data structures and local search. We also obtain a bound on the quality of the solutions by solving heuristically a dual relaxation of the problem, thus introducing an important capability to the solution process.

Solving large p -median clustering problems by primal–dual variable neighborhood search

We consider the two-level uncapacitated facility location problem with single-assignment constraints (TUFLP-S), a problem that arises in industrial applications in freight transportation and telecommunications. We present a new Lagrangian relaxation approach for the TUFLP-S, based on solving a single-level uncapacitated facility location problem (UFLP) as the Lagrangian subproblem. We also develop a Lagrangian heuristic that includes a mixed-integer programming–based large neighborhood search heuristic exploring neighborhoods by solving a series of small UFLPs. The dual and primal bounds thus obtained are used within an enumerative algorithm that implements specialized branching rules. Our computational experiments on instances derived from an industrial application in freight transportation as well as on large, hard, artificial instances confirm the efficiency of our specialized branch-and-bound algorithm.

/pdf/a-lagrangian-based-branch-and-bound-algorithm-for-the-two-206e5sgtvr.pdf

A Lagrangian-Based Branch-and-Bound Algorithm for the Two-Level Uncapacitated Facility Location Problem with Single-Assignment Constraints

We show how the performance of general purpose Mixed Integer Programming (MIP) solvers, can be enhanced by using the Semi-Lagrangian Relaxation (SLR) method. To illustrate this procedure we perform computational experiments on large-scale instances of the Uncapacitated Facility Location (UFL) problems with unknown optimal values. CPLEX solves 3 out of the 36 instances. By combining CPLEX with SLR, we manage to solve 18 out of the 36 instances and improve the best known lower bound for the other instances. The key point has been that, on average, the SLR approach, has reduced by more than 90% the total number of relevant UFL variables.

/pdf/semi-lagrangian-relaxation-applied-to-the-uncapacitated-unxhriwhf0.pdf

Semi-Lagrangian relaxation applied to the uncapacitated facility location problem

https://kar.kent.ac.uk/35143/1/Manuscript%20(final-april%202013).pdf

Lagrangian relaxation heuristics for the uncapacitated single-source multi-product facility location problem

Design and analysis of computationally difficult instances is one of the promising areas in combinatorial optimization. In this paper we present several new classes of benchmarks for the Uncapacitated Facility Location Problem. The first class is polynomially solvable. It has many strong local optima and large mutual pair distances. Two other classes have exponential number of strong local optima. Finally, three last classes have large duality gap and one of them is the most difficult for metaheuristics and the branch and bound method.

Computationally Difficult Instances for the Uncapacitated Facility Location Problem

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Computationally Difficult Instances for the Uncapacitated Facility Location Problem