Data aggregation for p-median problems

AlBdaiwi, Bader F.; Ghosh, Diptesh; Goldengorin, Boris

doi:10.1007/s10878-009-9251-8

Cited by 24 publications

(17 citation statements)

References 20 publications

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“…Considering the current capabilities of integer linear programming software, and possible formulation approaches (see, e.g., Church 2003Church , 2008Elloumi 2010;AlBdaiwi, Ghosh, and Goldengorin 2011), the absolute optimal solution to the VAPMP (when the assignment vector is not nonincreasing) cannot be found for problems of practical sizes using commercial solvers. Considering the current capabilities of integer linear programming software, and possible formulation approaches (see, e.g., Church 2003Church , 2008Elloumi 2010;AlBdaiwi, Ghosh, and Goldengorin 2011), the absolute optimal solution to the VAPMP (when the assignment vector is not nonincreasing) cannot be found for problems of practical sizes using commercial solvers.…”

Section: Discussionmentioning

confidence: 99%

“…The theoretical results proved in this article generate a rather large optimality set, and, in general, enumerating this set for large problems can be prohibitively time-consuming without development of further techniques for pruning invalid configurations. Considering the current capabilities of integer linear programming software, and possible formulation approaches (see, e.g., Church 2003Church , 2008Elloumi 2010;AlBdaiwi, Ghosh, and Goldengorin 2011), the absolute optimal solution to the VAPMP (when the assignment vector is not nonincreasing) cannot be found for problems of practical sizes using commercial solvers. The existence of this set has potential, however, if used within the context of specialized algorithms and heuristics.…”

Section: Discussionmentioning

confidence: 99%

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On the Finite Optimality Set of the Vector Assignment p‐Median Problem

Lei

Church

2014

Geographical Analysis

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The vector assignment p‐median problem (VAPMP) is one of the first discrete location problems to account for the service of a demand by multiple facilities, and has been used to model a variety of location problems in addressing issues such as system vulnerability and reliability. Specifically, it involves the location of a fixed number of facilities when the assumption is that each demand point is served a certain fraction of the time by its closest facility, a certain fraction of the time by its second closest facility, and so on. The assignment vector represents the fraction of the time a facility of a given closeness order serves a specific demand point. Weaver and Church showed that when the fractions of assignment to closer facilities are greater than more distant facilities, an optimal all‐node solution always exists. However, the general form of the VAPMP does not have this property. Hooker and Garfinkel provided a counterexample of this property for the nonmonotonic VAPMP. However, they do not conjecture as to what a finite set may be in general. The question of whether there exists a finite set of locations that contains an optimal solution has remained open to conjecture. In this article, we prove that a finite optimality set for the VAPMP consisting of “equidistant points” does exist. We also show a stronger result when the underlying network is a tree graph.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

Lei

Church

2014

Geographical Analysis

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“…If an optimal solution Q * is obtained, such that Proof: It has already been shown in the proof of Lemma 1 that j∈Q T SS j = qnσ 2 . Since this expression is constant, from equation (1) we have that j∈Q CSS j = qnσ 2 − j∈Q W SS j , therefore minimizing the latter is the same than maximizing the former.…”

Section: Theoremmentioning

confidence: 99%

“…The methodology was suggested long ago in [13], but only recently has been widely applied, see [1,5,6,14,15,21,22,31,40]. The methodology is connected to pseudo-Boolean representation and data aggregation for the p-median problem, see [1,11,12]. The radius formulation replaces assignment variables w ijk with radius variables h ijt .…”

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confidence: 99%

Mixed integer linear programming and heuristic methods for feature selection in clustering

Benati

García

Puerto

2018

Journal of the Operational Research Society

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This paper studies the problem of selecting relevant features in clustering problems, out of a data set in which many features are useless, or masking. The data set comprises a set U of units, a set V of features, a set R of (tentative) cluster centres and distances d ijk for every i ∈ U , k ∈ R, j ∈ V . The feature selection problem consists of finding a subset of features Q ⊆ V such that the total sum of the distances from the units to the closest centre is minimized. This is a combinatorial optimization problem that we show to be NP-complete, and we propose two mixed integer linear programming formulations to calculate the solution. Some computational experiments show that if clusters are well separated and the relevant features are easy to detect, then both formulations can solve problems with many integer variables. Conversely, if clusters overlap and relevant features are ambiguous, then even small problems are unsolved. To overcome this difficulty, we propose two heuristic methods to find that, most of the time, one of them, called q-vars, calculates the optimal solution quickly. Then, the q-vars heuristic is combined with the k-means algorithm to cluster some simulated data. We conclude that this approach outperforms other methods for clustering with variable selection that were proposed in the literature.

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“…It can be further reduced through monomial reduction. Interested readers could refer to [4] for details. Example 1 illustrates the PMP pseudo boolean formulation.…”

Section: A Pseudo Boolean Formulation Of the Pmpmentioning

confidence: 99%

A GPU-based genetic algorithm for the p-median problem

AlBdaiwi

AboElFotoh

2017

J Supercomput

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The p-median problem is a well-known NP-hard problem. Many heuristics have been proposed in the literature for this problem. In this paper, we exploit a GPGPU parallel computing platform to present a new genetic algorithm implemented in Cuda and based on a Pseudo Boolean formulation of the p-median problem. We have tested the effectiveness of our algorithm using a Tesla K40 (2880 Cuda cores) on 290 different benchmark instances obtained from OR-Library, discrete location problems benchmark library, and benchmarks introduced in recent publications. The algorithm succeeded in finding optimal solutions for all instances except for two OR-library instances, namely pmed30 and pmed40, where better than 99.9% approximations were obtained.

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Data aggregation for p-median problems

Cited by 24 publications

References 20 publications

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

Mixed integer linear programming and heuristic methods for feature selection in clustering

A GPU-based genetic algorithm for the p-median problem

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