Geometric interpretation of the optimality conditions in multifacility location and applications

Lefebvre, O.; Michelot, Christian; Plastria, Frank

doi:10.1007/bf00941161

Cited by 11 publications

(4 citation statements)

References 10 publications

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“…Since (P F ) is a convex optimization problem, it is possible to state necessary and sufficient optimality conditions. These conditions can be derived by using sub-differential calculus as done in Idrissi et al (1989) and Lefebvre et al (1990).…”

Section: Optimality Conditionsmentioning

confidence: 99%

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Complexity results on planar multifacility location problems with forbidden regions

Maier

Hamacher

2019

Math Meth Oper Res

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In this paper we deal with the planar location problem with forbidden regions. We consider the median objective with block norms and show that this problem is APXhard, even when considering the Manhattan metric as distance function and polyhedral forbidden areas. As direct consequence, the problem cannot be approximated in polynomial time within a factor of 1.0019, unless P = N P. In addition, we give a dominating set that contains at least one optimal solution. Based on this result an approximation algorithm is derived. For special instances it is possible to improve the algorithm. These instances include problems with bounded forbidden areas and a special structure as interrelation between the new facilities. For uniform weights, this algorithm becomes an FPTAS.

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Section: Optimality Conditionsmentioning

confidence: 99%

“…Theorem 3.4 [Optimality Conditions for (P F ), Idrissi et al (1989) and Lefebvre et al (1990)] The feasible point X…”

Section: Optimality Conditionsmentioning

confidence: 99%

Complexity results on planar multifacility location problems with forbidden regions

Maier

Hamacher

2019

Math Meth Oper Res

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“…Then (F, E) is a directed graph. Following e. g. [13,27], F represents the set of facilities (some of which may have fixed locations in IR n ), whereas E represents the interactions between these facilities.…”

Section: Multifacility Locationmentioning

confidence: 99%

“…We have shown in Theorem 10 that primal and dual optimal solutions (when they exist!) are related with each other as saddle point solutions of (27). However, the existence of optimal solutions for (P ) is not guaranteed when (P ) has a finite optimal value.…”

Section: Existence Of Primal and Dual Solutionsmentioning

confidence: 99%

Generalized Goal Programming: polynomial methods and applications

Carrizosa

Fliege

2002

Mathematical Programming

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In this paper we address a general Goal Programming problem with linear objectives, convex constraints, and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets. In particular, classical Linear Goal Programming problems, as well as several models in Location and Regression Analysis are modeled within this framework. In spite of its generality, this problem can be analyzed from a geometrical and a computational viewpoint, and a unified solution methodology can be given. Indeed, a dual is derived, enabling us to describe the set of optimal solutions geometrically. Moreover, Interior-Point methods are described which yield an ε-optimal solution in polynomial time.

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Applications and numerical convergence of the partial inverse method

Idrissi

Lefebvre

Michelot

Lecture Notes in Mathematics

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Geometric interpretation of the optimality conditions in multifacility location and applications

Cited by 11 publications

References 10 publications

Complexity results on planar multifacility location problems with forbidden regions

Complexity results on planar multifacility location problems with forbidden regions

Generalized Goal Programming: polynomial methods and applications

Applications and numerical convergence of the partial inverse method

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