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Hypercubes Latins maximin pour l’echantillonage de systèmes complexes

Kaourintin Le Guiban 1
1 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
LRI - Laboratoire de Recherche en Informatique
Abstract : A maximin Latin Hypercube Design (LHD) is a set of point in a hypercube which do not share a coordinate on any dimension and such that the minimal distance between two points, is maximal. Maximin LHDs are widely used in metamodeling thanks to their good properties for sampling. As most work concerning LHDs focused on heuristic algorithms to produce them, we decided to make a detailed study of this problem, including its complexity, approximability, and the design of practical heuristic algorithms.We generalized the maximin LHD construction problem by defining the problem of completing a partial LHD while respecting the maximin constraint. The subproblem where the partial LHD is initially empty corresponds to the classical LHD construction problem. We studied the complexity of the completion problem and proved its NP-completeness for many cases. As we did not determine the complexity of the subproblem, we searched for performance guarantees of algorithms which may be designed for both problems. On the one hand, we found that the completion problem is inapproximable for all norms in dimensions k ≥ 3. We also gave a weaker inapproximation result for norm L1 in dimension k = 2. On the other hand, we designed an approximation algorithm for the construction problem which we proved using two new upper bounds we introduced.Besides the theoretical aspect of this study, we worked on heuristic algorithms adapted for these problems, focusing on the Simulated Annealing metaheuristic. We proposed a new evaluation function for the construction problem and new mutations for both the construction and completion problems, improving the results found in the literature.
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Kaourintin Le Guiban. Hypercubes Latins maximin pour l’echantillonage de systèmes complexes. Autre. Université Paris Saclay (COmUE), 2018. Français. ⟨NNT : 2018SACLC008⟩. ⟨tel-01722842⟩

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