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Hybridation d’algorithmes évolutionnaires et de méthodes d’intervalles pour l’optimisation de problèmes difficiles

Abstract : Reliable global optimization is dedicated to finding a global minimum in the presence of rounding errors. The only approaches for achieving a numerical proof of optimality in global optimization are interval-based methods that interleave branching of the search-space and pruning of the subdomains that cannot contain an optimal solution. The exhaustive interval branch and bound methods have been widely studied since the 1960s and have benefittedfrom the development of refutation methods and filtering algorithms, stemming from the interval analysis and interval constraint programming communities. It is of the utmost importance: i) to compute sharp enclosures of the objective function and the constraints on a given subdomain; ii) to find a good approximation (an upper bound) of the globalminimum.State-of-the-art solvers are generally integrative methods, that is they embed local optimization algorithms to compute a good upper bound of the global minimum over each subspace. In this document, we propose a cooperativeframework in which interval methods cooperate with evolutionary algorithms. The latter are stochastic algorithms in which a population of individuals (candidate solutions) iteratively evolves in the search-space to reach satisfactory solutions. Evolutionary algorithms, endowed with operators that help individuals escape from local minima, are particularly suited for difficult problems on which traditional methods struggle to converge.Within our cooperative solver Charibde, the evolutionary algorithm and the interval- based algorithm run in parallel and exchange bounds, solutions and search-space via message passing. A strategy combining a geometric exploration heuristic and a domain reduction operator prevents premature convergence toward local minima and prevents theevolutionary algorithm from exploring suboptimal or unfeasible subspaces. A comparison of Charibde with state-of-the-art solvers based on interval analysis (GlobSol, IBBA, Ibex) on a benchmark of difficult problems shows that Charibde converges faster by an order of magnitude. New optimality results are provided for five multimodal problems, for which few solutions were available in the literature. We present an aeronautical application in which conflict solving between aircraft is modeled by an universally quantified constrained optimization problem, and solved by specific interval contractors. Finally, we certify the optimality of the putative solution to the Lennard-Jones cluster problem for five atoms, an open problem in molecular dynamics.
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Submitted on : Monday, March 16, 2015 - 4:31:45 PM
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  • HAL Id : tel-01132106, version 1

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Charlie Vanaret. Hybridation d’algorithmes évolutionnaires et de méthodes d’intervalles pour l’optimisation de problèmes difficiles. Recherche opérationnelle [cs.RO]. INP Toulouse, 2015. Français. ⟨tel-01132106⟩

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