Problemes d'optimisatión globale en statistique robuste

Abstract

Robust statistics is a branch of statistics dealing with the analysis of data containing contarninated observations. The robustness of an estirnator is rnea.sured notably by means of the br-eakrlown po'int. High-breakdown point estirnators are usua.lly clefined as global mínima. of a non-convex scale of the errors, hence their cornputa.tion is a challenging global optirnization problem. The objective of this dissertation is to investígate the potential contributions of modern global optimization methocls to this class of problems. The first part of this thesis is devotecl to the T-estimator for linear regression, which is defined as a global mínimum of a nonconvex clifi:'erentiable function. vVe investigate the impact of incorporating clustering techniques and stopping conditions in exist.ing stochastic a.lgorithms. The consequences of sorne phenomena involving the nea.rest neighbor in high dimension on clustering global optimization algorithms is thoroughly discussecl as well. The second part is clevoted to cleterministic a.lgorit.hms for computing the least trirmned squares regression estirna.tor, which is clefined through a. nonlinea.r mixedinteger program. Due to the combinatoria.l nature of this problem, we concentratecl on obtaining lower bouncls to be used in a branch-ancl-bound algorithm. In particular, we propose a second-order cone rela'"'<at.ion that can be complemented with concavity cuts that we obtain explicitly. Global optimality conditions are also provided.