Global optimization problems in robust statistics

Overview & Synopsis

Robust statistics is a branch of statistics dealing with the analysis of data containing contaminated observations. The robustness of an estimator is measured notably by means of the breakdown point. High-breakdown point estimators are usually defined as global minima of a non-convex scale of the errors, hence their computation is a challenging global optimization problem. The objective of this dissertation is to investigate the potential contributions of modern global optimization methods to this class of problems. The first part of this thesis is devoted to the tau -estimator for linear regression, which is defined as a global minimum of a nonconvex differentiable function. We investigate the impact of incorporating clustering techniques and stopping conditions in existing stochastic algorithms. The consequences of some phenomena involving the nearest neighbor in high dimension on clustering global optimization algorithms are thoroughly discussed as well. The second part is devoted to deterministic algorithms for computing the least trimmed squares regression estimator, which is defined through a nonlinear mixed-integer program. Due to the combinatorial nature of this problem, we concentrated on obtaining lower bounds to be used in a branch-and-bound algorithm. In particular, we propose a second-order cone relaxation that can be complemented with concavity cuts that we obtain explicitly. Global optimality conditions are also provided.