A quantitative study of the robustness properties of the 1 and the Huber M-estimator on finite samples is presented. The focus is on the linear model involving a fixed design matrix and additive errors restricted to the dependent variables consisting of noise and sparse outliers. We derive sharp error bounds for the 1 estimator in terms of the leverage constants of a design matrix introduced here. A similar analysis is performed for Huber's estimator using an equivalent problem formulation of independent interest. Our analysis considers outliers of arbitrary magnitude, and we recover breakdown point results as particular cases when outliers diverge. The practical implications of the theoretical analysis are discussed on two real datasets.
In previous works CO 2 emissions in oil refineries have been studied for production unit planning. In this manuscript the associated CO 2 mitigation costs are added to the scheduling of crude oil unloading and blending. Numerical simulations executed on literature cases show that the optimal scheduling may be affected, and thus CO 2 emissions may be greater than those predicted from production unit planning. Furthermore, the biobjective problem of maximizing profits and minimizing CO 2 emissions is studied; paretooptimal solutions and the lowest carbon pricing that induces the refinery to minimize CO 2 emissions are presented for each case.
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