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.
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.
Variational problems under uniform quasiconvex constraints on the gradient are studied. In particular, existence of solutions to such problems is proved as well as existence of lagrange multipliers associated to the uniform constraint. They are shown to satisfy an Euler-Lagrange equation and a complementarity property. Our technique consists in approximating the original problem by a one-parameter family of smooth unconstrained optimization problems. Numerical experiments confirm the ability of our method to accurately compute solutions and Lagrange multipliers. 2. Statement of the problem and main results. Let Ω be a bounded domain in R N with N ≥ 1 and T : Ω × R m×N → [0, ∞[ a Carathéodory function. Let s ≥ 1 and consider a functional J : W 1,s (Ω; R m ) → R ∪ {+∞}, which is supposed to be bounded from below and sequentially lower semicontinuous in the weak topology of W 1,s (Ω; R m ). We are interested in the minimization problem inf{J(v) | T (·, ∇v) ∞,Ω ≤ 1, v ∈ g + W 1,s 0 (Ω; R m )}, (2.1)where T (·, ∇v) ∞,Ω = ess-sup{T (x, ∇v(x)) | x ∈ Ω}, and g ∈ W 1,∞ (Ω; R m ) ∩ C(Ω; R m ) is a given function satisfying J(g) < +∞ and T (x, ∇g(x)) ≤ 1 for a.e. x ∈ Ω.Then (2.1) may be rewritten as inf J ∞ (v) | v ∈ g + W 1,s 0 (Ω; R m ) .(2.3)
This paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion, with an special emphasis on applications to computing the Least Trimmed Squares Estimator (LTSE) for robust regression. The computation of the LTSE is a challenging subset selection problem involving a nonlinear program with continuous and binary variables, linked in a highly nonlinear fashion. The selection of a globally optimal subset using the branch and bound (BB) algorithm is limited to problems in very low dimension, tipically d ≤ 5, as the complexity of the problem increases exponentially with d. We introduce a bold pruning strategy in the BB algorithm that results in a significant reduction in computing time, at the price of a negligeable accuracy lost. The novelty of our algorithm is that the bounds at nodes of the BB tree come from pseudo-convexifications derived using a linearization technique with approximate bounds for the nonlinear terms. The approximate bounds are computed solving an auxiliary semidefinite optimization problem. We show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the LTSE problem.
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