The Scientific Papers of Sir G. Darwin [V. 2 - Tidal Frict., by George Howard Darwin, Sir; Francis Darwin, Sir; Ernest W

By George Howard Darwin, Sir; Francis Darwin, Sir; Ernest W Brown; Frederick John Marrian Stratton; J Jackson

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YS ,x {ψ + 1 1−α ps ys : ys ≥ 0, x ω s − ψ − ys ≤ 0 ∀s, x ∈ X }. (19) s Let ψ ∗ (P ), x∗ (P ) be an optimal solution of (18) and denote ϕCα (P ) the optimal value. To get contamination bounds for the optimal value of (18) with P contaminated by a stress probability distribution Q it is sufficient to assume a compact set of optimal solutions of (18). An evident instance is compact X and bounded interval (14). The bounds follow the usual pattern, compare with (6): (1−λ)ϕCα (P )+λΦα (x∗ (P ), ψ ∗ (P ), Q) ≥ ϕCα (Pλ ) ≥ (1−λ)ϕCα (P )+λϕCα (Q).

5, and that samples are affected by current x and rare catastrophic events. In addition, the sample mean approximations FiN (x) may destroy the concavity (convexity) of functions Fi (x). For example, the expectation function ax2 , a = p1 ω1 + p2 ω2 > 0, ω1 > 0, ω2 < 0 is the convex function, but its samN k 2 ple mean approximation ( N1 k=1 ω )x may be the concave function even for rather large N in the case of a small probability p1 and a large impact ω1 > 0. In these cases, in general, only AMC optimization is applicable.

8 Sensitivity of Robust Strategies Robust strategies for global changes require a proper focus on potential extreme events. As a result, the robust strategy with a small ε > 0 probability of extreme events can be significantly different from the policy that ignores these events by using ε = 0. 3, when ε > 0 results in shifts of ranges fi (x, ω) to include potential catastrophic impacts (say, ranges of required emission reductions β in Example 4) that suddenly disappear for ε = 0. Informally speaking, the explicit introduction of extreme events with ε > 0 requires new sets of feasible decisions, new Facets of Robust Decisions 23 spatial, temporal, and social dimensions which suddenly disappear for ε = 0.

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