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By Gunter Lumer

ISBN-10: 3540046003

ISBN-13: 9783540046004

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Moreover, Campi and Garatti [9] also show that (5) holds with equality under mild conditions. t. (6) t∈T which does not retain the desirable continuity property of P. t. 1 0⎛n y x ⎞ y δ ⎝ x ⎠ ≥ 0, −1 δ ∈ Δ, where δ is a random vector taking values on Δ = [{1} × (−C) × {0}] ∪ {0} × St t∈T with probability P. Taking N values of δ at random on Δ with probability P, we get the corresponding scenario program and, maintaining the notation of the previous case, we have, again by [9], the tight bound n P {V (x∗N ) > ε} ≤ N i εi (1 − ε)N −i , i=0 for any given ε > 0 and for any x∗N optimal solution of the scenario program.

This chapter surveys the state of the art in sensitivity and stability analysis in LSIP. Key words: linear semi-infinite programming, linear inequality systems, stability analysis, sensitivity analysis 1 Introduction Let T be an infinite set, a : T → Rn , b : T → R, and c ∈ Rn . t. c x := ci xi i=1 at x ≥ bt , t ∈ T, (1) is called a (primal) linear semi-infinite programming (LSIP in brief) problem because the number of variables is finite whereas the set of constraints is infinite. The mappings a and b are called left- and right-hand side (LHS and RHS) functions whereas c is called cost vector.

The converse statement holds if π ∈ ΠP B. ϑP is usc at π ⇔ F is lsc at π. If π is Hwp, then the restriction of ϑP to ΠP B is continuous. If F ∗ is bounded, then π is Hwp ⇔ either F is lsc at π or |F | = 1. If F ∗ is unbounded and π is Hwp, then F is lsc at π. A similar analysis has been carried out in [22] with other Hwp concepts. In the particular case that π ∈ int ΠP anovas et al. [25] provide an S , C´ expression for α (called Lipschitz constant), in terms of the data, such that ϑP (π 1 ) − ϑP (π 2 ) ≤ αd∞ (π 1 , π 2 ) for all π 1 , π 2 in some neighborhood of π.

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Algebres de fonctions et espaces de Hardy by Gunter Lumer


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