Tractable approximations of sets defined with quantifiers
نویسنده
چکیده
Given a compact basic semi-algebraic set K ⊂ Rn×Rm, a simple set B (box or ellipsoid), and some semi-algebraic function f , we consider sets defined with quantifiers, of the form Rf := {x ∈ B : f(x,y) ≤ 0 for all y such that (x,y) ∈ K} Df := {x ∈ B : f(x,y) ≤ 0 for some y such that (x,y) ∈ K}. The former set Rf is particularly useful to qualify “robust” decisions x versus noise parameter y (e.g. in robust optimization) whereas the latter set Df (a projection) is useful in design optimization when one does not want to work with its lifted representation {(x,y) ∈ K : f(x,y) ≥ 0}. We provide a systematic procedure to obtain a sequence of explicit inner (resp. outer) approximations that converge to Rf (resp. Df ) in a strong sense. An additional feature is that each approximation is the sublevel set of a single polynomial whose vector of coefficients is an optimal solution of a semidefinite program. Several extensions are also proposed, and in particular, approximations for sets of the form RF := {x ∈ B : (x,y) ∈ F for all y such that (x,y) ∈ K} where F is some other basic-semi algebraic set, and also sets defined with two quantifiers.
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عنوان ژورنال:
- Math. Program.
دوره 151 شماره
صفحات -
تاریخ انتشار 2015