Efficient Algorithms for Approximate Smooth Selection

Abstract

Algorithm: C m Selection Algorithm Data: Real numbers M > 0, τ ∈ (0, 1), an N−element set E ∈ R n and a convex polytope K(x) ⊂ R D for each x ∈ E /* We suppose that each K(x) is specified by at most C linear constraints */ Result: One of the following two outcomes:for each x ∈ E. Moreover, we guarantee that there existsNo go: We guarantee that there exists noAlgorithm 2: C m selection algorithm description.The first step is to place the problem in a wider context. Instead of merely examining the values of F at points x ∈ E, we consider the (m − 1)-rst degree Taylor polynomial of F at x, which we denote by J x (F). We write P to denote the vector space of all such Taylor polynomials. Instead of families of convex sets K(x) ⊂ R D , we consider families of convex sets Γ (x, M, τ) ⊂ P (x ∈ E, M > 0, τ ∈ (0, 1)). We want to find F ∈ C m (R n , R D ) with norm at most CM, such that J x (F) ∈ Γ (x, M, τ) for all x ∈ E.Under suitable assumptions on the Γ (x, M, τ), we provide the following algorithm.Algorithm: Generalized Selection Algorithm Data: Real numbers M > 0, τ ∈ (0, 1). A suitable family of convex sets Γ (x, M, τ). Result: One of the following two outcomes: Success: We exhibit a polynomial P x ∈ Γ (x, CM, Cτ) for each x ∈ E. Moreover, we guarantee that there existsNo go: We guarantee that there exists noThe algorithm requires at most C(τ)N log N operations. Our previous C m selection algorithm is a special case of the Generalized selection algorithm.Once we are dealing with Γ 's, we can take D = 1 without loss of generality,