On Feasible Sets Defined Through Chebyshev Approximation

摘要

Let Z be a compact set of the real space R with at least n + 2 points; f, h1, h2: Z → R continuous functions, h1, h2 strictly positive and P(x,z), x: = (x_0,...,x_n)~τ ∈ R~(n+1), z ∈ R, a poly-nomial of degree at most n. Consider a feasible set M: = {x ∈ R~(n+1) | z ∈ Z, -h_2(z) ≤ P (x,z)-f(z) ≤ h_1 (z)}. Here it is proved the null vector 0 of R~(n+1) belongs to the compact convex hull of the gradients ± (1,z,...,z~n), where z ∈ Z are the index points in which the constraint functions are active for a given x~* ∈ M, if and only if M is a singleton.