Conditioning for optimization problems under general perturbations

Given a function f∈C1,1(B(0,r))f∈C1,1(B(0,r)), where B(0,r)B(0,r) denotes a ball of radius rr in a real Banach space EE, we provide the definition of a positive extended real number cˆ(f) defined through the function, that plays a role in the study of the sensitivity of the Argmin map of the perturbed function Fg(p,u)=f(u)−g(p,u)Fg(p,u)=f(u)−g(p,u). This number coincides with the number c2(f)c2(f) introduced by Zolezzi (2003) if linear perturbations g(p,u)=〈p,u〉g(p,u)=〈p,u〉 are considered.