Estimating Tangent and Normal Cones Without Calculus

1. The tangent cone. We can estimate the directional derivative and gradient of a smooth function quickly and easily using finite difference formulas. While rather inaccurate, such estimates have some appeal, needing neither calculus rules nor even a closed-form expression for the function. In the variational geometry of sets, the role of derivatives and gradients are played by the cones of “tangent” and “normal” vectors. We study here how we might estimate these cones, without any prior knowledge of the structure of the set (like convexity, for example), and without recourse either to the calculus rules of nonsmooth analysis or even to an analytic description of the set. We rely instead only on the most primitive description of the set, namely a membership oracle—an algorithm that decides whether or not any given point belongs to the set. Our interest is both philosophical and practical. Philosophically, are the tangent and normal cones in any sense computable from a primitive check on set membership? Practically, could we design a subroutine for estimating these cones without requiring any structural knowledge of the set from the user? Like differentiation, the idea of the tangent cone involves a limit, but one involving sets. We therefore present some notions of set convergence, before defining the tangent cone. Comprehensive presentations of variational analysis and nonsmooth optimization can be found in Clarke et al. [4] or Rockafellar and Wets [10]. We follow the notation and terminology of the latter, unless otherwise stated. Definition 1.1. Given a family of sets D ⊂Rn indexed by > 0, the outer and inner limits are defined, respectively, by