Cutting-Planes for Optimization of Convex Functions over Nonconvex Sets

We derive linear inequality characterizations for sets of the form conv{(x, q) ∈ R×R : q ≥ Q(x), x ∈ R − int(P )} where Q is convex and differentiable and P ⊂ R. We show that in several cases our characterization leads to polynomial-time separation algorithms that operate in the original space of variables, in particular when Q is a positive-definite quadratic and P is a polyhedron or an ellipsoid.