Global and local error bounds: characterizations via directional derivatives and tangent cones

Abstract

We develop new characterizations of both global and local error bounds for general functions, using directional derivatives and tangent cones without imposing convexity or linear structure. We first establish several equivalent conditions for the global error bound of a nonnegative lower semicontinuous function. These equivalences hold for general, possibly nonconvex and nonsmooth functions. We further link the error bound with perturbation stability, Hausdorff stability of sublevel sets, and an inverse-sublevel-set estimate. Turning to directional derivatives, we introduce the minimal unit-sphere directional derivative \(φ(x)\) on the tangent cone and clarify its exact relation with the global slope. For Lipschitz continuous functions we prove that \(\sup_{x\notin S_0} φ(x) < 0\) is sufficient for an error bound, and for convex functions on convex sets this condition is also necessary, In finite dimensions we obtain sharp local results: if \(φ(\bar{x}) > 0\) at a solution \(\bar{x}\), then a local error bound holds and the optimal constant is exactly \(1/φ(\bar{x})\); if \(φ(\bar{x}) = 0\) and a suitable direction exists outside the tangent cone of the solution set, the local error bound fails. A general estimate relating the directional derivative to the distance from the tangent cone of the solution set is also derived.