  This paper is devoted to the study of tilt stability in finite dimensional
optimization via the approach of using the subgradient graphical derivative. We
establish a new characterization of tilt-stable local minimizers for a broad
class of unconstrained optimization problems in terms of a uniform positive
definiteness of the subgradient graphical derivative of the objective function
around the point in question. By applying this result to nonlinear programming
under the metric subregularity constraint qualification, we derive a
second-order characterization and several new sufficient conditions for tilt
stability. In particular, we show that each stationary point of a nonlinear
programming problem satisfying the metric subregularity constraint
qualification is a tilt-stable local minimizer if the classical strong
second-order sufficient condition holds.
