Hölder Stable Minimizers, Tilt Stability, and Hölder metric Regularity of Subdifferentials

Using techniques of variational analysis and dual techniques for smooth conjugate functions, for a local minimizer of a proper lower semicontinuous function f on a Banach space, p ∈ (0, +∞) and q = 1+p p , we prove that the following two properties are always equivalent: (i) x̄ is a stable q-order minimizer of f and (ii) x̄ is a tilt-stable p-order minimizer of f . We also consider their relationships in conjunction with the p-order strong metric regularity of the subdifferential mapping ∂f .