  This paper sheds new light on regularity of multifunctions through various
characterizations of directional H\"older /Lipschitz metric regularity, which
are based on the concepts of slope and coderivative. By using these
characterizations, we show that directional H\"older /Lipschitz metric
regularity is stable, when the multifunction under consideration is perturbed
suitably. Applications of directional H\"older /Lipschitz metric regularity to
investigate the stability and the sensitivity analysis of parameterized
optimization problems are also discussed.
