  In this paper, we study relative metric regularity of set-valued mappings
with emphasis on directional metric regularity. We establish characterizations
of relative metric regularity without assuming the completeness of the image
spaces, by using the relative lower semicontinuous envelopes of the distance
functions to set-valued mappings. We then apply these characterizations to
establish a coderivative type criterion for directional metric regularity as
well as for the robustness of metric regularity.
