Some Sufficient Conditions Ensuring Calmness For Closed Multifunctions Over Constraint Sets

Calmness closely relates to the mathematical programming and optimization.Many scholars have done extensive research on the calmness.In this paper,we mainly study calmness and strong calmness of closed multifunctions over constraint sets in Banach spaces.Under the assumption of subsmooth,in terms of Clarke tangent cones and tan-gent derivatives,we provide some sufficient conditions ensuring calmness over constraint sets.For the strong calmness case,we further prove that the condition in terms of clarke tangent cone are also neccesary for the strong calmess.