First order and second order characterizations of metric subregularity and calmness of constraint set mappings.

*(English)*Zbl 1254.90246The author considers properties like metric regularity and subregularity, calmness, error bound property, etc. of multivalued mappings. In particular, a condition ensuring metric subregularity of general multifunctions between Banach spaces is derived. This condition is expressed in terms of the given data at the reference point and does not involve any information concerning the solution set of the corresponding inclusion given by the multifunction. In finite dimensions, this condition can be expressed in terms of a derivative which appears to be a combination of the coderivative and the contingent derivative. The author extends this condition under the additional assumption that one part of the multifunction is known to be subregular in advance. Second order conditions for metric subregularity, both sufficient and necessary, for multifunctions associated with constraint systems as they occur in optimization, have been derived.