First Order and Second Order Characterizations of Metric Subregularity and Calmness of Constraint Set Mappings
Helmut Gfrerer · SIAM Journal on Optimization · 2011
A condition ensuring metric subregularity (respectively, calmness) of general multifunctions between Banach spaces is derived. This condition is expressed solely 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. It is further shown that this sufficient condition is in some sense the weakest possible first order condition sufficient for subregularity. We extend this condition under the additional assumption that one part of the multifunction is known to be subregular in advance. We also derive second order conditions for metric subregularity, both sufficient and necessary, for multifunctions associated with constraint systems as they occur in optimization. We show that the main difference between the necessary and sufficient conditions is the replacement of an inequality by a strict inequality, just as in the case of “no gap” second order optimality conditions in optimization.