On Metric Pseudo-(sub)Regularity of Multifunctions and Optimality Conditions for Degenerated Mathematical Programs

This paper is devoted to the analysis of a special kind of regularity of a multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we derive conditions both necessary and sufficient for this property. As a byproduct we obtain a new sufficient condition for metric subregularity. Then we apply these results to multifunctions composed by a smooth mapping and a generalized polyhedral multifunction and obtain explicit formulas for this case. Finally we show how the theory can be used to obtain necessary optimality conditions when the constraint qualification condition of metric (sub)regularity is violated.