On metric and calmness qualification conditions in subdifferential calculus
Abstract.
The paper contains two groups of results. The first are criteria for calmness/subregularity
for set-valued mappings between finite-dimensional spaces. We give a
new sufficient condition whose subregularity part has the same form as the coderivative
criterion for "full" metric regularity but involves a different type of coderivative which is
introduced in the paper. We also show that the condition is necessary for mappings with
convex graphs.
The second group of results deals with the basic calculus rules of nonsmooth subdifferential
calculus. For each of the rules we state two qualification conditions: one in
terms of calmness/subregularity of certain set-valued mappings and the other as a metric
estimate (not necessarily directly associated with aforementioned calmness/subregularity
property). The conditions are shown to be weaker than the standard Mordukhovich-Rockafellar
subdifferential qualification condition; in particular they cover the cases of
convex polyhedral set-valued mappings and, more generally, mappings with semi-linear
graphs. Relative strength of the conditions is thoroughly analyzed. We also show, for each
of the calculus rules, that the standard qualification conditions are equivalent to "full"
metric regularity of precisely the same mappings that are involved in the subregularity
version of our calmness/subregularity condition.
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