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On Metric and Calmness Qualification Conditions in Subdifferential Calculus

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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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Authors and Affiliations

  1. Department of Mathematics, Technion, Haifa, 32000, Israel

    Alexander D. Ioffe

  2. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 18208, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. Alexander D. Ioffe
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  2. Jiří V. Outrata
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Corresponding author

Correspondence to Jiří V. Outrata.

Additional information

This paper is dedicated to Prof. B. S. Mordukhovich on the occasion of his 60th birthday.

The research of Jiří V. Outrata was supported by the grant A 107 5402 of the Grant Agency of the Academy of Sciences of the Czech Republic.

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Ioffe, A.D., Outrata, J.V. On Metric and Calmness Qualification Conditions in Subdifferential Calculus. Set-Valued Anal 16, 199–227 (2008). https://doi.org/10.1007/s11228-008-0076-x

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  • DOI: https://doi.org/10.1007/s11228-008-0076-x

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