, Volume 14, Issue 3, pp 249261
First online:
Supercalm Multifunctions For Convergence Analysis
 Adam B. LevyAffiliated withDepartment of Mathematics, Bowdoin College Email author
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Calmness of multifunctions is a wellstudied concept of generalized continuity in which singlevalued selections from the image sets of the multifunction exhibit a restricted type of local Lipschitz continuity where the base point is fixed as one point of comparison. Generalized continuity properties of multifunctions like calmness can be applied to convergence analysis when the multifunction appropriately represents the iterates generated by some algorithm. Since it involves an essentially linear relationship between input and output, calmness gives essentially linear convergence results when it is applied directly to convergence analysis. We introduce a new continuity concept called ‘supercalmness’ where arbitrarily small calmness constants can be obtained near the base point, which leads to essentially superlinear convergence results. We also explore partial supercalmness and use a wellknown generalized derivative to characterize both when a multifunction is supercalm and when it is partially supercalm. To illustrate the value of such characterizations, we explore in detail a new example of a general primal sequential quadratic programming method for nonlinear programming and obtain verifiable conditions to ensure convergence at a superlinear rate.
Key words
calmness multifunctions generalized continuity variational analysis convergence analysis sequential quadratic programmingAMS 2000 Subject Classification
Primary: 49J53 Secondary: 90C30 Title
 Supercalm Multifunctions For Convergence Analysis
 Journal

SetValued Analysis
Volume 14, Issue 3 , pp 249261
 Cover Date
 200609
 DOI
 10.1007/s1122800600228
 Print ISSN
 09276947
 Online ISSN
 1572932X
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 calmness
 multifunctions
 generalized continuity
 variational analysis
 convergence analysis
 sequential quadratic programming
 Primary: 49J53
 Secondary: 90C30
 Authors

 Adam B. Levy ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Bowdoin College, Brunswick, ME, 04011, USA