Characterizing generalized derivatives of set-valued maps: Extending the tangential and normal approaches

2011/06/12 by C. H. Jeffrey Pang, Pang, C. H. Jeffrey
Computer Science · Mathematics · #26E25 #47H04 #54C60 #58C06 #58C20 #90C31 #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #math.OC #msc:26E25 #msc:47H04 #msc:54C60 #msc:58C06 #msc:58C20 #msc:90C31

paper · pdf · doi:10.48550/arxiv.1106.2338

Accepted for publication at SIAM. J. Control Optim.. This submission is not the final version, but corrects the first version in various places

openalex publication_date 2011/06/12 · arxiv created 2012/11/19 · arxiv updated 2012/11/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [20]. This result generalizes the Aubin criterion in [9]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result.

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