Set-Valued and Variational Analysis

, Volume 27, Issue 3, pp 665–692 | Cite as

Continuous Outer Subdifferentials in Nonsmooth Optimization

  • Martin KnossallaEmail author


The theory of subdifferentials provides adequate methods and tools to put descent methods for nonsmooth optimization problems into practice. However, in applications it is often difficult to decide on a suitable subdifferential concept to construct a descent method. Therefore, we introduce subdifferentials in terms of their properties to indicate a selection of subdifferentials worth considering. This initials the first part of the construction of a continuous outer subdifferential (COS). Typically, methods based on e.g. the Clarke subdifferential are non-convergent without assumptions like semismoothness on the objective function. In cases in which only supersets of the Clarke subdifferential are known, semismoothness cannot be proved or is even violated. Therefore, in the second part of the construction, a previously selected subdifferential will be expanded to a continuous mapping, if necessary. This is also practicable for upper bounds of the subdifferential of current interest. Finally, based on COS we present a methodology for solving nonsmooth optimization problems. From a theoretical point of view, convergence is established through the construction of COS.


Descent method Nonsmooth optimization Set-valued mapping Subdifferentials 

Mathematics Subject Classification (2010)

49J52 49J53 90C26 


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The author wishes to thank the reviewers and the associate editor for their very grateful suggestions, which greatly have improve the presentation of the paper. Also, the author likes to thank W. Achtziger for his time to discuss several results.


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Authors and Affiliations

  1. 1.Department of MathematicsFriedrich-Alexander University Erlangen-NürnbergErlangenGermany

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