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The subdifferential of measurable composite max integrands and smoothing approximation

  • James V. Burke
  • Xiaojun ChenEmail author
  • Hailin Sun
Full Length Paper Series B
  • 63 Downloads

Abstract

The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential of the expectation equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about the calculation of Clarke subgradients for the expectation of non-regular integrands. The focus of this contribution is to approximate Clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand. A framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands. This class contains non-regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning.

Keywords

Stochastic optimization Clarke subgradient Smoothing Non-regular integrands 

Mathematics Subject Classification

90C15 

Notes

Acknowledgements

We would like to thank Associate Editor and two referees for their helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of Applied MathematicsHong Kong Polytechnic UniversityHong KongChina
  3. 3.Jiangsu Key Lab for NSLSCS, School of Mathematical SciencesNanjing Normal UniversityNanjingChina

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