The subdifferential of measurable composite max integrands and smoothing approximation

James V. Burke, Xiaojun Chen, Hailin Sun

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

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.
Original languageEnglish
Pages (from-to)229-264
Number of pages36
JournalMathematical Programming, Series B
Volume181
Issue number2
DOIs
Publication statusPublished - Jun 2020

Keywords

  • Clarke subgradient
  • Non-regular integrands
  • Smoothing
  • Stochastic optimization

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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