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Subdifferentiation and Smoothing of Nonsmooth Integral Functionals

James Burke(jvburke***at***uw.edu)
Xiaojun Chen(maxjchen***at***polyu.edu.hk)
Hailin Sun(hlsun***at***njust.edu.cn)

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 equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about calculation of Clarke subgradients for the expectation for 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 \emph{measurable composite max integrands}, or CM 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

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: May, 2017

Download: [PDF]

Entry Submitted: 05/12/2017
Entry Accepted: 05/12/2017
Entry Last Modified: 05/12/2017

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