The subdifferential of measurable composite max integrands and smoothing approximation

• 成果类型: Article
• 署名作者: Burke, James V.; Chen, Xiaojun; Sun, Hailin
• 署名单位: University of Washington; University of Washington Seattle; Hong Kong Polytechnic University; Nanjing Normal University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01441-9
• 发表日期: 2020
• 页码: 229-264
• 关键词: 2-stage stochastic programs; stationary-points; CONVERGENCE; optimality; nonsmooth

摘要

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.