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Penalty methods with stochastic approximation for stochastic nonlinear programming


Authors: Xiao Wang, Shiqian Ma and Ya-xiang Yuan
Journal: Math. Comp. 86 (2017), 1793-1820
MSC (2010): Primary 90C15, 90C30, 62L20, 90C60
DOI: https://doi.org/10.1090/mcom/3178
Published electronically: October 12, 2016
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Abstract: In this paper, we propose a class of penalty methods with stochastic approximation for solving stochastic nonlinear programming problems. We assume that only noisy gradients or function values of the objective function are available via calls to a stochastic first-order or zeroth-order oracle. In each iteration of the proposed methods, we minimize an exact penalty function which is nonsmooth and nonconvex with only stochastic first-order or zeroth-order information available. Stochastic approximation algorithms are presented for solving this particular subproblem. The worst-case complexity of calls to the stochastic first-order (or zeroth-order) oracle for the proposed penalty methods for obtaining an $ \epsilon $-stochastic critical point is analyzed.


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Additional Information

Xiao Wang
Affiliation: School of Mathematical Sciences, University of Chinese Academy of Sciences; Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, People’s Republic of China
Email: wangxiao@ucas.ac.cn

Shiqian Ma
Affiliation: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong
Email: sqma@se.cuhk.edu.hk

Ya-xiang Yuan
Affiliation: State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, People’s Republic of China
Email: yyx@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/mcom/3178
Keywords: Stochastic programming, nonlinear programming, stochastic approximation, penalty method, global complexity bound
Received by editor(s): April 8, 2015
Received by editor(s) in revised form: December 1, 2015
Published electronically: October 12, 2016
Additional Notes: The research of the first author was supported in part by Postdoc Grant 119103S175, UCAS President Grant Y35101AY00 and NSFC Grant 11301505.
The research of the second author was supported in part by a Direct Grant of the Chinese University of Hong Kong (Project ID: 4055016) and the Hong Kong Research Grants Council General Research Fund Early Career Scheme (Project ID: CUHK 439513)
The research of the third author was supported in part by NSFC Grants 11331012, 11321061 and 11461161005
Article copyright: © Copyright 2016 American Mathematical Society