Penalty methods with stochastic approximation for stochastic nonlinear programming

Wang, Xiao; Ma, Shiqian; Yuan, Ya-xiang

doi:10.1090/mcom/3178

Penalty methods with stochastic approximation for stochastic nonlinear programming
HTML articles powered by AMS MathViewer

by Xiao Wang, Shiqian Ma and Ya-xiang Yuan;

Math. Comp. 86 (2017), 1793-1820

DOI: https://doi.org/10.1090/mcom/3178

Published electronically: October 12, 2016

PDF | Request permission

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.

References

Fabian Bastin, Cinzia Cirillo, and Philippe L. Toint, Convergence theory for nonconvex stochastic programming with an application to mixed logit, Math. Program. 108 (2006), no. 2-3, Ser. B, 207–234. MR 2238700, DOI 10.1007/s10107-006-0708-6
Wei Bian and Xiaojun Chen, Worst-case complexity of smoothing quadratic regularization methods for non-Lipschitzian optimization, SIAM J. Optim. 23 (2013), no. 3, 1718–1741. MR 3093871, DOI 10.1137/120864908
John R. Birge and François Louveaux, Introduction to stochastic programming, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2011. MR 2807730, DOI 10.1007/978-1-4614-0237-4
D. Brownstone, D. S. Bunch, and K. Train, Joint mixed logit models of stated and revealed preferences for alternative-fuel vehicles, Transportation Research B 34 (2000), no. 5, 315–338.
Coralia Cartis, Nicholas I. M. Gould, and Philippe L. Toint, On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming, SIAM J. Optim. 21 (2011), no. 4, 1721–1739. MR 2869514, DOI 10.1137/11082381X
K. L. Chung, On a stochastic approximation method, Ann. Math. Statistics 25 (1954), 463–483. MR 64365, DOI 10.1214/aoms/1177728716
Cong D. Dang and Guanghui Lan, Stochastic block mirror descent methods for nonsmooth and stochastic optimization, SIAM J. Optim. 25 (2015), no. 2, 856–881. MR 3341135, DOI 10.1137/130936361
John C. Duchi, Peter L. Bartlett, and Martin J. Wainwright, Randomized smoothing for stochastic optimization, SIAM J. Optim. 22 (2012), no. 2, 674–701. MR 2968871, DOI 10.1137/110831659
Yuri Ermoliev, Stochastic quasigradient methods and their application to system optimization, Stochastics 9 (1983), no. 1-2, 1–36. MR 703846, DOI 10.1080/17442508308833246
Michael C. Fu, Optimization for simulation: theory vs. practice, INFORMS J. Comput. 14 (2002), no. 3, 192–215. MR 1918923, DOI 10.1287/ijoc.14.3.192.113
A. A. Gaĭvoronskiĭ, Nonstationary stochastic programming problems, Kibernetika (Kiev) 4 (1978), 89–92 (Russian, with English summary). MR 509843
R. Garmanjani and L. N. Vicente, Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization, IMA J. Numer. Anal. 33 (2013), no. 3, 1008–1028. MR 3081492, DOI 10.1093/imanum/drs027
Saeed Ghadimi and Guanghui Lan, Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization I: A generic algorithmic framework, SIAM J. Optim. 22 (2012), no. 4, 1469–1492. MR 3023780, DOI 10.1137/110848864
Saeed Ghadimi and Guanghui Lan, Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, II: Shrinking procedures and optimal algorithms, SIAM J. Optim. 23 (2013), no. 4, 2061–2089. MR 3118261, DOI 10.1137/110848876
Saeed Ghadimi and Guanghui Lan, Stochastic first- and zeroth-order methods for nonconvex stochastic programming, SIAM J. Optim. 23 (2013), no. 4, 2341–2368. MR 3134439, DOI 10.1137/120880811
Saeed Ghadimi and Guanghui Lan, Accelerated gradient methods for nonconvex nonlinear and stochastic programming, Math. Program. 156 (2016), no. 1-2, Ser. A, 59–99. MR 3459195, DOI 10.1007/s10107-015-0871-8
Saeed Ghadimi, Guanghui Lan, and Hongchao Zhang, Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization, Math. Program. 155 (2016), no. 1-2, Ser. A, 267–305. MR 3439803, DOI 10.1007/s10107-014-0846-1
D. A. Hensher and W. H. Greene, The mixed logit model: The state of practice, Transportation 30 (2003), 133–176.
A. Juditsky, P. Rigollet, and A. B. Tsybakov, Learning by mirror averaging, Ann. Statist. 36 (2008), no. 5, 2183–2206. MR 2458184, DOI 10.1214/07-AOS546
Anton J. Kleywegt, Alexander Shapiro, and Tito Homem-de-Mello, The sample average approximation method for stochastic discrete optimization, SIAM J. Optim. 12 (2001/02), no. 2, 479–502. MR 1885572, DOI 10.1137/S1052623499363220
Guanghui Lan, An optimal method for stochastic composite optimization, Math. Program. 133 (2012), no. 1-2, Ser. A, 365–397. MR 2921104, DOI 10.1007/s10107-010-0434-y
Guanghui Lan, Arkadi Nemirovski, and Alexander Shapiro, Validation analysis of mirror descent stochastic approximation method, Math. Program. 134 (2012), no. 2, Ser. A, 425–458. MR 2961314, DOI 10.1007/s10107-011-0442-6
J. Mairal, F. Bach, J. Ponce, and G. Sapiro, Online dictionary learning for sparse coding, In ICML, 2009.
A. Nemirovski and R.Y. Rubinstein, An efficient stochastic approximation algorithm for stochastic saddle point problems, in Modeling Uncertainty, Springer, 2005, pp. 156–184.
A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM J. Optim. 19 (2008), no. 4, 1574–1609. MR 2486041, DOI 10.1137/070704277
A. S. Nemirovsky and D. B. Yudin, Problem complexity and method efficiency in optimization, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. Translated from the Russian and with a preface by E. R. Dawson; A Wiley-Interscience Publication. MR 702836
Yu. E. Nesterov, A method for solving the convex programming problem with convergence rate $O(1/k^{2})$, Dokl. Akad. Nauk SSSR 269 (1983), no. 3, 543–547 (Russian). MR 701288
Y. E. Nesterov, Random gradient-free minimization of convex functions, Technical report, Center for Operation Research and Econometrics (CORE), Catholic University of Louvain, 2010.
Jorge Nocedal and Stephen J. Wright, Numerical optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. MR 2244940
B. T. Polyak, A new method of stochastic approximation type, Avtomat. i Telemekh. 7 (1990), 98–107 (Russian); English transl., Automat. Remote Control 51 (1990), no. 7, 937–946 (1991). MR 1071220
B. T. Polyak and A. B. Juditsky, Acceleration of stochastic approximation by averaging, SIAM J. Control Optim. 30 (1992), no. 4, 838–855. MR 1167814, DOI 10.1137/0330046
Herbert Robbins and Sutton Monro, A stochastic approximation method, Ann. Math. Statistics 22 (1951), 400–407. MR 42668, DOI 10.1214/aoms/1177729586
Andrzej Ruszczyński and Wojciech Syski, A method of aggregate stochastic subgradients with on-line stepsize rules for convex stochastic programming problems, Math. Programming Stud. 28 (1986), 113–131. Stochastic programming 84. II. MR 836764, DOI 10.1007/bfb0121128
Jerome Sacks, Asymptotic distribution of stochastic approximation procedures, Ann. Math. Statist. 29 (1958), 373–405. MR 98427, DOI 10.1214/aoms/1177706619
Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczyński, Lectures on stochastic programming, MPS/SIAM Series on Optimization, vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2009. Modeling and theory. MR 2562798, DOI 10.1137/1.9780898718751
Mengdi Wang and Dimitri P. Bertsekas, Stochastic first-order methods with random constraint projection, SIAM J. Optim. 26 (2016), no. 1, 681–717. MR 3472017, DOI 10.1137/130931278
Y. Yuan, Conditions for convergence of trust region algorithms for nonsmooth optimization, Math. Programming 31 (1985), no. 2, 220–228. MR 777292, DOI 10.1007/BF02591750