Mathematical Programming

, Volume 165, Issue 1, pp 391–431 | Cite as

On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

  • Farzad Yousefian
  • Angelia Nedić
  • Uday V. Shanbhag
Full Length Paper Series B


Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA\(_r\), in which we employ a weighted iterate-averaging, parameterized by a scalar r where \(r = 1\) provides us with the standard averaging scheme. The main contributions are threefold: (i) when \(r<1\) and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate \(\mathcal{O}({1}\slash {\root 6 \of {k}})\) after k steps; (ii) when \(r<1\), and smoothing and regularization are suppressed, the gap function admits the rate \(\mathcal{O}({1}\slash {\sqrt{k}})\), thus improving the rate \(\mathcal{O}(\ln (k)/\sqrt{k})\) under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for \(r < 1\). Notably, we prove the superiority of the scheme with \(r < 1\) with its counterpart with \(r=1\) in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.


  1. 1.
    Bertsekas, D.P.: Stochastic optimization problems with nondifferentiable cost functionals. J. Optim. Theory Appl. 12(2), 218–231 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming: Springer Series in Operations Research. Springer, Berlin (1997)MATHGoogle Scholar
  3. 3.
    Cicek, D., Broadie, M., Zeevi, A.: General bounds and finite-time performance improvement for the Kiefer–Wolfowitz stochastic approximation algorithm. Oper. Res. 59, 1211–1224 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Duchi, J.C., Bartlett, P.L., Martin, J.: Wainwright, Randomized smoothing for stochastic optimization. SIAM J. Optim. 22(2), 674–701 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ermoliev, Y.M.: Stochastic quasigradient methods and their application to system optimization. Stochastics 9, 1–36 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vols. I, II, Springer Series in Operations Research. Springer, New York (2003)Google Scholar
  7. 7.
    Ghadimi, S., Lan, G.: Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, part I: a generic algorithmic framework. SIAM J. Optim. 22(4), 1469–1492 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Golshtein, E.G., Tretyakov, N.V.: Modified Lagrangians and Monotone Maps in Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1996), Translated from the 1989 Russian original by Tretyakov, A Wiley-Interscience PublicationGoogle Scholar
  9. 9.
    Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53(6), 1462–1475 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Juditsky, A., Nemirovski, A., Tauvel, C.: Solving variational inequalities with stochastic mirror-prox algorithm. Stoch. Syst. 1(1), 17–58 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kannan, A., Shanbhag, U.V.: Distributed computation of equilibria in monotone Nash games via iterative regularization techniques. SIAM J. Optim. 22(4), 1177–1205 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kannan, A., Shanbhag, U.V., Kim, H.M.: Addressing supply-side risk in uncertain power markets: stochastic Nash models, scalable algorithms and error analysis. Optim. Methods Softw. 28(5), 1095–1138 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Knopp, K.: Theory and Applications of Infinite Series. Blackie & Son Ltd., Glasgow, Great Britain (1951)MATHGoogle Scholar
  14. 14.
    Koshal, J., Nedić, A., Shanbhag, U.V.: Regularized iterative stochastic approximation methods for variational inequality problems. IEEE Trans. Autom. Control 58(3), 594–609 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kushner, H.J., Yin, G.G.: Stochastic Approximation and Recursive Algorithms and Applications. Springer, New York (2003)MATHGoogle Scholar
  16. 16.
    Lakshmanan, H., Farias, D.: Decentralized recourse allocation in dynamic networks of agents. SIAM J. Optim. 19(2), 911–940 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Larsson, T., Patriksson, M.: A class of gap functions for variational inequalities. Math. Program. 64, 53–79 (1994)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lu, S.: Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities. SIAM J. Optim. 24(3), 1458–1484 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38(3), 545–568 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Marcotte, P., Zhu, D.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9(1), 179–189 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mayne, D.Q., Polak, E.: Nondifferential optimization via adaptive smoothing. J. Optim. Theory Appl. 43(4), 601–613 (1984)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Metzler, C., Hobbs, B.F., Pang, J.-S.: Nash–Cournot equilibria in power markets on a linearized dc network with arbitrage: formulations and properties. Netw. Spatial Econ. 3(2), 123–150 (2003)CrossRefGoogle Scholar
  23. 23.
    Nedić, A., Lee, S.: On stochastic subgradient mirror-descent algorithm with weighted averaging. SIAM J. Optim. 24(1), 84–107 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574–1609 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Norkin, V.I.: The analysis and optimization of probability functions. Technical report, International Institute for Applied Systems Analysis technical report, 1993, WP-93-6Google Scholar
  26. 26.
    Polyak, B.T.: Introduction to Optimization. Optimization Software Inc, New York (1987)MATHGoogle Scholar
  27. 27.
    Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30(4), 838–855 (1992)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ravat, U., Shanbhag, U.V.: On the characterization of solution sets in smooth and nonsmooth stochastic convex Nash games. SIAM J. Optim. 21(3), 1046–1081 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22, 400–407 (1951)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  31. 31.
    Shapiro, A.: Monte Carlo Sampling Methods, Handbook in Operations Research and Management Science, vol. 10. Elsevier Science, Amsterdam (2003)Google Scholar
  32. 32.
    Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. The society for industrial and applied mathematics and the mathematical programming society, Philadelphia, USA (2009)Google Scholar
  33. 33.
    Steklov, V.A.: Sur les expressions asymptotiques decertaines fonctions dfinies par les quations diffrentielles du second ordre et leers applications au problme du dvelopement d’une fonction arbitraire en sries procdant suivant les diverses fonctions. Commun. Charkov Math. Soc. 2(10), 97–199 (1907)Google Scholar
  34. 34.
    Xu, H.: Adaptive smoothing method, deterministically computable generalized jacobians, and the newton method. J. Optim. Theory Appl. 109(1), 215–224 (2001)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Xu, H.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pacific J. Oper. Res. 27(1), 103–119 (2010)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yousefian, F., Nedić, A., Shanbhag, U.V.: On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequalities. arXiv:1411.0209v2
  37. 37.
    Yousefian, F., Nedić, A., Shanbhag, U.V.: Self-tuned stochastic approximation schemes for non-Lipschitzian stochastic multi-user optimization and Nash games. IEEE Trans. Autom. Control 61(7), 1753–1766. doi:10.1109/TAC.2015.2478124.
  38. 38.
    Yousefian, F., Nedić, A., Shanbhag, U.V.: On stochastic gradient and subgradient methods with adaptive steplength sequences. automatica 48(1), 56–67 (2012), An extended version of the paper available at arXiv:1105.4549
  39. 39.
    Yousefian, F., Nedić, A., Shanbhag, U.V.: A Regularized smoothing stochastic approximation (RSSA) algorithm for stochastic variational inequality problems. In: Proceedings of the 2013 Winter Simulation Conference (2013), pp. 933–944Google Scholar

Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.School of Industrial Engineering and ManagementOklahoma State UniversityStillwaterUSA
  2. 2.School of Electrical, Computer and Energy EngineeringArizona State UniversityTempeUSA
  3. 3.Industrial and Manufacturing EngineeringPennsylvania State UniversityState CollegeUSA

Personalised recommendations