# An Infeasible Stochastic Approximation and Projection Algorithm for Stochastic Variational Inequalities

## Abstract

In this paper, we consider a stochastic variational inequality, in which the mapping involved is an expectation of a given random function. Inspired by the work of He (Appl Math Optim 35:69–76, 1997) and the extragradient method proposed by Iusem et al. (SIAM J Optim 29:175–206, 2019), we propose an infeasible projection algorithm with line search scheme, which can be viewed as a modification of the above-mentioned method of Iusem et al. In particular, in the correction step, we replace the projection by computing search direction and stepsize, that is, we need only one projection at each iteration, while the method of Iusem et al. requires two projections at each iteration. Moreover, we use dynamic sampled scheme with line search to cope with the absence of Lipschitz constant and choose the stepsize to be bounded away from zero and the direction to be a descent direction. In the process of stochastic approximation, we iteratively reduce the variance of a stochastic error. Under appropriate assumptions, we derive some properties related to convergence, convergence rate, and oracle complexity. In particular, compared with the method of Iusem et al., our method uses less projections and has the same iteration complexity, which, however, has a higher oracle complexity for a given tolerance in a finite dimensional space. Finally, we report some numerical experiments to show its efficiency.

## Keywords

Stochastic variational inequality Infeasible projection algorithm Stochastic approximation Convergence rate Oracle complexity## Mathematics Subject Classification

65K15 90C33 90C15## Notes

### Acknowledgements

This work was supported in part by NSFC (Nos. 11671250, 11431004, 71831008) and Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034). The authors are grateful to an anonymous referee for his/her helpful comments and suggestions, which have led to much improvement of the paper.

## References

- 1.Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
- 2.Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev.
**39**(4), 669–713 (1997)MathSciNetzbMATHGoogle Scholar - 3.Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample-path solution of stochastic variational inequalities. Math. Program.
**84**(2), 313–333 (1999)MathSciNetzbMATHGoogle Scholar - 4.Chen, X.J., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res.
**30**(4), 1022–1038 (2005)MathSciNetzbMATHGoogle Scholar - 5.Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl.
**140**(1), 103–116 (2009)MathSciNetzbMATHGoogle Scholar - 6.Rockafellar, R.T., Wets, R.J.B.: Stochastic variational inequalities: single-stage to multistage. Math. Program.
**165**(1), 331–360 (2017)MathSciNetzbMATHGoogle Scholar - 7.Chen, X.J., Pong, T.K., Wets, R.J.B.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program.
**165**(1), 71–111 (2017)MathSciNetzbMATHGoogle Scholar - 8.Lin, G.H., Fukushima, M.: Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: a survey. Pac. J. Optim.
**6**(3), 455–482 (2010)MathSciNetzbMATHGoogle Scholar - 9.Shanbhag, U.V.: Stochastic variational inequality problems: applications, analysis, and algorithms. INFORMS Tutor. Oper. Res. 71–107, (2013). https://doi.org/10.1287/educ.2013.0120
- 10.Xu, H.F.: Sample average approximation methods for a class of stochastic variational inequality problems. Asia-Pac. J. Oper. Res.
**27**(1), 103–119 (2010)MathSciNetzbMATHGoogle Scholar - 11.Wang, M.Z., Lin, G.H., Gao, Y.L., Ali, M.M.: Sample average approximation method for a class of stochastic variational inequality problems. J. Syst. Sci. Complex.
**24**(6), 1143–1153 (2011)MathSciNetzbMATHGoogle Scholar - 12.Yang, Z.P., Zhang, J., Zhu, X., Lin, G.H.: Infeasible interior-point algorithms based on sampling average approximations for a class of stochastic complementarity problems and theirapplications. J. Comput. Appl. Math.
**352**, 382–400 (2019)MathSciNetzbMATHGoogle Scholar - 13.Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program.
**53**(1), 99–110 (1992)MathSciNetzbMATHGoogle Scholar - 14.Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat.
**22**, 400–407 (1951)MathSciNetzbMATHGoogle Scholar - 15.Jiang, H.Y., Xu, H.F.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control
**53**(6), 1462–1475 (2008)MathSciNetzbMATHGoogle Scholar - 16.Yousefian, F., Nedić, A., Shanbhag, U.V.: Distributed adaptive steplength stochastic approximation schemes for Cartesian stochastic variational inequality problems. IEEE Trans. Autom. Control
**61**(3), 1753–1766 (2016)zbMATHGoogle Scholar - 17.Koshal, J., Nedic, A., Shanbhag, U.V.: Regularized iterative stochastic approximation methods for stochastic variational inequality problems. IEEE Trans. Autom. Control
**58**(3), 594–609 (2013)MathSciNetzbMATHGoogle Scholar - 18.Iusem, A.N., Jofré, A., Thompson, P.: Incremental constraint projection methods for monotone stochastic variational inequalities. Math. Oper. Res.
**44**(1), 236–263 (2018)MathSciNetGoogle Scholar - 19.Yousefian, F., Nedić, A., Shanbhag, U.V.: On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems. Math. Program.
**165**(1), 391–431 (2017)MathSciNetzbMATHGoogle Scholar - 20.Kannan, A., Shanbhag, U.V.: The pseudomonotone stochastic variational inequality problem: analytical statements and stochastic extragradient schemes. In: The 2014 Conference on American Control Conference, pp. 2930–2935 (2014)Google Scholar
- 21.Yousefian, F., Nedić, A., Shanbhag, U.V.: Optimal robust smoothing extragradient algorithms for stochastic variational inequality problems. In: The 2014 IEEE Annual Conference on Decision and Control, pp. 5831–5836 (2014)Google Scholar
- 22.Iusem, A.N., Jofré, A., Oliveira, R.I., Thompson, P.: Extragradient method with variance reduction for stochastic variational inequalities. SIAM J. Optim.
**27**(2), 686–724 (2017)MathSciNetzbMATHGoogle Scholar - 23.Iusem, A.N., Jofré, A., Oliveira, R.I., Thompson, P.: Variance-based extragradient methods with line search for stochastic variational inequalities. SIAM J. Optim.
**29**(1), 175–206 (2019)MathSciNetzbMATHGoogle Scholar - 24.He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim.
**35**(1), 69–76 (1997)MathSciNetzbMATHGoogle Scholar - 25.Burkholder, D.L., Davis, B.J., Gundy, R.F.: Integral inequalities for convex functions of operators on martingales. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 223–240 (1972)Google Scholar
- 26.Robbins, H., Siegmund, D.: A convergence theorem for non negative almost supermartingales and some applications. In: Optimizing Methods in Statistics, pp. 233–257 (1971)Google Scholar
- 27.Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim.
**43**(2), 147–168 (2001)MathSciNetzbMATHGoogle Scholar - 28.Dang, C.D., Lan, G.: On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators. Comput. Optim. Appl.
**60**(2), 277–310 (2015)MathSciNetzbMATHGoogle Scholar - 29.Cai, X., Gu, G., He, B.: On the \(O(1/t)\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl.
**57**(2), 339–363 (2014)MathSciNetzbMATHGoogle Scholar - 30.Kannan, A., Shanbhag, U.V.: Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants (2019). Preprint arXiv:1410.1628v3