The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.
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The analog in numerical analysis lies in the theory of overdetermined systems of equations generated by noise in measurements of parameters.
In part, this has recently been taken up by Chen et al. in  in terms of discrete approximations to two-stage stochastic variational inequality problems based on “continuous” probability instead of “discrete” probability, as here.
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Chen, X., Fukushima, M.: Expected residual minimization method or stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)
Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)
Chen, X., Pong, T.K., Wets, R.: Two-stage stochastic variational inequalities: an ERM-solution procedure. Math. Program. 165, 71–111 (2017)
Fang, H., Chen, X., Fukushima, M.: Stochastic \(R_0\) matrix linear complementarity problem. SIAM J. Optim. 18, 482–506 (2007)
Jiang, H., Xu, H.: Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Autom. Control 53, 1462–1475 (2008)
Lu, S.: Confidence regions for stohastic variational inequalities. Math. Oper. Res. 38, 545–568 (2013)
Xu, H.: Sample average approximation methods for a class of variational inequality problems. Asia Pac. J. Oper. Res. 27, 103–119 (2010)
Gürkan, G., Özge, A.Y., Robinson, S.M.: Sample path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)
Iusem, A., Jofré, A., Thompson, P.: Incremental constraint projection methods for monotone stochastic variational inequalities, Math. Oper. Res. (to appear)
Iusem, A., Jofré, A., Oliveira, R.I., Thompson, P.: Extragradient method with variance reduction for stochastic variational inequalities. SIAM J. Optim. 27, 686–724 (2017)
Iusem, A., Jofré, A., Oliveira, R.I., Thompson, P.: Variance-based stochastic extragradient methods with line search for stochastic variational inequalies (submitted)
Gabriel, S.A., Zhuang, J., Egging, R.: Solving stochastic complementarity problems in energy market modeling using scenario reduction. Eur. J. Oper. Res. 197, 1028–1040 (2009)
Gabriel, S.A., Fuller, J.D.: A Benders decomposition method for solving stochastic complementarity problems with an application in energy. Comput. Econ. 35, 301–329 (2010)
Philpott, A.B., Ferris, M.C., Wets, R.J.-B.: Equilibrium, uncertainty and risk in hydro-thermal electricity systems. Math. Program. B 157, 483–513 (2016)
Rockafellar, R.T., Wets, R.J.-B.: Stochastic variational inequalities: single-stage to multistage. Math. Program. B 165, 291–330 (2017)
Rockafellar, R.T., Wets, R.J.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119–147 (1991)
Chen, X., Sun, H., Xu, H.: Discrete approximation of two-stage stochastic and distributionally robust linear complementarity problems. arXiv:1796.06847
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Spingarn, J.E.: Partial inverse of a monotone operator. Appl. Math. Optim. 10, 247–265 (1983)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Cambridge (1992). (republished by SIAM in 2009)
Qi, Liqun, Sun, Jie: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
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Rockafellar, R.T., Sun, J. Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging. Math. Program. 174, 453–471 (2019) doi:10.1007/s10107-018-1251-y
- Progressive hedging algorithm
- Stochastic variational inequality problems
- Stochastic complementarity problems
- Stochastic programming problems
- Maximal monotone mappings
- Proximal point algorithm
- Problem decomposition
Mathematics Subject Classification