Advertisement

Mathematical Programming

, Volume 165, Issue 1, pp 71–111 | Cite as

Two-stage stochastic variational inequalities: an ERM-solution procedure

  • Xiaojun ChenEmail author
  • Ting Kei Pong
  • Roger J-B. Wets
Full Length Paper Series B

Abstract

We propose a two-stage stochastic variational inequality model to deal with random variables in variational inequalities, and formulate this model as a two-stage stochastic programming with recourse by using an expected residual minimization solution procedure. The solvability, differentiability and convexity of the two-stage stochastic programming and the convergence of its sample average approximation are established. Examples of this model are given, including the optimality conditions for stochastic programs, a Walras equilibrium problem and Wardrop flow equilibrium. We also formulate stochastic traffic assignments on arcs flow as a two-stage stochastic variational inequality based on Wardrop flow equilibrium and present numerical results of the Douglas–Rachford splitting method for the corresponding two-stage stochastic programming with recourse.

Keywords

Stochastic variational inequalities Stochastic program with recourse Wardrop equilibrium Expected residual minimization Regularized gap function Splitting method 

Mathematics Subject Classification

90C33 90C15 

References

  1. 1.
    Agdeppa, R., Yamashita, N., Fukushima, M.: Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem. Pac. J. Optim. 6, 3–19 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Attouch, H., Wets, R.: Epigraphical processes: laws of large numbers for random lsc functions. Sémin. d’Anal. Convexe 13, 1–29 (1990)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Beckmann, M., McGuire, C., Winsten, C.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)Google Scholar
  5. 5.
    Cruz, J.Y.B., Iusem, A.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46, 247–263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, London (1997)zbMATHGoogle Scholar
  7. 7.
    Brown, D., DeMarzo, X., Eaves, B.C.: Computing equilibria when asset markets are incomplete. Econometrica 64, 1–27 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30, 1022–1038 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, X., Sun, H., Wets, R.: Regularized mathematical programs with stochastic equilibrium constraints: estimating structural demand models. SIAM J. Optim. 25, 53–75 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, X., Wets, R., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing. SIAM J. Optim. 22, 649–673 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X., Ye, Y.: On homotopy-smoothing methods for box constrained variational inequalities. SIAM J. Control Optim. 37, 589–616 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117, 51–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cominetti, R.: Equilibrium routing under uncertainty. Math. Program. 151, 117–151 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Correa, J.R., Stier-Moses, N.E.: Wardrop Equilibria. Wiley Online Library (2011)Google Scholar
  15. 15.
    Dang, C., Ye, Y., Zhu, Z.: An interior-point path-following algorithm for computing a Leontief economy equilibrium. Computat. Optim. Appl. 50, 213–236 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Demarzo, P., Eaves, B.C.: Computing equilibria of GEI by relocalization on a grassmann manifold. J. Math. Econ. 26, 479–497 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Deride, J., Jofré, A., Wets, R.: Solving deterministic and stochastic equilibrium problems via augmented Walrasian. Comput. Econ. (2017)Google Scholar
  18. 18.
    Dirkse, S., Ferris, M., Munson, T.: http://pages.cs.wisc.edu/ferris/path.html, 2015—the PATH solver. Technical report, University of Wisconsin (2015)
  19. 19.
    Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eckstein, J., Bertsekas, D.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fachinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)Google Scholar
  22. 22.
    Fang, H., Chen, X., Fukushima, M.: Stochastic R\(_0\) matrix linear complementarity problems. SIAM J. Optim. 18, 482–506 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ferris, M., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ferris, M., Wets, R.: MOPEC: a computationally amiable formulation of multi-agent optimization problems with global equilibrium constraints. Department of Mathematics, University of California-Davis 63, 309–345 (2017)Google Scholar
  25. 25.
    Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gürkan, G., Özge, A., Robinson, S.: Sample-path solution of stochastic variational inequalities with applications to option pricing. In: Charnes, J., Morrice, D., Brunner, D., Swain, J. (eds) Proceedings of the 1966 Winter Simulation Conference, pp. 337–344. INFORMS (1966)Google Scholar
  28. 28.
    Gürkan, G., Özge, A., Robinson, S.: Sample-path solution of stochastic variational inequalities. Math. Program. 84, 313–333 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    He, B., Yuan, X.: On the O\((1/n)\) convergence rate of the Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. 50, 700–709 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Iusem, A., Jofré, A., Thompson, P.: Approximate projection methods for monotone stochastic variational inequalities. Technical report IMPA, Rio de Janeiro (2015)Google Scholar
  31. 31.
    Iusem, A., Svaiter, B.: A variant of Kopelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Jiang, H., Xu, H.: Stochastic approximation approaches to stochastic variational inequalities. IEEE Trans. Autom. Control 53, 1462–1475 (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Jofré, A., Rockafellar, R.T., Wets, R.: Variational inequalities and economic equilibrium. Math. Oper. Res. 32, 32–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jofré, A., Rockafellar, R.T., Wets, R.: General economic equilibirum with financial market and retainability. Economic Theory 63, 309–345 (2017)Google Scholar
  35. 35.
    Juditsky, A., Nemirovski, A., Tauvel, C.: Solving variational inequalities with stochastic mirror-prox algorithm. Stoch. Syst. 1, 17–58 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kall, P., Wallace, S.: Stochastic Programming. Wiley, London (1995)zbMATHGoogle Scholar
  37. 37.
    Korf, L., Wets, R.: Random lsc functions: an ergodic theorem. Math. Oper. Res. 26, 421–445 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lamm, M., Lu, S., Budhiraja, A: Individual confidence intervals for solutions to expected value formulations of stochastic variational inequalities. Math. Program. doi: 10.1007/s10107-016-1046-y
  39. 39.
    Li, G., Pong, T.K.: Douglas–Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Math. Program. 159, 371–401 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lu, S.: Confidence regions for stochastic variational inequalities. Optimization 63, 1431–1443 (2012)CrossRefzbMATHGoogle Scholar
  41. 41.
    Lu, S.: Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities. SIAM J. Optim. 24, 1458–1484 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Luo, M., Lin, G.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140, 103–116 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Luo, Z., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  44. 44.
    Marcotte, P. Patriksson, P.: Transportation, Handbooks in Operations Research and Management Science, vol. 14 of Handbooks in Operations Research and Management Science, Chapter Traffic Equilibrium, pp. 623–713. Elsevier (2007)Google Scholar
  45. 45.
    Norkin, V., Wets, R.: On strong graphical law of large numbers for random semicontinuous mappings. Vestn. St.-Petersbg. Univ. 10, 102–110 (2013)Google Scholar
  46. 46.
    Norkin, V., Wets, R., Xu, H.: Graphical Convergence of Sample Average Random Set-Valued Mappings. Mathematics, University of California, Davis (2010)Google Scholar
  47. 47.
    Pang, J.-S., Su, C.-L., Lee, Y.: A constructive approach to estimating pure characteristics demand models with pricing. Oper. Res. 63, 639–659 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30, 245–256 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pennanen, T., Koivu, M.: Epi-convergent discretization of stochastic programs via integration quadratures. Numer. Math. 100, 141–163 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Philpott, A., Ferris, M., Wets, R.: Equilibrium, uncertainty and risk in hydro-thermal electricity systems. Math. Program. 157, 483–513 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar
  52. 52.
    Rockafellar, R.T., Wets, R.: Stochastic convex programming: Kuhn–Tucker conditions. J. Math. Econ. 2, 349–370 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Rockafellar, R.T., Wets, R.: Nonanticipativity and \({\cal{L}}^1\)-martingales in stochastic optimization problems. Math. Program. Study 6, 170–187 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Rockafellar, R.T., Wets, R.: Stochastic convex programming: basic duality. Pac. J. Math. 62, 173–195 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Rockafellar, R.T., Wets, R.: Stochastic convex programming: relatively complete recourse and induced feasibility. SIAM J. Control Optim. 14, 574–589 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Rockafellar, R.T., Wets, R.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119–147 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Rockafellar, R.T., Wets, R.: A dual strategy for the implementation of the aggregation principle in decision making under uncertainty. Appl. Stoch. Models Data Anal. 8, 245–255 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  59. 59.
    Rockafellar, R.T., Wets, R.: Stochastic variational inequalities: single-stage to multistage. Math. Program. doi: 10.1007/s10107-016-0995-5
  60. 60.
    Shanbhag, U.V., Infanger, G., Glynn, P.: A complementarity framework for forward contracting under uncertainty. Oper. Res. 59, 810–834 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Shapiro, A.: Lectures on Stochastic Programming, Chapter 5: Statistical Inference, pp. 155–252. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  62. 62.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming-Modeling and Theory. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  63. 63.
    Shapiro, A., Xu, H.: Uniform laws of large numbers for set-valued mappings and subdifferentials of random functions. J. Math. Anal. Appl. 325, 1390–1399 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Sun, D., Toh, K.-C., Yang, L.: A convergent 3-block semi-proximal alternating irection method of multipliers for conic programming with 4-type of constraints. SIAM J. Optim. 25(2), 882–915 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Van Slyke, R., Wets, R.: L-shaped linear programs with application to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Wang, M., Bertsekas, D.: Incremental constraint projection methods for variational inequalities. Math. Program. 150, 321–363 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Xie, Y., Shanbhag, U.V.: On robust solutions to uncertain linear complementarity problems and their variants. SIAM J. Optim. 26(4), 2120–2159 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Yang, H.: Sensitivity analysis for the elastic-demand network equilibrium problem with applications. Transp. Res. B 31, 55–70 (1997)CrossRefGoogle Scholar
  69. 69.
    Yin, Y., Madanat, S., Lu, X.-Y.: Robust improvement schemes for road networks under demand uncertainty. Eur. J. Oper. Res. 198, 470–479 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Zhang, C., Chen, X., Sumalee, A.: Robust Wardrop’s user equilibrium assignment under stochastic demand and supply: expected residual minimization approach. Transp. Res. B 45, 534–552 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  • Xiaojun Chen
    • 1
    Email author
  • Ting Kei Pong
    • 1
  • Roger J-B. Wets
    • 2
  1. 1.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  2. 2.Department of MathematicsUniversity of California, DavisDavisUSA

Personalised recommendations