Skip to main content

Advertisement

SpringerLink
Log in

An approximation scheme for a class of risk-averse stochastic equilibrium problems

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a single-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions of all the agents. An important characteristic of our approach is that the agents hedge risk in the objective functions (on costs or profits) of their optimization problems, which has a clear economic interpretation. This feature is obviously desirable, but in the risk-averse case it leads to variational inequalities with set-valued operators—a class of problems for which no established software is currently available. To overcome this difficulty, we define a sequence of approximating differentiable variational inequalities based on smoothing the nonsmooth risk measure in the agents’ problems, such as average or conditional value-at-risk. The smoothed variational inequalities can be tackled by the PATH solver, for example. The approximation scheme is shown to converge, including the case when smoothed problems are solved approximately. An interesting by-product of our proposal is that the smoothing approach allows us to show existence of an equilibrium for the original problem. To assess the proposed technique, numerical results are presented. The first set of experiments is on randomly generated equilibrium problems, for which we compare the proposed methodology with the standard smooth reformulation of average value-at-risk minimization (using additional variables to rewrite the associated max-function). The second set of experiments deals with a part of the real-life European gas network, for which Dantzig–Wolfe decomposition can be combined with the smoothing approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Baldick, R., Helman, U., Hobbs, B., O’Neill, R.: Design of efficient generation markets. Proc. IEEE 93(11), 1998–2012 (2005). doi:10.1109/JPROC.2005.857484

    Article  Google Scholar 

  2. Burke, J., Hoheisel, T.: Epi-convergent smoothing with applications to convex composite functions. SIAM J. Optim. 23, 1457–1479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C., Mangasarian, O.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, X.: Smoothing methods for nonsmooth, novonvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MathSciNet  Google Scholar 

  5. Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022–1038 (2005). doi:10.1287/moor.1050.0160

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X., Wets, R.J.B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649–673 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117(1–2), 51–80 (2009). doi:10.1007/s10107-007-0163-z

    Article  MathSciNet  MATH  Google Scholar 

  8. Collado, R.A., Powell, W.B.: Threshold risk measures part 1: finite horizon http://castlelab.princeton.edu/Papers/Collado-Threshold-Measures-FINITE_ONLY-April-22-2013.pdf (2013)

  9. Conejo, A., Nogales, F., Arroyo, J., Garcia-Bertrand, R.: Risk-constrained self-scheduling of a thermal power producer. IEEE Trans. Power Syst. 19(3), 1569–1574 (2004). doi:10.1109/TPWRS.2004.831652

    Article  Google Scholar 

  10. Conejo, A.J., Carrión, M., Morales, J.M.: Decision Making Under Uncertainty in Electricity Markets. International Series in Operations Research & Management Science. Springer, New York (2010)

    Book  MATH  Google Scholar 

  11. David, A., Wen, F.: Strategic bidding in competitive electricity markets: a literature survey. In: Power Engineering Society Summer Meeting, 2000. IEEE, vol. 4, pp. 2168–2173 (2000). doi:10.1109/PESS.2000.866982

  12. Dentcheva, D., Ruszczyński, A., Shapiro, A.: Lectures on Stochastic Programming. SIAM, Philadelphia (2009)

    MATH  Google Scholar 

  13. Dirkse, S., Ferris, M.: The PATH solver : a nonmonotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)

    Article  Google Scholar 

  14. Egging, R.: Benders decomposition for multi-stage stochastic mixed complementarity problems applied to a global natural gas market model. Eur. J. Oper. Res. 226(2), 341–353 (2013). doi:10.1016/j.ejor.2012.11.024

    Article  MathSciNet  MATH  Google Scholar 

  15. Egging, R.G., Gabriel, S.A.: Examining market power in the European natural gas market. Energy Policy 34(17), 2762–2778 (2006). doi:10.1016/j.enpol.2005.04.018

    Article  Google Scholar 

  16. Ehrenmann, A., Neuhoff, K.: A comparison of electricity market designs in networks. Oper. Res. 57(2), 274–286 (2009)

    Article  MATH  Google Scholar 

  17. Ehrenmann, A., Smeers, Y.: Generation capacity expansion in risky environment: a stochastic equilibrium analysis. Oper. Res. 59(6), 1332–1346 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Facchinei, F., Kanzow, C.: Generalized nash equilibrium problems. Ann. OR 175(1), 177–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. II. Springer, New York (2003)

    MATH  Google Scholar 

  21. Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12, 207–227 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fuller, J.D., Chung, W.: Dantzig–Wolfe decomposition of variational inequalities. Comput. Econ. 25, 303–326 (2005). doi:10.1007/s10614-005-2519-x

    Article  MATH  Google Scholar 

  23. Gabriel, S.A., Zhuang, J., Egging, R.: Solving stochastic complementarity problems in energy market modeling using scenario reduction. Eur. J. Oper. Res. 197(3), 1028–1040 (2009). doi:10.1016/j.ejor.2007.12.046

    Article  MathSciNet  MATH  Google Scholar 

  24. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. No. 305–306 in Grund. der math. Wiss. Springer, Berlin (1993). (two volumes)

  25. Hobbs, B., Metzler, C., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15(2), 638–645 (2000). doi:10.1109/59.867153

    Article  Google Scholar 

  26. Hu, X., Ralph, D.: Using EPECs to model bilevel games in restructured electricity markets with locational prices. Oper. Res. 55(5), 809–827 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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). doi:10.1080/10556788.2012.676756

    Article  MathSciNet  MATH  Google Scholar 

  28. Kulkarni, A., Shanbhag, U.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45–55 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kulkarni, A.A., Shanbhag, U.V.: Revisiting generalized Nash games and variational inequalities. J. Optim. Theory Appl. 154, 175–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lin, G.H., Chen, X., Fukushima, M.: Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Program. 116(1–2), 343–368 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Luna, J.P., Sagastizábal, C., Solodov, M.: A class of Dantzig–Wolfe type decomposition methods for variational inequality problems. Math. Program. 143, 177–209 (2014). doi:10.1007/s10107-012-0599-7

    Article  MathSciNet  MATH  Google Scholar 

  32. Luna, J.P., Sagastizábal, C., Solodov, M.: Complementarity and game-theoretical models for equilibria in energy markets: deterministic and risk-averse formulations. In: Kovacevic, R., Pflug, G., Vespucci, M. (eds.) International Series in Operations Research and Management Science, vol. 199, Chap. 10, pp. 237–264. Springer, Berlin (2014)

    Google Scholar 

  33. Meng, F., Sun, J., Goh, M.: A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure. Comput. Optim. Appl. 50(2), 379–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pang, J.S.: Three modeling paradigms in mathematical programming. Math. Program. 125(2), 297–323 (2010). doi:10.1007/s10107-010-0395-1

    Article  MathSciNet  MATH  Google Scholar 

  35. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21–56 (2005). doi:10.1007/s10287-004-0010-0

    Article  MathSciNet  MATH  Google Scholar 

  36. Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 6(3), 373–375 (2009). doi:10.1007/s10287-009-0093-8

    Article  MathSciNet  MATH  Google Scholar 

  37. Patriksson, M., Wynter, L.: Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25(4), 159–167 (1999). doi:10.1016/S0167-6377(99)00052-8

    Article  MathSciNet  MATH  Google Scholar 

  38. Ralph, D., Smeers, Y.: EPECs as models for electricity markets. In: Power Systems Conference and Exposition, 2006. PSCE ’06. 2006 IEEE PES, pp. 74–80 (2006). doi:10.1109/PSCE.2006.296252

  39. Ralph, D., Smeers, Y.: Pricing risk under risk measures: an introduction to stochastic-endogenous equilibria. Social Sci. Res. Netw. http://ssrn.com/abstract=1903897 (2011)

  40. Ralph, D., Xu, H.: Implicit smoothing and its application to optimization with piecewise smooth equality constraints1. J. Optim. Theory Appl. 124(3), 673–699 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21(3), 1168–1199 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rockafellar, R., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  43. Rockafellar, R., Wets, J.B.: Variational Analysis. Springer, New York (1997)

    MATH  Google Scholar 

  44. Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 128(1), 223243 (2006). doi:10.1007/s10957-005-7566-x

    Article  MathSciNet  Google Scholar 

  45. Wets, R.J.B.: Stochastic programming models: wait-and-see versus here-and-now. In: Greengard, C., Ruszczynski, A. (eds.) Decision Making Under Uncertainty, The IMA Volumes in Mathematics and Its Applications, vol. 128, pp. 1–15. Springer, New York (2002). doi:10.1007/978-1-4684-9256-9_1

    Chapter  Google Scholar 

  46. Xu, H., Zhang, D.: Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications. Math. Program. 119(2), 371–401 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, H., Zhang, D.: Stochastic Nash equilibrium problems: sample average approximation and applications. Comput. Optim. Appl. 55(3), 597–645 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Yin, H., Shanbhag, U., Mehta, P.: Nash equilibrium problems with scaled congestion costs and shared constraints. IEEE Trans. Autom. Control 56(7), 1702–1708 (2011). doi:10.1109/TAC.2011.2137590

    Article  MathSciNet  Google Scholar 

  49. Zhao, J., Hobbs, B.F., Pang, J.S.: Long-run equilibrium modeling of emissions allowance allocation systems in electric power markets. Oper. Res. 58(3), 529–548 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors acknowledge insightful and significant comments and remarks from the reviewers and the editors, which led to a substantially improved version of this work. The authors also thank Alexey Izmailov on the issue of computing the second derivatives described in Proposition 2.

Author information

Author notes

  1. Juan Pablo Luna

    Present address: COPPE-UFRJ, Engenharia de Produção, Cidade Universitária, Rio de Janeiro, RJ, 21941-972, Brazil

Authors and Affiliations

  1. IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, 22460-320, Brazil

    Juan Pablo Luna

  2. IMPA, Rio de Janeiro, RJ, Brazil

    Claudia Sagastizábal & Mikhail Solodov

Authors
  1. Juan Pablo Luna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Claudia Sagastizábal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikhail Solodov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Pablo Luna.

Additional information

Juan Pablo Luna was supported by CNPq post-doctoral scholarship 503441/2012-0. Claudia Sagastizábal is partially supported by Grants CNPq 303840/2011-0, AFOSR FA9950-11-1-0139, as well as by PRONEX-Optimization and FAPERJ. Mikhail Solodov is supported in part by CNPq Grants 302637/2011-7 and PVE 401119/2014-9, by PRONEX-Optimization and by FAPERJ.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luna, J.P., Sagastizábal, C. & Solodov, M. An approximation scheme for a class of risk-averse stochastic equilibrium problems. Math. Program. 157, 451–481 (2016). https://doi.org/10.1007/s10107-016-0988-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-016-0988-4

Keywords

Mathematics Subject Classification

Search

Navigation