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A class of Dantzig–Wolfe type decomposition methods for variational inequality problems

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Abstract

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems may be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretic models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise intractable if the widely used PATH solver is applied directly, without decomposition.

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References

  1. Ben Amor, H., Desrosiers, J., Frangioni, A.: On the choice of explicit stabilizing terms in column generation. Discret. Appl. Math. 157, 1167–1184 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  3. Briant, O., Lemaréchal, C., Meurdesoif, P., Michel, S., Perrot, N., Vanderbeck, F.: Comparison of bundle and classical column generation. Math. Program. 113, 299–344 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, G., Teboulle, M.: A proximal-based decomposition method for convex minimization problems. Math. Program. 64, 81–101 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chung, W., Fuller, J.D.: Subproblem approximation in Dantzig–Wolfe decomposition of variational inequality models with an application to a multicommodity economic equilibrium model. Oper. Res. 58, 1318–1327 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Inc., Boston (1992)

    MATH  Google Scholar 

  7. Crainic, T.G., Frangioni, A., Gendron, B.: Bundle-based relaxation methods for multi-commodity capacitated fixed charge metwork design problems. Discret. Appl. Math. 112, 73–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dantzig, G.B., Wolfe, P.: The decomposition algorithm for linear programs. Econometric 29, 767–778 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dirkse, S.P., Ferris, M.C.: The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)

    Article  Google Scholar 

  10. Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Eckstein, J.: Some saddle-function splitting methods for convex programming. Optim. Methods Softw. 4, 75–83 (1994)

    Article  MathSciNet  Google Scholar 

  12. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Fuller, J.D., Chung, W.: Dantzig–Wolfe decomposition of variational inequalities. Comput. Econ. 25, 303–326 (2005)

    Article  MATH  Google Scholar 

  17. He, B., Liao, L.Z., Han, D., Yang, H.: A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92, 103–118 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hobbs, B.F., Pang, J.S.: Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints. Oper. Res. 55(1), 113–127 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jones, K.L., Lustig, I.J., Farwolden, J.M., Powell, W.B.: Multicommodity network flows: the impact of formulation on decomposition. Math. Program. 62, 95–117 (1993)

    Article  MATH  Google Scholar 

  20. Josephy, N.H.: Newton’s Method for Generalized Equations. Technical Summary Report no. 1965. Mathematics Research Center, University of Wisconsin, Madison (1979)

  21. Kannan, A., Zavala, V.M.: A Game-Theoretical Dynamic Model for Electricity Markets. Argonne National Aboratory, Technical, Report ANL/MCS-P1792-1010 (2011)

  22. Kannan, A., Shanbhag, U.V., Kim, H.M.: Addressing supply-side risk in uncertain power markets: stochastic generalized Nash models, scalable algorithms and error analysis. Optim. Methods Softw. www.tandfonline.com, doi:10.1080/10556788.2012.676756

  23. Kannan, A., Shanbhag, U.V., Kim, H.M.: Strategic behavior in power markets under uncertainty. Energy Syst. 2, 115–141 (2011)

    Article  Google Scholar 

  24. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Lecture Notes in Economics and Mathematical Systems, vol. 495. Springer, Berlin (2001)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  27. Lotito, P.A., Parente, L.A., Solodov, M.V.: A class of variable metric decomposition methods for monotone variational inclusions. J. Convex Anal. 16, 857–880 (2009)

    MATH  MathSciNet  Google Scholar 

  28. Luo, Z.-Q., Tseng, P.: Error bound and convergence analysis of matrix splitting algorithm for the affine variational inequality problems. SIAM J. Optim. 2, 43–54 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Martinet, B.: Regularisation d’inequations variationelles par approximations successives. Revue Française d’Informatique et de Recherche Opérationelle 4, 154–159 (1970)

    MATH  MathSciNet  Google Scholar 

  30. Parente, L.A., Lotito, P.A., Solodov, M.V.: A class of inexact variable metric proximal point algorithms. SIAM J. Optim. 19, 240–260 (2008)

    Article  MathSciNet  Google Scholar 

  31. Pennanen, T.: A splitting method for composite mappings. Numer. Funct. Anal. Optim 23, 875–890 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rockafellar, R.T.: Local boundedness of nonlinear, monotone operators. Mich. Math. J. 16, 397–407 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  34. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  35. Shanbhag, U.V., Infanger, G., Glynn, P.W.: A complementarity framework for forward contracting under uncertainty. Oper. Res. 59, 810–834 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Solodov, M.V.: A class of globally convergent algorithms for pseudomonotone variational inequalities. In: Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.) Complementarity: Applications, Algorithms and Extensions, Applied Optimization 50, Chapter 14, pp. 297–315. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  37. Solodov, M.V.: A class of decomposition methods for convex optimization and monotone variational inclusions via the hybrid inexact proximal point framework. Optim. Methods Softw. 19, 557–575 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Solodov, M.V.: Constraint qualifications. In: Cochran, J.J., et al. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)

  39. Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. Solodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34, 1814–1830 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  41. Spingarn, J.E.: Applications of the method of partial inverses to convex programming. Math. Program. 32, 199–223 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  42. Tseng, P.: Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. SIAM J. Control Optim. 29, 119–138 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  43. Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951–965 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  44. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We thank the two referees for their constructive comments which helped us to improve the original version.

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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, Claudia Sagastizábal & Mikhail Solodov

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  1. Juan Pablo Luna
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  2. Claudia Sagastizábal
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  3. Mikhail Solodov
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Correspondence to Mikhail Solodov.

Additional information

The first author is supported by FAPERJ scholarship 100.491/2011. The second author is partially supported by Grants CNPq 303840/2011-0, AFOSR FA9550-08-1-0370, NSF DMS 0707205, as well as by PRONEX-Optimization and FAPERJ. The third author is supported in part by CNPq Grant 302637/2011-7, by PRONEX-Optimization and by FAPERJ. C. Sagastizábal: Visiting researcher at IMPA, on leave from INRIA Rocquencourt, France.

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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). https://doi.org/10.1007/s10107-012-0599-7

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  • DOI: https://doi.org/10.1007/s10107-012-0599-7

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