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Subproblem Approximation in Dantzig-Wolfe Decomposition of Variational Inequality Models with an Application to a Multicommodity Economic Equilibrium Model

Author

Listed:
  • William Chung

    (Department of Management Sciences, City University of Hong Kong, Hong Kong, China)

  • J. David Fuller

    (Department of Management Sciences, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

Abstract

We present a modification to Dantzig-Wolfe decomposition of variational inequality (VI) problems that allows for approximation of the VI mapping in the subproblem. The approximation is parameterized by the most recent master problem solution, and it must satisfy two simple requirements. In an electronic companion (online appendix), we show that the proofs of convergence and other important properties go through with subproblem approximation. The approximation procedure is illustrated by an application to a class of multicommodity economic equilibrium models (MCEEMs): the standard Dantzig-Wolfe decomposition by commodity does not allow the subproblem to be decomposed into separate subproblems for each commodity, but we show two ways to approximate the subproblem's inverse demand function, and both ways allow the subproblem to be broken into separate single-commodity problems. A further approximation is combined with each of the inverse demand approximations; in effect, an approximate supply or demand curve is introduced into each commodity's subproblem for transfers of commodities between different subproblems, thus allowing the subproblems to produce better proposals. An illustration is included for an MCEEM that represents energy markets in Canada.

Suggested Citation

  • William Chung & J. David Fuller, 2010. "Subproblem Approximation in Dantzig-Wolfe Decomposition of Variational Inequality Models with an Application to a Multicommodity Economic Equilibrium Model," Operations Research, INFORMS, vol. 58(5), pages 1318-1327, October.
    • Handle: RePEc:inm:oropre:v:58:y:2010:i:5:p:1318-1327
    DOI: 10.1287/opre.1090.0803
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    References listed on IDEAS

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    1. Frederic H. Murphy & Murthy V. Mudrageda, 1998. "A Decomposition Approach for a Class of Economic Equilibrium Models," Operations Research, INFORMS, vol. 46(3), pages 368-377, June.
    2. William Chung & J. D. Fuller & Y. June Wu, 2006. "A New Decomposition Method for Multiregional Economic Equilibrium Models," Operations Research, INFORMS, vol. 54(4), pages 643-655, August.
    3. Wu, Y.June & Chung, William, 1997. "Assessing the control of energy-related CO2 emissions with a dynamic energy process model," Energy, Elsevier, vol. 22(7), pages 693-704.
    4. Byong-Hun Ahn & William W. Hogan, 1982. "On Convergence of the PIES Algorithm for Computing Equilibria," Operations Research, INFORMS, vol. 30(2), pages 281-300, April.
    5. Frederic H. Murphy, 1973. "Column Dropping Procedures for the Generalized Programming Algorithm," Management Science, INFORMS, vol. 19(11), pages 1310-1321, July.
    6. Frederic H. Murphy & John J. Conti & Susan H. Shaw & Reginald Sanders, 1988. "Modeling and Forecasting Energy Markets with the Intermediate Future Forecasting System," Operations Research, INFORMS, vol. 36(3), pages 406-420, June.
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    Cited by:

    1. William Chung, 2024. "Truncated Dantzig–Wolfe Decomposition for a Class of Constrained Variational Inequality Problems," Computational Economics, Springer;Society for Computational Economics, vol. 64(1), pages 81-104, July.
    2. Juan Pablo Luna & Claudia Sagastizábal & Mikhail Solodov, 2020. "A class of Benders decomposition methods for variational inequalities," Computational Optimization and Applications, Springer, vol. 76(3), pages 935-959, July.
    3. Pedro Borges & Claudia Sagastizábal & Mikhail Solodov, 2021. "Decomposition Algorithms for Some Deterministic and Two-Stage Stochastic Single-Leader Multi-Follower Games," Computational Optimization and Applications, Springer, vol. 78(3), pages 675-704, April.

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