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No Curse of Dimensionality for Contraction Fixed Points Even in the Worst Case

Author

Listed:
  • John Rust

    (Department of Economics Yale University)

  • Joseph Traub

    (Computer Science, Columbia University)

  • Henryk Wozniakowski

    (Computer Science, Columbia University)

Abstract

We consider the problem of computing approximations to fixed points of quasilinear contraction mappings defined on the space of continuous functions of $d$ variables. Our main emphasis is on large d. Examples of such mappings include the Bellman operator from the theory of dynamic programming. This paper proves that there exist deterministic algorithms for computing approximations to fixed points for some classes of quasilinear contraction mappings which are strongly tractable, i.e., in the worst case the number of function evaluations needed to compute an e-approximation to the solution at any finite number of points in its domain is bounded by C/e^p where both C and p are independent of d. This is done by using relations between the quasilinear contraction problem and the conditional expectation and approximation problems. The conditional expectation problem is equivalent to weighted multivariate integration. This allows us to apply recent proof technique and results on the strong tractability of weighted multivariate integration and approximation to establish strong tractability for the quasilinear fixed point problem. In particular, this holds when the fixed points belong to a Sobolev space for a specific weighted norm.

Suggested Citation

  • John Rust & Joseph Traub & Henryk Wozniakowski, 1999. "No Curse of Dimensionality for Contraction Fixed Points Even in the Worst Case," Computational Economics 9902001, University Library of Munich, Germany.
  • Handle: RePEc:wpa:wuwpco:9902001
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    References listed on IDEAS

    as
    1. Rust, John, 1985. "Stationary Equilibrium in a Market for Durable Assets," Econometrica, Econometric Society, vol. 53(4), pages 783-805, July.
    2. Tauchen, George & Hussey, Robert, 1991. "Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models," Econometrica, Econometric Society, vol. 59(2), pages 371-396, March.
    3. Spassimir H. Paskov & Joseph F. Traub, 1995. "Faster Valuation of Financial Derivatives," Working Papers 95-03-034, Santa Fe Institute.
    4. John Rust, 1997. "Using Randomization to Break the Curse of Dimensionality," Econometrica, Econometric Society, vol. 65(3), pages 487-516, May.
    5. John Rust, 1997. "A Comparison of Policy Iteration Methods for Solving Continuous-State, Infinite-Horizon Markovian Decision Problems Using Random, Quasi-random, and Deterministic Discretizations," Computational Economics 9704001, University Library of Munich, Germany.
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    Cited by:

    1. Hall, George & Rust, John, 2000. "An empirical model of inventory investment by durable commodity intermediaries," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 52(1), pages 171-214, June.

    More about this item

    JEL classification:

    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs

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