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Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?

Author

Listed:
  • J. Rust

    () (Department of Economics, University of Maryland, USA)

  • J. F. Traub

    () (Department of Computer Science, Columbia University, New York, USA)

  • H. Wozniakowski

    () (Department of Computer Science, Columbia University, New York, USA and Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland)

Abstract

This paper analyzes the complexity of the "contraction fixed point problem": compute an epsilon-approximation to the fixed point "V"*Gamma("V"*) of a contraction mapping Gamma that maps a Banach space "B-sub-d" of continuous functions of "d" variables into itself. We focus on "quasi linear contractions" where Gamma is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Gamma, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an epsilon-approximation to a fixed point "V"* is an element of "B-sub-d" increases exponentially in "d". We show that the curse of dimensionality disappears if the domain of Gamma has additional special structure. We identify a particular type of special structure for which the problem is "strongly tractable" even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an epsilon-approximation of "V"* is bounded by "C"epsilon-super- - "p" where "C" and "p" are constants independent of "d". We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent "p"1 is nearly achieved. Copyright The Econometric Society 2002.

Suggested Citation

  • J. Rust & J. F. Traub & H. Wozniakowski, 2002. "Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?," Econometrica, Econometric Society, vol. 70(1), pages 285-329, January.
  • Handle: RePEc:ecm:emetrp:v:70:y:2002:i:1:p:285-329
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    Cited by:

    1. Alexei Onatski & Noah Williams, 2003. "Modeling Model Uncertainty," Journal of the European Economic Association, MIT Press, vol. 1(5), pages 1087-1122, September.
    2. George Hall and John Rust, Yale University, 2001. "Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market," Computing in Economics and Finance 2001 274, Society for Computational Economics.
    3. John Rust, 2014. "The Limits of Inference with Theory: A Review of Wolpin (2013)," Journal of Economic Literature, American Economic Association, vol. 52(3), pages 820-850, September.
    4. Aguirregabiria, Victor, 2005. "Nonparametric identification of behavioral responses to counterfactual policy interventions in dynamic discrete decision processes," Economics Letters, Elsevier, vol. 87(3), pages 393-398, June.
    5. Joao Macieira, 2010. "Oblivious Equilibrium in Dynamic Discrete Games," 2010 Meeting Papers 680, Society for Economic Dynamics.
    6. Yongyang Cai & Kenneth Judd, 2015. "Dynamic programming with Hermite approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 245-267, June.
    7. Yongyang Cai & Kenneth Judd & Greg Thain & Stephen Wright, 2015. "Solving Dynamic Programming Problems on a Computational Grid," Computational Economics, Springer;Society for Computational Economics, vol. 45(2), pages 261-284, February.
    8. Manuel Santos & John Rust, "undated". "Convergence Properties of Policy Iteration," Working Papers 2133377, Department of Economics, W. P. Carey School of Business, Arizona State University.

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