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Approximation Schemes for Functional Optimization Problems

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  • S. Giulini

    (University of Genoa)

  • M. Sanguineti

    (University of Genoa)

Abstract

Approximation schemes for functional optimization problems with admissible solutions dependent on a large number d of variables are investigated. Suboptimal solutions are considered, expressed as linear combinations of n-tuples from a basis set of simple computational units with adjustable parameters. Different choices of basis sets are compared, which allow one to obtain suboptimal solutions using a number n of basis functions that does not grow “fast” with the number d of variables in the admissible decision functions for a fixed desired accuracy. In these cases, one mitigates the “curse of dimensionality,” which often makes unfeasible traditional linear approximation techniques for functional optimization problems, when admissible solutions depend on a large number d of variables.

Suggested Citation

  • S. Giulini & M. Sanguineti, 2009. "Approximation Schemes for Functional Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 33-54, January.
    • Handle: RePEc:spr:joptap:v:140:y:2009:i:1:d:10.1007_s10957-008-9471-6
    DOI: 10.1007/s10957-008-9471-6
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    References listed on IDEAS

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    1. A. Alessandri & C. Cervellera & M. Sanguineti, 2007. "Functional Optimal Estimation Problems and Their Solution by Nonlinear Approximation Schemes," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 445-466, September.
    2. R. Zoppoli & M. Sanguineti & T. Parisini, 2002. "Approximating Networks and Extended Ritz Method for the Solution of Functional Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 403-440, February.
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    Cited by:

    1. Mauro Gaggero & Giorgio Gnecco & Marcello Sanguineti, 2014. "Approximate dynamic programming for stochastic N-stage optimization with application to optimal consumption under uncertainty," Computational Optimization and Applications, Springer, vol. 58(1), pages 31-85, May.
    2. G. Gnecco & M. Sanguineti, 2010. "Estimates of Variation with Respect to a Set and Applications to Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 53-75, April.
    3. G. Gnecco & M. Sanguineti, 2010. "Suboptimal Solutions to Dynamic Optimization Problems via Approximations of the Policy Functions," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 764-794, September.
    4. Giorgio Gnecco, 2016. "On the Curse of Dimensionality in the Ritz Method," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 488-509, February.

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