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Solving Infinite-dimensional Optimization Problems by Polynomial Approximation

  • Olivier Devolder
  • François Glineur
  • Yurii Nesterov
Conference paper

Summary

We solve a class of convex infinite-dimensional optimization problems using a numerical approximation method that does not rely on discretization. Instead, we restrict the decision variable to a sequence of finite-dimensional linear subspaces of the original infinite-dimensional space and solve the corresponding finite-dimensional problems in a efficient way using structured convex optimization techniques.We prove that, under some reasonable assumptions, the sequence of these optimal values converges to the optimal value of the original infinite-dimensional problem and give an explicit description of the corresponding rate of convergence.

Keywords

Linear Subspace Polynomial Approximation Functional Space Regularity Theorem Normed Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R.A. Adams (2003) Sobolev Spaces: 2nd edition. Academic Press.Google Scholar
  2. 2.
    E.J. Anderson and P. Nash (1987) Linear Programming in infinite-dimensional spaces. Wiley.Google Scholar
  3. 3.
    S. Boyd and L. Vandenbergh (2009) Convex Optimization (7th printing with corrections). Cambdrige University Press.Google Scholar
  4. 4.
    E.W. Cheney (1982) Introduction to Approximation Theory (second edition). AMS Chelsea Publishing.Google Scholar
  5. 5.
    Z. Ditzian and V. Totik (1987) Moduli of smoothness. Springer-Verlag.Google Scholar
  6. 6.
    A. Gil, J. Segura and N.M. Temme (2007) Numerical Methods for special functions. SIAM.Google Scholar
  7. 7.
    M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich (2009) Optimization with PDE constraints. Springer-Verlag.Google Scholar
  8. 8.
    D.G. Luenberger (1969) Optimization by vector space methods. Wiley.Google Scholar
  9. 9.
    Y.E. Nesterov and A.S. Nemirovskii (1994) Interior-Point Polynomial Algorithms in Convex Programming. SIAM.Google Scholar
  10. 10.
    Y.E. Nesterov (2000) Squared functional systems and optimization problems. In High performance Optimization, pages 405-440. Kluwer Academic Publishers.Google Scholar
  11. 11.
    A.F. Timan (1963) Theory of approximation of functions of a real variable. Pergamon Press.Google Scholar
  12. 12.
    E. Zeidler (1985) Nonlinear Functional Analysis and its Applications (Part 3): Variational Methods and Optimization. Springer-Verlag.Google Scholar

Copyright information

© Springer -Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Olivier Devolder
    • 1
  • François Glineur
    • 1
  • Yurii Nesterov
    • 1
  1. 1.ICTEAM & IMMAQUniversité catholique de Louvain, CORELouvain-la-NeuveBelgium

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