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Multivariate approximation by polynomial and generalized rational functions.

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Titel: Multivariate approximation by polynomial and generalized rational functions.
Autoren: Díaz Millán, R., Peiris, V., Sukhorukova, N., Ugon, J.
Quelle: Optimization; Apr2022, Vol. 71 Issue 4, p1171-1187, 17p
Schlagwörter: POLYNOMIAL approximation, CHEBYSHEV approximation, LINEAR programming, POLYNOMIAL time algorithms
Abstract: In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. [ABSTRACT FROM AUTHOR]
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  Data: Multivariate approximation by polynomial and generalized rational functions.
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  Data: Optimization; Apr2022, Vol. 71 Issue 4, p1171-1187, 17p
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  Data: <searchLink fieldCode="DE" term="%22POLYNOMIAL+approximation%22">POLYNOMIAL approximation</searchLink><br /><searchLink fieldCode="DE" term="%22CHEBYSHEV+approximation%22">CHEBYSHEV approximation</searchLink><br /><searchLink fieldCode="DE" term="%22LINEAR+programming%22">LINEAR programming</searchLink><br /><searchLink fieldCode="DE" term="%22POLYNOMIAL+time+algorithms%22">POLYNOMIAL time algorithms</searchLink>
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  Data: In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of Optimization is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/02331934.2022.2044478
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      – Code: eng
        Text: English
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        PageCount: 17
        StartPage: 1171
    Subjects:
      – SubjectFull: POLYNOMIAL approximation
        Type: general
      – SubjectFull: CHEBYSHEV approximation
        Type: general
      – SubjectFull: LINEAR programming
        Type: general
      – SubjectFull: POLYNOMIAL time algorithms
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      – TitleFull: Multivariate approximation by polynomial and generalized rational functions.
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              Text: Apr2022
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