Abstract
The purpose of this paper is to present two algorithms for global minimization of multivariate polynomials. For a multivariate real polynomial f, we provide an effective algorithm for deciding whether or not the infimum of f is finite. In the case of f having a finite infimum, the infimum of f can be accurately coded as (h; a, b), where h is a real polynomial in one variable, a and b is two rational numbers with a < b, and (h, a, b) stands for the only real root of h in the open interval ]a, b[. Moreover, another algorithm is provided to decide whether or not the infimum of f is attained when the infimum of f is finite. Our methods are called “nonstandard”, because an infinitesimal element is introduced in our arguments.
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Zeng, G., Xiao, S. Global minimization of multivariate polynomials using nonstandard methods. J Glob Optim 53, 391–415 (2012). https://doi.org/10.1007/s10898-011-9718-x
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DOI: https://doi.org/10.1007/s10898-011-9718-x
Keywords
- Polynomial optimization
- Infimum
- Global minimum
- Connected component
- Infinitesimal element
- Transfer principle