New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

Guoyin Li University of New South Wales B.S. Mordukhovich Wayne States University T.S. Pham University of Dalat

Numerical Analysis and Scientific Computing Optimization and Control mathscidoc:1904.25002

Mathematical Programming, 153, (2), 333–362, 2015
In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.
error bound, polynomial systems, holderian stability, spectral theory of tensors
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@inproceedings{guoyin2015new,
  title={New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors},
  author={Guoyin Li, B.S. Mordukhovich, and T.S. Pham},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190414192326197977236},
  booktitle={Mathematical Programming},
  volume={153},
  number={2},
  pages={333–362},
  year={2015},
}
Guoyin Li, B.S. Mordukhovich, and T.S. Pham. New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors. 2015. Vol. 153. In Mathematical Programming. pp.333–362. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190414192326197977236.
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