Mathematical Programming

, Volume 153, Issue 2, pp 333–362 | Cite as

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

Full Length Paper Series A

Abstract

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.

Keywords

Error bounds Polynomials Variational analysis   Generalized differentiation Łojasiewicz’s inequality  Hölderian stability Polynomial optimization and complementarity 

Mathematics Subject Classification

90C26 90C31 49J52 49J53 26D10 

Notes

Acknowledgments

The authors are gratefully indebted to the referees and the handling Associate Editor for their helpful remarks, which allowed us to significantly improved the original presentation.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  4. 4.Center of Research and DevelopmentDuy Tan UniversityDanangVietnam
  5. 5.Department of MathematicsUniversity of DalatDalatVietnam

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