Complementary Problems with Polynomial Data

Abstract

Given polynomial maps \(f, g \colon \mathbb {R}^{n} \to \mathbb {R}^{n}\), we consider the polynomial complementary problem of finding a vector \(x \in \mathbb {R}^{n}\) such that

$$ f(x) \ge 0, \quad g(x) \ge 0, \quad \text{ and } \quad \langle f(x), g(x) \rangle = 0. $$

In this paper, we present various properties on the solution set of the problem, including genericity, nonemptiness, compactness, uniqueness as well as error bounds with explicit exponents. These strengthen and generalize some previously known results.

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Notes

  1. 1.

    We would like to thank Hongjin He for showing us this reference.

  2. 2.

    A multifunction is polyhedral if its graph is the union of finitely many polyhedral convex sets.

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Acknowledgements

The first author is partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), grant 101.04-2019.302

The authors are grateful to three anonymous referees for careful reading of the original submission and for helpful suggestions and kind remarks.

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Correspondence to Tien-Son Pham.

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Dedicated to Professor Boris Mordukhovich on the occasion of his 70th birthday.

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Pham, TS., Nguyen, C.H. Complementary Problems with Polynomial Data. Vietnam J. Math. (2021). https://doi.org/10.1007/s10013-020-00467-3

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Keywords

  • Polynomial complementarity problem
  • Existence
  • Boundedness
  • Uniqueness
  • Error bound
  • Genericity

Mathematics Subject Classification (2010)

  • 90C33