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Error Bounds for the Solution Sets of Quadratic Complementarity Problems

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Abstract

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.

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Notes

  1. Actually, an arbitrary system of polynomial equalities and inequalities can be reduced to a system of quadratic polynomial equalities and inequalities via polynomial reduction. This technique will be presented in Sect. 4.

  2. During the second revision, one referee kindly pointed out that an improvement can be given, if instead using Theorem 3.3 in the recent monograph by Hà and Pham [27]. The exponent in (2) can be improved to \(\frac{1}{R(n+r+s-1,d+1)}\)

  3. The first version only gave the proof for \({\mathbf {x}}\in V(\epsilon )\). One referee suggested the current version.

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Acknowledgements

We are grateful for the two referees and the editor for helpful comments and valuable suggestions. In particular, one referee’s suggestion improves Theorem 3.1. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11171328 and 11431002), and Innovation Research Foundation of Tianjin University (Grant Nos. 2017XZC-0084 and 2017XRG-0015).

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Shenglong Hu

  2. College of Science, China Jiliang University, Hangzhou, 310018, China

    Jie Wang

  3. School of Mathematics, Tianjin University, Tianjin, 300350, China

    Zheng-Hai Huang

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  1. Shenglong Hu
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  2. Jie Wang
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  3. Zheng-Hai Huang
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Correspondence to Shenglong Hu.

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Communicated by Guoyin Li.

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Hu, S., Wang, J. & Huang, ZH. Error Bounds for the Solution Sets of Quadratic Complementarity Problems. J Optim Theory Appl 179, 983–1000 (2018). https://doi.org/10.1007/s10957-018-1356-8

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