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Error bounds for analytic systems and their applications

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Abstract

Using a 1958 result of Lojasiewicz, we establish an error bound for analytic systems consisting of equalities and inequalities defined by real analytic functions. In particular, we show that over any bounded region, the distance from any vectorx in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated atx. For quadratic systems satisfying certain nonnegativity assumptions, we show that this exponent is equal to 1/2. We apply the error bounds to the Karush—Kuhn—Tucker system of a variational inequality, the affine variational inequality, the linear and nonlinear complementarity problem, and the 0–1 integer feasibility problem, and obtain new error bound results for these problems. The latter results extend previous work for polynomial systems and explain why a certain square-root term is needed in an error bound for the (monotone) linear complementarity problem.

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Authors and Affiliations

  1. Communications Research Laboratory, McMaster University, Room 225, L8S 4K1, Hamilton, Ontario, Canada

    Zhi-Quan Luo

  2. Department of Mathematical Sciences, The Johns Hopkins University, 21218, Baltimore, Maryland, USA

    Jong-Shi Pang

Authors
  1. Zhi-Quan Luo
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  2. Jong-Shi Pang
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Additional information

The research of this author is based on work supported by the Natural Sciences and Engineering Research Council of Canada under grant OPG0090391.

The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and by the Office of Naval Research under grant 4116687-01.

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Luo, ZQ., Pang, JS. Error bounds for analytic systems and their applications. Mathematical Programming 67, 1–28 (1994). https://doi.org/10.1007/BF01582210

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  • DOI: https://doi.org/10.1007/BF01582210

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