Error bounds for analytic systems and their applications

Zhi Quan Luo, Jong Shi Pang

Research output: Contribution to journalArticlepeer-review

51 Scopus citations

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 vector x in the region to the solution set of an analytic system is bounded by a residual function, raised to a certain power, evaluated at x. 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.

Original languageEnglish (US)
Pages (from-to)1-28
Number of pages28
JournalMathematical Programming
Volume67
Issue number1
DOIs
StatePublished - Oct 1994

Keywords

  • Affine variational inequality
  • Analytic systems
  • Complementarity problem
  • Error bound
  • Integer feasibility problem
  • Karush-Kuhn-Tucker conditions

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