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The error and perturbation bounds of the general absolute value equations

Shi-Liang Wu, Cui-Xia Li

TL;DR

This work addresses error and perturbation analysis for two general AVEs, $Ax-B|x|=b$ and $Ax-|Bx|=b$, by introducing a class of absolute value functions to build a matrix-agnostic error-bound framework based on the natural residual $r(x)=Ax-B|x|-b$. It derives upper and lower error bounds in terms of diagonal sign matrices and establishes three practical regimes (Cases I–III) to bound the quantities $\overline{\alpha},\overline{\beta},$ and $\overline{\gamma}$, providing a perturbation bound that reduces to classical linear-system results when the nonlinear term vanishes. The authors apply this AVE framework to the linear complementarity problem via the modulus method, obtaining perturbation bounds for LCP solutions and connecting to existing results in Chen07. Numerical experiments on AVEs derived from LCP demonstrate that the proposed relative perturbation bounds are accurate for small perturbations and remain informative for large-scale problems. Overall, the paper delivers a general, computable theory for error and perturbation analysis of AVEs and shows its practical relevance to LCP perturbations.

Abstract

To our knowledge, the error and perturbation bounds of the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of two types of the general absolute value equations (AVEs): $Ax-B|x|=b$ and $Ax-|Bx|=b$. Some useful error bounds and perturbation bounds of the above two types of absolute value equations are provided. Without limiting the matrix type, some computable estimates for the above upper bounds are given. By applying the absolute value equations, a new approach for some existing perturbation bounds of the linear complementarity problem (LCP) in (SIAM J. Optim., 18 (2007), pp. 1250-1265) is provided. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.

The error and perturbation bounds of the general absolute value equations

TL;DR

This work addresses error and perturbation analysis for two general AVEs, and , by introducing a class of absolute value functions to build a matrix-agnostic error-bound framework based on the natural residual . It derives upper and lower error bounds in terms of diagonal sign matrices and establishes three practical regimes (Cases I–III) to bound the quantities and , providing a perturbation bound that reduces to classical linear-system results when the nonlinear term vanishes. The authors apply this AVE framework to the linear complementarity problem via the modulus method, obtaining perturbation bounds for LCP solutions and connecting to existing results in Chen07. Numerical experiments on AVEs derived from LCP demonstrate that the proposed relative perturbation bounds are accurate for small perturbations and remain informative for large-scale problems. Overall, the paper delivers a general, computable theory for error and perturbation analysis of AVEs and shows its practical relevance to LCP perturbations.

Abstract

To our knowledge, the error and perturbation bounds of the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of two types of the general absolute value equations (AVEs): and . Some useful error bounds and perturbation bounds of the above two types of absolute value equations are provided. Without limiting the matrix type, some computable estimates for the above upper bounds are given. By applying the absolute value equations, a new approach for some existing perturbation bounds of the linear complementarity problem (LCP) in (SIAM J. Optim., 18 (2007), pp. 1250-1265) is provided. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds.
Paper Structure (11 sections, 20 theorems, 116 equations, 4 tables)

This paper contains 11 sections, 20 theorems, 116 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Error bound
  3. Framework of error bounds for AVEs
  4. Estimations of $\overline{\alpha}$, $\overline{\beta}$ and $\overline{\gamma}$
  5. Case I
  6. Case II
  7. Case III
  8. Perturbation bound
  9. An application
  10. Numerical examples
  11. Conclusion

Key Result

Lemma 2.1

Let $a=(a_{1},a_{2},\ldots,a_{n})^{T}, b=(b_{1},b_{2},\ldots,b_{n})^{T}$ be any two vectors in $\mathbb{R}^{n}$. Then there exist $d_{i}\in [-1,1]$ such that

Theorems & Definitions (27)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Remark 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 2.1
  • Theorem 2.5
  • Theorem 2.6
  • ...and 17 more