Skip to main content
SpringerLink
Log in

A globally and quadratically convergent method for absolute value equations

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We investigate the NP-hard absolute value equation (AVE) Ax−|x|=b, where A is an arbitrary n×n real matrix. In this paper, we propose a smoothing Newton method for the AVE. When the singular values of A exceed 1, we show that this proposed method is globally convergent and the convergence rate is quadratic. Preliminary numerical results show that this method is promising.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chung, S.J.: NP-completeness of the linear complementarity problem. J. Optim. Theory Appl. 60, 393–399 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cottle, R.W., Dantzig, G.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)

    MATH  Google Scholar 

  4. Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. Qi, L., Sun, D.: Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113, 121–147 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Rohn, J.: A theorem of the alternatives for the equation Ax+B|x|=b. Linear Multilinear Algebra 52, 421–426 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Stewart, G.W.: Introduction to Matrix Computations. Academic Press, San Diego (1973)

    MATH  Google Scholar 

  12. Sun, D., Han, J.: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7, 463–480 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Western Australian Centre of Excellence in Industrial Optimisation (WACEIO), Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, WA, 6845, Australia

    Louis Caccetta & Guanglu Zhou

  2. Institute of Operations Research, Qufu Normal University, Rizhao, 276826, Shandong, P.R. China

    Biao Qu

Authors
  1. Louis Caccetta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Biao Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guanglu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guanglu Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caccetta, L., Qu, B. & Zhou, G. A globally and quadratically convergent method for absolute value equations. Comput Optim Appl 48, 45–58 (2011). https://doi.org/10.1007/s10589-009-9242-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-009-9242-9

Keywords

Search

Navigation