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A verification method for enclosing solutions of absolute value equations

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Abstract

In this paper, we investigate the NP-hard absolute value equations (AVE) Ax − |x| = b and establish two computational enclosures for the solution of AVE. The condition for enclosing the solution and guaranteeing the existence of solutions of AVE is presented in a given domain. Based on this condition, new algorithms are designed to compute an interval to enclose the unknown solution of AVE. Numerical results are reported to support the theoretical analysis in the paper.

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References

  1. Rohn J.: A theorem of the alternatives for the equation Ax + B|x| = b. Linear Multilinear Algebra 52(6), 421–426 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Mangasarian O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 43–53 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Oleg, P.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. doi:10.1007/s10589-007-9158-1 (2007)

  5. Rohn J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. doi:10.1007/s10589-009-9242-9 (2009)

  9. Rohn J.: An algorithm for solving the absolute value equation. Electr. J. Linear Algebra 18, 589–599 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Zhang C., Wei Q.J.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143, 391–403 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Moore R.E.: A computational test for convergence of iterative methods for nonlinear systems. SIAM J. Numer. Anal. 15, 1194–1196 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  12. Qi, L.: A generalization of the Krawczyk-Moore algorithm. In: Nickel, K. (ed.) Interval Mathematics (1980)

  13. Zhang D., Li W., Shen Z.: Solving underdetermined system with interval methods. Reliab. Comput. 5, 23–33 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Alefeld G.E., Mayer G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang Z.: Validation and enclosure of solution of linear complementarity problems. Computing 79, 61–77 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang H., Cao D.: Interval expansion method for nonlinear equation in several variables. Appl. Math. Comput. 212, 153–161 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Varga R.: Matrix Iterative Analysis, 2nd revised and expanded edition. Springer, Berlin-Heidelberg-New York (2000)

    Google Scholar 

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Authors and Affiliations

  1. College of Science, China University of Mining and Technology, Xuzhou, 221008, People’s Republic of China

    Haijun Wang & Suyu Cao

  2. College of Science, Nanjing University of Technology, Nanjing, 210016, People’s Republic of China

    Hao Liu

Authors
  1. Haijun Wang
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  2. Hao Liu
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  3. Suyu Cao
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Corresponding author

Correspondence to Haijun Wang.

Additional information

This work is partially supported by the National Natural Science Foundation of China (11071117) and the Fundamental Research Funds for the Central Universities (2010LKSX01).

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Wang, H., Liu, H. & Cao, S. A verification method for enclosing solutions of absolute value equations. Collect. Math. 64, 17–38 (2013). https://doi.org/10.1007/s13348-011-0057-5

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  • DOI: https://doi.org/10.1007/s13348-011-0057-5

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