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On unique solvability of the absolute value equation

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Abstract

It is proved that the singular value condition σ max(|B|) < σ min(A) implies unique solvability of the absolute value equation Ax + B|x| = b for each right-hand side b. This is a generalization of an earlier result by Mangasarian and Meyer proved for the special case of B = −I.

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References

  1. VERABSVALEQN: Verified solution of the equation A x + B|x| = b (2008). Available at http://www.cs.cas.cz/rohn/matlab/verabsvaleqn.html

  2. VERSOFT: Verification software in MATLAB/INTLAB (2009). Available at http://www.cs.cas.cz/rohn/matlab

  3. Mangasarian O.: Absolute value equation solution via concave minimization. Optim. Lett. 1(1), 3–8 (2007). doi:10.1007/s11590-006-0005-6

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Mangasarian O.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009). doi:10.1007/s11590-008-0094-5

    Article  MATH  MathSciNet  Google Scholar 

  6. Mangasarian O.L., Meyer R.R.: Absolute value equations. Linear Algebra Appl. 419(2–3), 359–367 (2006). doi:10.1016/j.laa.2006.05.004

    Article  MATH  MathSciNet  Google Scholar 

  7. Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. (accepted)

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Rohn J.: Description of all solutions of a linear complementarity problem. Electron. J. Linear Algebra 18, 246–252 (2009)

    Google Scholar 

  10. Schäfer U.: Das lineare Komplementaritätsproblem, Eine Einführung (The Linear Complementarity Problem, An Introduction). Springer, Berlin (2008)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Prague, Czech Republic

    Jiri Rohn

  2. School of Business Administration, Anglo-American University, Prague, Czech Republic

    Jiri Rohn

Authors
  1. Jiri Rohn
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Correspondence to Jiri Rohn.

Additional information

Supported by the Czech Republic Grant Agency under grants 201/09/1957 and 201/08/J020, and by the Institutional Research Plan AV0Z10300504.

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Rohn, J. On unique solvability of the absolute value equation. Optim Lett 3, 603–606 (2009). https://doi.org/10.1007/s11590-009-0129-6

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  • DOI: https://doi.org/10.1007/s11590-009-0129-6

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