Skip to main content
SpringerLink
Log in

A new concave minimization algorithm for the absolute value equation solution

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we study the absolute value equation (AVE) \(Ax-b=|x|\). One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable.

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. Clarke, F.H.: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  2. Cottle, R.W., Pang, J.S.: On solving linear complementarity problems as linear programs. In: Balinski, M.L., Cottle, R.W. (eds.) Complementarity and Fixed Point Problems, Mathematics Program of Studies, vol. 7, pp. 88–107. Springer, Berlin (1978)

    Chapter  Google Scholar 

  3. Cruz, J.B., Ferreira, O.P., Prudente, L.: On the global convergence of the inexact semi-smooth Newton method for absolute value equation. Comput. Optim. Appl. 65(1), 93–108 (2016)

    Article  MathSciNet  Google Scholar 

  4. Dür, M., Horst, R., Locatelli, M.: Necessary and sufficient global optimality conditions for convex maximization revisited. J. Math. Anal. Appl. 217(2), 637–649 (1998)

    Article  MathSciNet  Google Scholar 

  5. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2007)

    MATH  Google Scholar 

  6. Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217–269. Springer, Berlin (1995)

    Chapter  Google Scholar 

  7. Gabriel, S.A., Moré, J.J.: Smoothing of mixed complementarity problems. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems: State of the Art, pp. 105–116. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

  8. Hiriart-Urruty, J.: Mean value theorems in nonsmooth analysis. Numer. Funct. Anal. Optim. 2(1), 1–30 (1980)

    Article  MathSciNet  Google Scholar 

  9. Hiriart-Urruty, J., Ledyaev, Y.S.: A note on the characterization of the global maxima of a (tangentially) convex function over a convex set. J. Convex Anal. 3, 55–62 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Hladík, M.: Bounds for the solutions of absolute value equations. Comput. Optim. Appl. 69(1), 243–266 (2018)

    Article  MathSciNet  Google Scholar 

  11. ILOG: Cplex 9.0 reference manual. ILOG CPLEX Division (2003)

  12. Kuttler, J.R.: A fourth-order finite-difference approximation for the fixed membrane eigenproblem. Math. Comput. 25(114), 237–256 (1971)

    Article  MathSciNet  Google Scholar 

  13. Le Thi, H.A., Dinh, T.P.: On solving linear complementarity problems by dc programming and dca. Comput. Optim. Appl. 50(3), 507–524 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Mangasarian, O.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)

    Article  MathSciNet  Google Scholar 

  16. Mangasarian, O., Meyer, R.: Absolute value equations. Linear Algebra Appl. 419(2–3), 359–367 (2006)

    Article  MathSciNet  Google Scholar 

  17. Mangasarian, O.L.: Linear complementarity problems solvable by a single linear program. Math. Program. 10(1), 263–270 (1976)

    Article  MathSciNet  Google Scholar 

  18. Mangasarian, O.L.: Characterization of linear complementarity problems as linear programs. In: Balinski, M.L., Cottle, R.W. (eds.) Complementarity and Fixed Point Problems, Mathematics Program of Studies, vol. 7, pp. 74–87. Springer, Berlin (1978)

    Chapter  Google Scholar 

  19. Mangasarian, O.L.: Solution of general linear complemetarity problems via nondifferentiable concave minimization. Acta Math. Vietnam. 22(1), 199–205 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9(7), 1469–1474 (2015)

    Article  MathSciNet  Google Scholar 

  22. Mangasarian, O.L.: Sufficient conditions for the unsolvability and solvability of the absolute value equation. Optim. Lett. 11(7), 1469–1475 (2017)

    Article  MathSciNet  Google Scholar 

  23. Nguyen, V.T., Ty, H.: Solving the linear complementarity problem through concave programming. USSR Comput. Math. Math. Phys. 23(3), 602–608 (1983)

    MathSciNet  Google Scholar 

  24. Palais, R.S.: Natural operations on differential forms. Trans. Am. Math. Soc. 92(1), 125–141 (1959)

    Article  MathSciNet  Google Scholar 

  25. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  26. Rohn, J.: Forty necessary and sufficient conditions for regularity of interval matrices: a survey. Electron. J. Linear Algebra 18, 500–512 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Schrijver, A.: Theory of Linear and Integer Programming Report. Wiley, Chichester (1998)

    MATH  Google Scholar 

  28. Tuy, H.: Convex Analysis and Global Optimization (Springer Optimization and Its Applications Book 110). Springer, Berlin (2016)

    Book  Google Scholar 

  29. Wu, S.L., Li, C.X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)

    Article  MathSciNet  Google Scholar 

  30. Zamani, M., Hladík, M.: Error bounds and a condition number for the absolute value equations (2019). arXiv:1912.12904

  31. Zhang, C., Wei, Q.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143(2), 391–403 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

M. Hladík was supported by the Czech Science Foundation Grant P403-18-04735S.

Author information

Authors and Affiliations

  1. Parametric MultiObjective Optimization Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Moslem Zamani

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Moslem Zamani

  3. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 11800, Prague, Czech Republic

    Milan Hladík

Authors
  1. Moslem Zamani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milan Hladík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moslem Zamani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zamani, M., Hladík, M. A new concave minimization algorithm for the absolute value equation solution. Optim Lett 15, 2241–2254 (2021). https://doi.org/10.1007/s11590-020-01691-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-020-01691-z

Keywords

Search

Navigation