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The Horizontal Linear Complementarity Problem and Robustness of the Related Matrix Classes

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Optimization and Learning (OLA 2021)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1443))

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Abstract

We consider the horizontal linear complementarity problem and we assume that the input data have the form of intervals, representing the range of possible values. For the classical linear complementarity problem, there are known various matrix classes that identify interesting properties of the problem (such as solvability, uniqueness, convexity, finite number of solutions or boundedness). Our aim is to characterize the robust version of these properties, that is, to check them for all possible realizations of interval data. We address successively the following matrix classes: nonnegative matrices, Z-matrices, semimonotone matrices, column sufficient matrices, principally nondegenerate matrices, \(R_0\)-matrices and R-matrices. The reduction of the horizontal linear complementarity problem to the classical one, however, brings complicated dependencies between interval parameters, resulting in some cases to higher computational complexity.

Supported by the Czech Science Foundation Grants P403-18-04735S (M. Hladík) and P403-20-17529S (M. Rada).

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References

  1. Alefeld, G., Schäfer, U.: Iterative methods for linear complementarity problems with interval data. Computing 70(3), 235–259 (2003). https://doi.org/10.1007/s00607-003-0014-6

    Article  MathSciNet  MATH  Google Scholar 

  2. Chung, S.J.: NP-completeness of the linear complementarity problem. J. Optim. Theor. Appl. 60(3), 393–399 (1989)

    Article  MathSciNet  Google Scholar 

  3. Cottle, R.W.: Linear complementarity since 1978. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications, NOIA, vol. 79, pp. 239–257. Springer, Boston (2005). https://doi.org/10.1007/0-387-24276-7_18

    Chapter  Google Scholar 

  4. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. revised ed of the 1992 original edn. SIAM, Philadelphia, PA (2009)

    Google Scholar 

  5. Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, New York (2006)

    MATH  Google Scholar 

  6. Garloff, J., Adm, M., Titi, J.: A survey of classes of matrices possessing the interval property and related properties. Reliab. Comput. 22, 1–10 (2016)

    MathSciNet  Google Scholar 

  7. Gowda, M.: Reducing a monotone horizontal LCP to an LCP. Appl. Math. Lett. 8(1), 97–100 (1995)

    Article  MathSciNet  Google Scholar 

  8. Hladík, M.: Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl. 438(11), 4156–4165 (2013)

    Article  MathSciNet  Google Scholar 

  9. Hladík, M.: Transformations of interval linear systems of equations and inequalities. Linear Multilinear Algebra 65(2), 211–223 (2017)

    Article  MathSciNet  Google Scholar 

  10. Hladík, M.: Stability of the linear complementarity problem properties under interval uncertainty. Cent. Eur. J. Oper. Res. 29, 875–889 (2021)

    Google Scholar 

  11. Hladík, M.: Tolerances, robustness and parametrization of matrix properties related to optimization problems. Optimization 68(2–3), 667–690 (2019)

    Article  MathSciNet  Google Scholar 

  12. Hladík, M.: An overview of polynomially computable characteristics of special interval matrices. In: Kosheleva, O., et al. (eds.) Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications, Studies in Computational Intelligence, vol. 835, pp. 295–310. Springer, Cham (2020)

    Google Scholar 

  13. Horáček, J., Hladík, M., Černý, M.: Interval linear algebra and computational complexity. In: Bebiano, N. (ed.) MAT-TRIAD 2015. SPMS, vol. 192, pp. 37–66. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-49984-0_3

    Chapter  MATH  Google Scholar 

  14. Kreinovich, V.: Why intervals? a simple limit theorem that is similar to limit theorems from statistics. Reliab. Comput. 1(1), 33–40 (1995)

    Article  MathSciNet  Google Scholar 

  15. Kreinovich, V.: Why intervals? why fuzzy numbers? towards a new justification. In: Mendel, J.M., Omori, T., Ya, X. (eds.) 2007 IEEE Symposium on Foundations of Computational Intelligen, pp. 113–119 (2007)

    Google Scholar 

  16. Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1998)

    Book  Google Scholar 

  17. Ma, H., Xu, J., Huang, N.: An iterative method for a system of linear complementarity problems with perturbations and interval data. Appl. Math. Comput. 215(1), 175–184 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Mezzadri, F., Galligani, E.: Splitting methods for a class of horizontal linear complementarity problems. J. Optim. Theor. Appl. 180(2), 500–517 (2019)

    Article  MathSciNet  Google Scholar 

  19. Mezzadri, F., Galligani, E.: A modulus-based nonsmooth Newton’s method for solving horizontal linear complementarity problems. Optimization Letters 15(5), 1785–1798 (2019). https://doi.org/10.1007/s11590-019-01515-9

    Article  MathSciNet  MATH  Google Scholar 

  20. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia, PA (2009)

    Book  Google Scholar 

  21. Murty, K.G., Yu, F.T.: Linear Complementarity, Internet Linear and Nonlinear Programming (1997)

    Google Scholar 

  22. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  23. Rex, G., Rohn, J.: Sufficient conditions for regularity and singularity of interval matrices. SIAM J. Matrix Anal. Appl. 20(2), 437–445 (1998)

    Article  MathSciNet  Google Scholar 

  24. 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 

  25. Samelson, H., Thrall, R.M., Wesler, O.: A partition theorem for euclidean \(n\)-spaces. Proc. Am. Math. Soc. 9, 805–807 (1958)

    MathSciNet  MATH  Google Scholar 

  26. Sznajder, R., Gowda, M.: Generalizations of P\({}_0\)- and P-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223–224, 695–715 (1995)

    Article  MathSciNet  Google Scholar 

  27. Tseng, P.: Co-NP-completeness of some matrix classification problems. Math. Program. 88(1), 183–192 (2000)

    Article  MathSciNet  Google Scholar 

  28. Tütüncü, R.H., Todd, M.J.: Reducing horizontal linear complementarity problems. Linear Algebra Appl. 223–224, 717–729 (1995)

    Article  MathSciNet  Google Scholar 

  29. Zheng, H., Vong, S.: On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of \(H_+\)-matrices. Appl. Math. Comput. 369(124890), 1–6 (2020)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Milan Hladík

  2. Faculty of Informatics and Statistics, Department of Econometrics, University of Economics, W. Churchill’s Sq. 4, 130 67, Prague, Czech Republic

    Miroslav Rada

Authors
  1. Milan Hladík
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  2. Miroslav Rada
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Correspondence to Milan Hladík .

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Editors and Affiliations

  1. University of Cádiz, Cádiz, Spain

    Bernabé Dorronsoro

  2. University of Technology of Troyes, Troyes, France

    Lionel Amodeo

  3. University of Catania, Catania, Italy

    Mario Pavone

  4. University of Cádiz, Cádiz, Spain

    Patricia Ruiz

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Hladík, M., Rada, M. (2021). The Horizontal Linear Complementarity Problem and Robustness of the Related Matrix Classes. In: Dorronsoro, B., Amodeo, L., Pavone, M., Ruiz, P. (eds) Optimization and Learning. OLA 2021. Communications in Computer and Information Science, vol 1443. Springer, Cham. https://doi.org/10.1007/978-3-030-85672-4_26

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  • DOI: https://doi.org/10.1007/978-3-030-85672-4_26

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