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On Generalized Positive Subdefinite Matrices and Interior Point Algorithm

  • A. K. Das
  • R. Jana
  • Deepmala
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 225)

Abstract

In this paper, we propose an iterative and descent type interior point method to compute solution of linear complementarity problem LCP(qA) given that A is real square matrix and q is a real vector. The linear complementarity problem includes many of the optimization problems and applications. In this context, we consider the class of generalized positive subdefinite matrices (GPSBD) which is a generalization of the class of positive subdefinite (PSBD) matrices. Though Lemke’s algorithm is frequently used to solve small and medium-size LCP(qA), Lemke’s algorithm does not compute solution of all problems. It is known that Lemke’s algorithm is not a polynomial time bound algorithm. We show that the proposed algorithm converges to the solution of LCP(qA) where A belongs to GPSBD class. We provide the complexity analysis of the proposed algorithm. A numerical example is illustrated to show the performance of the proposed algorithm.

Keywords

Interior point algorithm Generalized positive subdefinite matrices (GPSBD) Positive subdefinite matrices (PSBD) Linear complementarity problem 

Notes

Acknowledgements

The second author R. Jana is thankful to the Department of Science and Technology, Govt. of India, INSPIRE Fellowship Scheme for financial support. The research work of the third author Deepmala is supported by the Science and Engineering Research Board (SERB), Government of India under SERB N-PDF scheme, File Number: PDF/2015/000799.

References

  1. 1.
    Crouzeix, J.P., Komlósi, S.: The linear complementarity problem and the class of generalized positive subdefinite matrices. In: Optimization Theory, pp. 45–63. Springer, US (2001)Google Scholar
  2. 2.
    Den Hertog, D.: Interior Point Approach to Linear, Quadratic and Convex Programming: Algorithms and Complexity, vol. 277. Springer Science & Business Media (2012)Google Scholar
  3. 3.
    Fathi, Y.: Computational complexity of LCPs associated with positive definite symmetric matrices. Math. Program. 17(1), 335–344 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Khachiyan, L.G.: Polynomial algorithms in linear programming. USSR Comput. Math. Math. Phys. 20(1), 53–72 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kojima, M., Megiddo, N., Ye, Y.: An interior point potential reduction algorithm for the linear complementarity problem. Math. Program. 54(1–3), 267–279 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Martos, B.: Subdefinite matrices and quadratic forms. SIAM J. Appl. Math. 17(6), 1215–1223 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mohan, S.R., Neogy, S.K., Das, A.K.: More on positive subdefinite matrices and the linear complementarity problem. Linear Algebra Appl. 338(1), 275–285 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Monteiro, R.C., Adler, I.: An \({\rm O}(n^3L)\) Primal-Dual Interior Point Algorithm for Linear Programming. Report ORC 87-4, Dept. of Industrial Engineering and Operations Research, University of California, Berkeley, CA (1987)Google Scholar
  9. 9.
    Neogy, S.K., Das, A.K.: Some properties of generalized positive subdefinite matrices. SIAM J. Matrix Anal. Appl. 27(4), 988–995 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pang, J.S.: Iterative descent algorithms for a row sufficient linear complementarity problem. SIAM J. Matrix Anal. Appl. 12(4), 611–624 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Todd, M.J., Ye, Y.: A centered projective algorithm for linear programming. Math. Oper. Res. 15(3), 508–529 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wang, G.Q., Yu, C.J., Teo, K.L.: A full-Newton step feasible interior-point algorithm for \({\rm P}_{*}(\kappa )\)-linear complementarity problems. J. Glob. Optim. 59(1), 81–99 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ye, Y.: An \({\rm O}(n^3 L)\) potential reduction algorithm for linear programming. Math. Program. 50(1–3), 239–258 (1991)CrossRefGoogle Scholar
  14. 14.
    Ye, Y.: Interior Point Algorithms: Theory and Analysis, vol. 44. Wiley (2011)Google Scholar
  15. 15.
    Ye, Y., Pardalos, P.M.: A class of linear complementarity problems solvable in polynomial time. Linear Algebra Appl. 152, 3–17 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Jadavpur UniversityKolkataIndia
  3. 3.Mathematics DisciplinePDPM Indian Institute of Information Technology, Design and ManufacturingJabalpurIndia

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