Mathematical Programming

, Volume 134, Issue 2, pp 349–364 | Cite as

Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix

  • Ibtihel Ben Gharbia
  • J. Charles GilbertEmail author
Full Length Paper Series A


The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) \({0\leq x\perp(Mx+q)\geq0}\) can be viewed as a semismooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, Mx + q) = 0, which is equivalent to the LCP. When M is an M-matrix of order n, the algorithm is known to converge in at most n iterations. We show in this paper that this result no longer holds when M is a P-matrix of order ≥ 3, since then the algorithm may cycle. P-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary q. Incidentally, convergence occurs for a P-matrix of order 1 or 2.


Linear complementarity problem Newton’s method Nonconvergence Nonsmooth function M-matrix P-matrix 

Mathematics Subject Classification (2010)

49J52 49M15 90C33 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aganagić M.: Newton’s method for linear complementarity problems. Math. Program. 28, 349–362 (1984)zbMATHCrossRefGoogle Scholar
  2. 2.
    Ben Gharbia, I., Gilbert, J.: Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix—the full report. Rapport de Recherche 7160, INRIA, BP 105, 78153 Le Chesnay, France (2009).
  3. 3.
    Ben Gharbia, I., Gilbert, J.: (2011) (in preparation)Google Scholar
  4. 4.
    Bergounioux M., Haddou M., Hintermüller M., Kunisch K.: A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11, 495–521 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bergounioux M., Ito K., Kunisch K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176–1194 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bonnans J., Gilbert J., Lemaréchal C., Sagastizábal C.: Numerical Optimization—Theoretical and Practical Aspects, 2nd edn. Universitext. Springer, Berlin (2006)Google Scholar
  7. 7.
    Buchholzer, H., Kanzow, C., Knabner, P., Kräutle, S.: Solution of Reactive transport problems including mineral precipitation-dissolution reactions by a semismooth Newton method. Technical Report 288, Institute of Mathematics, University of Würzburg, Würzburg (2009)Google Scholar
  8. 8.
    Chandrasekaran R.: A special case of the complementary pivot problem. Opsearch 7, 263–268 (1970)MathSciNetGoogle Scholar
  9. 9.
    Clarke F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  10. 10.
    Conn, A., Gould, N., Toint, P.: Trust-region methods. MPS-SIAM Series on Optimization 1. SIAM and MPS, Philadelphia (2000)Google Scholar
  11. 11.
    Cottle R., Dantzig G.: Complementarity pivot theory of mathematical programming. Linear Algebra Appl. 1, 103–125 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cottle, R., Pang, J.S., Stone, R.: The linear complementarity problem. No. 60 in Classics in Applied Mathematics. SIAM, Philadelphia, PA, USA (2009)Google Scholar
  13. 13.
    Coxson G.: The P-matrix problem is co-NP-complete. Math. Program. 64, 173–178 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems (two volumes). Springer Series in Operations Research, Springer (2003)Google Scholar
  15. 15.
    Fiedler M., Pták V.: On matrices with nonpositive off-diagonal elements and principal minors. Czech. Math. J. 12, 382–400 (1962)Google Scholar
  16. 16.
    Fischer A., Kanzow C.: On finite termination of an iterative method for linear complementarity problems. Math. Program. 74, 279–292 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Golub G., Loan C.V.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore, Maryland (1996)zbMATHGoogle Scholar
  18. 18.
    Harker, P., Pang, J.S.: A damped-Newton method for the linear complementarity problem. In: Allgower, E., Georg, K. (eds.) Computational Solution of Nonlinear Systems of Equations, No. 26 in Lecture in Applied Mathematics. AMS, Providence, RI (1990)Google Scholar
  19. 19.
    Hintermüller M., Ito K., Kunisch K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2003)zbMATHCrossRefGoogle Scholar
  20. 20.
    Ito K., Kunisch K.: On a semi-smooth Newton method and its globalization. Math. Program. 118, 347–370 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kanzow C.: Inexact semismooth Newton methods for large-scale complementarity problems. Optim. Methods Softw. 19, 309–325 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Karmarkar N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kellogg R.: On complex eigenvalues of M and P matrices. Numer. Math. 19, 170–175 (1972)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. No. 538 in Lecture Notes in Computer Science. Springer, Berlin (1991)Google Scholar
  25. 25.
    Kojima M., Shindo S.: Extension of Newton and quasi-Newton methods to systems of PC 1 equations. J. Oper. Res. Soc. Jpn. 29, 352–375 (1986)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kostreva, M.: Direct algorithms for complementarity problems. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, New York (1976)Google Scholar
  27. 27.
    Kräutle, S.: The semismooth Newton method for multicomponent reactive transport with minerals. Technical Report, Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany (2010)Google Scholar
  28. 28.
    Lemke C.: Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11, 681–689 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Mangasarian O.: Solution of symmetric linear complementarity problems by iterative methods. J. Optim. Theory Appl. 22, 465–485 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Megiddo, N.: A note on the complexity of P-matrix LCP and computing an equilibrium. Technical Report RJ 6439 (62557), IBM Research, Almaden Research Center, 650 Harry Road, San Jose, CA, USA (1988)Google Scholar
  31. 31.
    Metla, N.: The sequential quadratic programming method for elliptic optimal control problems with mixed control-state constraints. Ph.D. Thesis, Johann Radon Institute for Computational and Applied Mathematics, Johannes Kepler Universität, Linz, Austria (2008)Google Scholar
  32. 32.
    Morris W.: Randomized pivot algorithms for P-matrix linear complementarity problems. Math. Program. 92A, 285–296 (2002)CrossRefGoogle Scholar
  33. 33.
    Murty K.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988)zbMATHGoogle Scholar
  34. 34.
    Qi L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Samelson H., Thrall R., Wesler O.: A partition theorem for the Euclidean n-space. Proc. Am. Math. Soc. 9, 805–807 (1958)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Schäfer U.: A linear complementarity problem with a P-matrix. SIAM Rev. 46, 189–201 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Solow, D., Stone, R., Tovey, C.: Solving LCP on P-matrices is probably not NP-hard. Unpublished note (1987)Google Scholar
  38. 38.
    Tseng P.: Co-NP-completeness of some matrix classification problems. Math. Program. 88, 183–192 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.Eddimo, CeremadeUniversity Paris-DauphineParis Cedex 16France
  2. 2.INRIA Paris-RocquencourtLe ChesnayFrance

Personalised recommendations