Mathematical Programming

, Volume 168, Issue 1–2, pp 93–121 | Cite as

A structure-preserving pivotal method for affine variational inequalities

  • Youngdae Kim
  • Olivier Huber
  • Michael C. FerrisEmail author
Full Length Paper Series B


Affine variational inequalities (AVI) are an important problem class that subsumes systems of linear equations, linear complementarity problems and optimality conditions for quadratic programs. This paper describes PathAVI, a structure-preserving pivotal approach, that can efficiently process (solve or determine infeasible) large-scale sparse instances of the problem with theoretical guarantees and at high accuracy. PathAVI implements a strategy known to process models with good theoretical properties without reducing the problem to specialized forms, since such reductions may destroy sparsity in the models and can lead to very long computational times. We demonstrate formally that PathAVI implicitly follows the theoretically sound iteration paths, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction. We also extend the class of problems that PathAVI can process. The paper illustrates the effectiveness of our approach by comparison to the Path solver used on a complementarity reformulation of the AVI in the context of applications in friction contact and Nash Equilibria. PathAVI is a general purpose solver, and freely available under the same conditions as Path.


Affine variational inequality Normal map Path-following algorithm 

Mathematics Subject Classification

90C33 90C49 65K10 65K15 



This work is supported in part by the Air Force Office of Scientific Research and the Department of Energy. The authors are grateful to Steven Dirkse and Todd Munson for comments and suggestions leading to improved computational performance.


  1. 1.
    Acary, V., Brémond, M., Koziara, T., Pérignon, F.: FCLIB: a collection of discrete 3D Frictional Contact problems. Technical Report RT-0444, INRIA. (2014)
  2. 2.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics. Springer, Berlin (2008)zbMATHGoogle Scholar
  3. 3.
    Acary V., Brémond M., Huber O., Pérignon F., Sinclair, S.: An introduction to SICONOS, Rapport Technique, INRIA (2017)Google Scholar
  4. 4.
    Bixby, R.E.: Implementing the simplex method: the initial basis. ORSA J. Comput. 4(3), 267–284 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, M., Ferris, M.C.: Lineality removal for copositive-plus normal maps. Commun. Appl. Nonlinear Anal. 2, 1–10 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cao, M., Ferris, M.C.: A pivotal method for affine variational inequalities. Math. Oper. Res. 21(1), 44–64 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cottle, R., Pang, J.S., Stone, R.: The Linear Complementarity Problem. No. 60 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefGoogle Scholar
  8. 8.
  9. 9.
    Dirkse, S.P., Ferris, M.C.: The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5(2), 123–156 (1995)CrossRefGoogle Scholar
  10. 10.
    Dirkse, S.P., Ferris, M.C.: A pathsearch damped newton method for computing general equilibria. Ann. Oper. Res. 68(2), 211–232 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dirkse, S.P., Ferris, M.C.: Crash techniques for large-scale complementarity problems. In: Ferris, M.C., Pang J.S. (eds.) Complementarity and Variational Problems: State of the Art. SIAM, Philadelphia (1997)Google Scholar
  12. 12.
    Dubois, F., Jean, M., Renouf, M., Mozul, R., Martin, A., Bagneris, M.: LMGC90. In: 10e colloque national en calcul des structures. Giens. (2011)
  13. 13.
    Eaves, B.C.: A short course in solving equations with PL homotopies. Nonlinear Programming. In: SIAM-AMS Proceedings, vol. 9, pp. 73–143 (1976)Google Scholar
  14. 14.
    Eldersveld, S.K., Saunders, M.A.: A block-lu update for large-scale linear programming. SIAM J. Matrix Anal. Appl. 13(1), 191–201 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)zbMATHGoogle Scholar
  16. 16.
    Ferris, M.C., Dirkse, S.P., Jagla, J.H., Meeraus, A.: An extended mathematical programming framework. Comput. Chem. Eng. 33(12), 1973–1982 (2009)CrossRefGoogle Scholar
  17. 17.
    Ferris, M.C., Munson, T.S.: PATH 4.7. (2013)
  18. 18.
    Huber, O., Ferris, M.C.: Friction contact problems from a variational inequality perspective (2016) (in preparation)Google Scholar
  19. 19.
    Klarbring, A., Pang, J.S.: Existence of solutions to discrete semicoercive frictional contact problems. SIAM J. Optim. 8(2), 414–442 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Manag. Sci. 11(7), 681–689 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu, J.: Strong stability in variational inequalities. SIAM J. Control Optim. 33(3), 725–749 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maros, I.: QP examples. (1998)
  23. 23.
    Murty, K.G.: Linear and Combinatorial Programming. Wiley, New York (1976)zbMATHGoogle Scholar
  24. 24.
    Robinson, S.M.: Generalized equations and their solutions, part I: basic theory. Math. Program. Study 10, 128–141 (1979)CrossRefzbMATHGoogle Scholar
  25. 25.
    Robinson, S.M.: Normal maps induced by linear transformations. Math. Oper. Res. 17(3), 691–714 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Robinson, S.M.: Convexity in Finite-Dimensional Spaces (2015) (unpublished manuscript)Google Scholar
  27. 27.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Wisconsin Institute for Discovery and Department of Computer SciencesUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations