Journal of Global Optimization

, Volume 53, Issue 1, pp 29–51 | Cite as

On linear programs with linear complementarity constraints



The paper is a manifestation of the fundamental importance of the linear program with linear complementarity constraints (LPCC) in disjunctive and hierarchical programming as well as in some novel paradigms of mathematical programming. In addition to providing a unified framework for bilevel and inverse linear optimization, nonconvex piecewise linear programming, indefinite quadratic programs, quantile minimization, and 0 minimization, the LPCC provides a gateway to a mathematical program with equilibrium constraints, which itself is an important class of constrained optimization problems that has broad applications. We describe several approaches for the global resolution of the LPCC, including a logical Benders approach that can be applied to problems that may be infeasible or unbounded.


Linear programs with linear complementarity constraints Inverse programming Hierarchical programming Piecewise linear programming Quantile minimization Cross-validated support vector regression 


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  1. 1.
    Ahmed S., Shapiro A.: Solving chance-constrained stochastic programs via sampling and integer programming. In: Chen, Z.L., Raghavan, S. (eds) TutORials in Operations Research, chapter 12, pp. 261–269. INFORMS, London (2008)Google Scholar
  2. 2.
    Ahuja R.K., Orlin J.B.: Inverse optimization. Oper. Res. 49(5), 771–783 (2001)CrossRefGoogle Scholar
  3. 3.
    Anandalingam G., Friesz T.L.: Hierarchical optimization: an introduction. Ann. Oper. Res. 34(1), 1–11 (1992)CrossRefGoogle Scholar
  4. 4.
    Audet C., Savard G., Zghal W.: New branch-and-cut algorithm for bilevel linear programming. J. Optim. Theory Appl. 38(2), 353–370 (2007)CrossRefGoogle Scholar
  5. 5.
    Balas E.: Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)CrossRefGoogle Scholar
  6. 6.
    Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58, 295–324 (1993)CrossRefGoogle Scholar
  7. 7.
    Balas E., Perregaard M.: Lift-and-project for mixed 0-1 programming: recent progress. Dis. Appl. Math. 123(1–3), 129–154 (2002)CrossRefGoogle Scholar
  8. 8.
    Balas E., Perregaard M.: A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0-1 programming. Math. Program. 94(2-3), 221–245 (2003)CrossRefGoogle Scholar
  9. 9.
    Bennett, K.P., Ji, X., Hu, J., Kunapuli, G., Pang, J.-S.: Model selection via bilevel programming. In: Proceedings of the International Joint Conference on Neural Networks (IJCNN’06) Vancouver, pp. 1922–1929. B.C. Canada, July 16–21 (2006)Google Scholar
  10. 10.
    Burer S., Vandenbussche D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2), 259–282 (2008)CrossRefGoogle Scholar
  11. 11.
    Burer S., Vandenbussche D.: Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound. Comput. Optim. Appl. 43(2), 181–195 (2009)CrossRefGoogle Scholar
  12. 12.
    Chang M.W., Lin C.J.: Leave-one-out bounds for support vector regression model selection. Neural Comput. 17, 1188–1222 (2005)CrossRefGoogle Scholar
  13. 13.
    Chapelle O., Vapnik V., Bousquet O., Mukherjee S.: Choosing multiple parameters with support vector machines. Mach. Learn. 46, 131–159 (2002)CrossRefGoogle Scholar
  14. 14.
    Chung K.W., Kao W.C., Sun C.L., Wang L.L., Lin C.J.: Radius margin bounds for support vector machines with the RBF kernel. Neural Comput. 15, 2643–2681 (2003)CrossRefGoogle Scholar
  15. 15.
    Colson B., Marcotte P., Savard G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)CrossRefGoogle Scholar
  16. 16.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The linear complementarity problem. SIAM Classics in Applied Mathematics 60, Philadelphia (2009) [Originally published by Academic Press, Boston (1992)]Google Scholar
  17. 17.
    Dempe S.: Foundations of Bilevel Programming. Kluwer, Dordrecht (2002)Google Scholar
  18. 18.
    Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program (2010) (online first)Google Scholar
  19. 19.
    Facchinei F., Pang J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems: vols. I and II. Springer, New York (2003)Google Scholar
  20. 20.
    Floudas C.A., Pardalos P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms, vol. 455 of Lecture Notes in Computer Science. Springer, New York (1990)CrossRefGoogle Scholar
  21. 21.
    Floudas C.A., Pardalos P.M., Adjiman C., Esposito W.R., Gümüs Z.H., Harding S.T., Klepeis J.L., Meyer C.A., Schweiger C.A.: Handbook of Test Problems in Local and Global Optimization, vol. 33 of Nonconvex Optimization and its Applications. Kluwer, Dordrecht (1999)Google Scholar
  22. 22.
    Giannessi, F., Tomasin, E.: Nonconvex quadratic programs, linear complementarity problems, and integer linear programs. In: Conti, R., Ruberti, A. (eds.) Fifth Conference on Optimization Techniques (Rome 1973), Part I, vol. 3 of Lecture Notes in Computer Science, pp. 437–449. Springer, Berlin (1973)Google Scholar
  23. 23.
    Grone B., Johnson C.R., Marques de Sa E., Wolkowicz H.: Positive definite completions of partial Hermitian matrices. Lin. Alg. Appl. 58, 109–124 (1984)CrossRefGoogle Scholar
  24. 24.
    Horn R.A., Johnson C.: Matrix Analysis. Cambridge University Press, Cambridge (1985)Google Scholar
  25. 25.
    Horst R., Pardalos P.M., Thoai N.V.: Introduction to Global Optimization, 2nd edn, Vol. 48 of Nonconvex Optimization and its Applications. Kluwer, Dordrecht (2000)Google Scholar
  26. 26.
    Hu, J.: On linear programs with linear complementarity constraints. PhD thesis, Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, August (2009)Google Scholar
  27. 27.
    Hu, J., Mitchell, J.E., Pang, J.S.: An LPCC approach to nonconvex quadratic programs. Technical report, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180. Accepted for publication in Mathematical Programming. May (2008)Google Scholar
  28. 28.
    Hu J., Mitchell J.E., Pang J.S., Bennett K.P., Kunapuli G.: On the global solution of linear programs with linear complementarity constraints. SIAM J. Optim. 19(1), 445–471 (2008)CrossRefGoogle Scholar
  29. 29.
    Iyengar G., Kang W.: Inverse conic programming with applications. Oper. Res. Lett. 33(3), 319–330 (2005)CrossRefGoogle Scholar
  30. 30.
    Keha, A.B.: A polyhedral study of nonconvex piecewise linear optimization. PhD thesis, Industrial Engineering, Georgia Institute of Technology, Atlanta, GA, August (2003)Google Scholar
  31. 31.
    Keha A.B., de Farias I.R. Jr, Nemhauser G.L.: Models for representing piecewise linear cost functions. Oper. Res. Lett. 32(1), 44–48 (2004)CrossRefGoogle Scholar
  32. 32.
    Keha A.B., de Farias I.R. Jr, Nemhauser G.L.: A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54(5), 847–858 (2006)CrossRefGoogle Scholar
  33. 33.
    Kunapuli, G., Hu, J., Bennett, K.P., Pang, J.S.: Bilevel model selection for support vector machines. In: Data Mining and Mathematical Programming, vol 45 of CRM Proceedings and Lecture Notes, pp. 129–158. Centre de Recherches Mathématiques (2008)Google Scholar
  34. 34.
    Kunapuli G., Hu J., Bennett K.P., Pang J.S.: Classification model selection via bilevel programming. Optim. Methods Softw. 23(4), 475–489 (2008)CrossRefGoogle Scholar
  35. 35.
    Krishnan K., Mitchell J.E.: A unifying framework for several cutting plane methods for semidefinite programming. Optim. Methods Softw. 21(1), 57–74 (2006)CrossRefGoogle Scholar
  36. 36.
    Lovász L., Schrijver A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)CrossRefGoogle Scholar
  37. 37.
    Luedtke, J.: Integer programming approaches for some non-convex and stochastic optimization problems. PhD thesis, Industrial and Systems Engineering, Georgia Institute of Technology (2007)Google Scholar
  38. 38.
    Luo Z.-Q., Pang J.-S., Ralph D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  39. 39.
    Pang J.-S., Fukushima M.: Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13(1–3), 111–136 (1999)CrossRefGoogle Scholar
  40. 40.
    Pang J.-S., Leyffer S.: On the global minimization of the Value-at-Risk. Optim. Methods Softw. 19(5), 611–631 (2004)CrossRefGoogle Scholar
  41. 41.
    Pardalos, P.M.: The linear complementarity problem. In: Gomez, S., Hennart, J.P. (eds.) Advances in Optimization and Numerical Analysis, pp. 39–49 (1994)Google Scholar
  42. 42.
    Prékopa A.: Probabilistic programming. In: Ruszczynski, A., Shapiro, A. (eds) Stochastic Programming, chapter 5, Handbooks in Operations Research and Management Science, vol. 10., Elsevier, Amsterdam (2003)Google Scholar
  43. 43.
    Qualizza, A., Belotti, P., Margot, F.: Linear programming relaxations of quadratically constrained quadratic programs. IMA Volume Series Springer, accepted (2010)Google Scholar
  44. 44.
    Rockafellar R.T., Uryasev S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)Google Scholar
  45. 45.
    Rockafellar R.T., Uryasev S.: Conditional value-at-risk for general loss distribution. J. Bank. Finance 26, 1443–1471 (2002)CrossRefGoogle Scholar
  46. 46.
    Saxena A., Bonami P., Lee J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124(1–2), 383–411 (2010)CrossRefGoogle Scholar
  47. 47.
    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Technical Report RC24695, IBM. Accepted for publication in Mathematical Programming August (2008)Google Scholar
  48. 48.
    Scheel H., Scholtes S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)CrossRefGoogle Scholar
  49. 49.
    Scholtes, S.: Active set methods for inverse linear complementarity problems. Technical report, Judge Institute of Management Science, Cambridge University, Cambridge CB2 1AG, November (1999)Google Scholar
  50. 50.
    Sherali H.D., Adams W.P.: A hierarchy of relaxation between the continuous and convex hull representations for zero-one programming problems. SIAM J. Dis. Math. 3, 411–430 (1990)CrossRefGoogle Scholar
  51. 51.
    Sherali H.D., Fraticelli B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Global Optim. 22, 233–261 (2002)CrossRefGoogle Scholar
  52. 52.
    Vandenbussche D., Nemhauser G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)CrossRefGoogle Scholar
  53. 53.
    Vandenbussche D., Nemhauser G.L.: A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 102(3), 531–557 (2005)CrossRefGoogle Scholar
  54. 54.
    Vapnik V.: Statistical Learning Theory. Wiley-Interscience, New York (1998)Google Scholar
  55. 55.
    Vielma J.P., Ahmed S., Nemhauser G.L.: Mixed-integer models for nonseparable piecewise linear optimization: Unifying framework and extensions. Oper. Res. 58(2), 303–315 (2010)CrossRefGoogle Scholar
  56. 56.
    Vielma J.P., Keha A.B., Nemhauser G.L.: Nonconvex, lower semicontinuous piecewise linear optimization. Dis. Optim. 5(2), 467–488 (2008)CrossRefGoogle Scholar
  57. 57.
    Xiao X., Zhang L., Zhang J.: A smoothing Newton method for a type of inverse semi-definite quadratic programming problem. J. Comput. Appl. Math. 223(1), 485–498 (2009)CrossRefGoogle Scholar
  58. 58.
    Zhang J., Zhang L.: An augmented Lagrangian method for a type of inverse quadratic programming problems. Appl. Math. Optim. 61(1), 57–83 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Jing Hu
    • 1
  • John E. Mitchell
    • 2
  • Jong-Shi Pang
    • 3
  • Bin Yu
    • 4
  1. 1.Market Analytics, Inc.EvanstonUSA
  2. 2.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA
  3. 3.Department of Industrial and Enterprise Systems EngineeringUniversity of IllinoisUrbanaUSA
  4. 4.Department of Decision Sciences and Engineering SystemsRensselaer Polytechnic InstituteTroyUSA

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