Journal of Global Optimization

, Volume 62, Issue 2, pp 263–297 | Cite as

An algorithm for global solution to bi-parametric linear complementarity constrained linear programs

  • Yu-Ching Lee
  • Jong-Shi Pang
  • John E. Mitchell


A linear program with linear complementarity constraints (LPCC) is among the simplest mathematical programs with complementarity constraints. Yet the global solution of the LPCC remains difficult to find and/or verify. In this work we study a specific type of the LPCC which we term a bi-parametric LPCC. Reformulating the bi-parametric LPCC as a non-convex quadratically constrained program, we develop a domain-partitioning algorithm that solves a series of the linear subproblems and/or convex quadratically constrained subprograms obtained by the relaxations of the complementarity constraint. The choice of an artificial constants-pair allows us to control the domain on which the partitioning is done. Numerical results of robustly solving 105 randomly generated bi-parametric LPCC instances of different structures associated with different numbers of complementarity constraints by the algorithm are presented.


Mathematical program with complementarity constraints Bi-parametric program Domain partitioning Global optimization algorithm 



Thanks to Dr. Bin Yu for his valuable comments on the numerical experiments.


  1. 1.
    Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, alphabb, for general twice-differentiable constrained NLPs—I. Theoretical advances—II. Application of theory and test problems. Comput. Chem. Eng. 22(9), 1137–1158 (1998)CrossRefGoogle Scholar
  2. 2.
    Al-Khayyal, F.A.: Generalized bilinear programming: part I. Models, applications and linear programming relaxation. Eur. J. Oper. Res. 60(3), 306–314 (1992)CrossRefMATHGoogle Scholar
  3. 3.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Anitescu, M.: On using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints. SIAM J. Optim. 15, 1203–1236 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Audet, C., Savard, G., Zghal, W.: New branch-and-cut algorithm for bilevel linear programming. J. Optim. Theory Appl. 134, 353–370 (2007)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bai, L., Mitchell, J., Pang, J.S.: Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints. Optim. Lett. 8(3), 811–822 (2014)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bai, L., Mitchell, J.E., Pang, J.S.: On convex quadratic programs with linear complementarity constraints. Comput. Optim. Appl. 54(3), 517–554 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bard, J.F., Moore, J.T.: A branch and bound algorithm for the bilevel programming problem. SIAM J. Sci. Stat. Comput. 11(2), 281–292 (1990)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Benson, H., Sen, A., Shanno, D., Vanderbei, R.: Interior-point algorithms, penalty methods and equilibrium problems. Comput. Optim. Appl. 34, 155–182 (2006)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bialas, W.F., Karwan, M.H.: Two-level linear programming. Manag. Sci. 30(8), 1004–1020 (1984)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large scale nonlinear programming. SIAM J. Optim. 9(4), 877–900 (1999)Google Scholar
  12. 12.
    Candler, W., Townsley, R.: A linear two-level programming problem. Comput. Oper. Res. 9(1), 59–76 (1982)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen, L., Goldfarb, D.: An active-set method for mathematical programs with linear complementarity constraints. Manuscript, Department of Industrial Engineering and Operations Research, Columbia University (2007)Google Scholar
  14. 14.
    Columbano, S., Fukuda, K., Jones, C.N.: An output-sensitive algorithm for multi-parametric LCPs with sufficient matrices. In: CRM Proceedings and Lecture Notes, vol. 48, pp. 73–102 (2009)Google Scholar
  15. 15.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Society for Industrial and Applied Mathematics, Philadephia, PA (2009)CrossRefMATHGoogle Scholar
  16. 16.
    Czyzyk, J., Mesnier, M.P., More, J.: The NEOS server. IEEE J. Comput. Sci. Eng. 5(3), 68–75 (1998)CrossRefGoogle Scholar
  17. 17.
    Dolan, E.: The NEOS Server 4.0 Administrative Guide. Technical Report, Argonne National Laboratory (2001)Google Scholar
  18. 18.
    Esposito, W.R., Floudas, C.A.: Global optimization in parameter estimation of nonlinear algebraic models via the error-in-variables approach. Ind. Eng. Chem. Res. 37(5), 1841–1858 (1998)CrossRefGoogle Scholar
  19. 19.
    Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optim. Methods Softw. 19, 15–40 (2004)Google Scholar
  20. 20.
    Fletcher, R., Leyffer, S., Toint, P.L.: On the global convergence of a filter-SQP algorithm. SIAM J. Optim. 13(1), 44–59 (2002)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Floudas, C.A., Gounaris, C.: A review of recent advances in global optimization. J. Glob. Optim. 45, 3–38 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Floudas, C.A., Visweswaran, V.: Primal-relaxed dual global optimization approach. J. Optim. Theory Appl. 78, 187–225 (1993)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Ghaffari-Hadigheh, A., Ghaffari-Hadigheh, H., Terlaky, T.: Bi-parametric optimal partition invariancy sensitivity analysis in linear optimization. Cent. Eur. J. Oper. Res. 16(2), 215–238 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Ghaffari-Hadigheh, A., Romanko, O., Terlaky, T.: Bi-parametric convex quadratic optimization. Optim. Methods Softw. 25, 229–245 (2010)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12(4), 979–1006 (2002)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Gropp, W., More, J.: Optimization environments and the NEOS server. In: Buhmann, M.D., Iserles, A. (eds.) Approximation Theory and Optimization: Tributes to M. J. D. Powell, pp. 167–182. Cambridge University Press, Cambridge, MA (1997)Google Scholar
  27. 27.
    Gumus, Z.H., Floudas, C.A.: Global optimization of nonlinear bilevel programming problems. J. Glob. Optim. 20, 1–31 (2001)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13, 1194–1217 (1992)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    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, 445–471 (2008)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
  31. 31.
    Kunapuli, G., Bennett, K.P., Hu, J., Pang, J.S.: Classification model selection via bilevel programming. Optim. Methods Softw. 23(4), 475–489 (2008)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    McCormick, G.: Computability of global solutions to factorable nonconvex programs: part I convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Nocedal, J., Wächter, A., Waltz, R.A.: Adaptive barrier update strategies for nonlinear interior methods. SIAM J. Optim. 19(4), 1674–1693 (2009)CrossRefMATHGoogle Scholar
  35. 35.
    Paris, Q.: Multiple optimal solutions in quadratic programming models. West. J. Agric. Econ. 8(2), 141–154 (1983)Google Scholar
  36. 36.
    Patrinos, P., Sarimveis, H.: A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings. Automatica 46(9), 1405–1418 (2010)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Patrinos, P., Sarimveis, H.: Convex parametric piecewise quadratic optimization: theory and algorithms. Automatica 47(8), 1770–1777 (2011)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Tondel, P., Johansen, T.A., Bemporad, A.: An algorithm for multi-parametric quadratic programming and explicit mpc solutions. Automatica 39(3), 489–497 (2003)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Van Voorhis, T., Al-Khayyal, F.A.: Difference of convex solution of quadratically constrained optimization problems. Eur. J. Oper. Res. 148(2), 349–362 (2003)CrossRefMATHGoogle Scholar
  40. 40.
    Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1997)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Waltz, R.A., Plantenga, T.D., Ziena Optimization, I.: Knitro user’s manual version 7.0 (2010)Google Scholar
  42. 42.
    Yu, B.: A Branch and Cut Approach to Linear Programs with Linear Complementarity Constraints. Ph.D. thesis, Rensselaer Polytechnic Institute (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yu-Ching Lee
    • 1
  • Jong-Shi Pang
    • 2
  • John E. Mitchell
    • 3
  1. 1.Department of Industrial and Enterprise Systems EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Daniel J Epstein Department of Industrial and Systems EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

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