Convexification Techniques for Linear Complementarity Constraints

  • Trang T. Nguyen
  • Mohit Tawarmalani
  • Jean-Philippe P. Richard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6655)

Abstract

We develop convexification techniques for linear programs with linear complementarity constraints (LPCC). In particular, we generalize the reformulation-linearization technique of [9] to complementarity problems and discuss how it reduces to the standard technique for binary mixed-integer programs. Then, we consider a class of complementarity problems that appear in KKT systems and show that its convex hull is that of a binary mixed-integer program. For this class of problems, we study further the case where a single complementarity constraint is imposed and show that all nontrivial facet-defining inequalities can be obtained through a simple cancel-and-relax procedure. We use this result to identify special cases where McCormick inequalities suffice to describe the convex hull and other cases where these inequalities are not sufficient.

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References

  1. 1.
    Balas, E.: Disjunctive programming: Properties of the convex hull of feasible points. Discrete Applied Mathematics 89(1-3), 3–44 (1998); original manuscript was published as a technical report in 1974MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Mathematical Programming 58, 295–324 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Farias, I.R., Johnson, E.L., Nemhauser, G.L.: Facets of the complementarity knapsack polytope. Mathematics of Operations Research 27, 210–226 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Review 39, 669–713 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hu, J., Mitchell, J.E., Pang, J.S., Yu, B.: On linear programs with linear complementarity constraints. Journal of Global Optimization (to appear)Google Scholar
  6. 6.
    Jeroslow, R.G.: Cutting-planes for complementarity constraints. SIAM Journal on Control and Optimization 16(1), 56–62 (1978)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  8. 8.
    Rockafellar, R.T., Wets, R.J.B.: Variational analysis. A Series of Comprehensive Studies in Mathematics. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3, 411–430 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sherali, H.D., Krishnamurthy, R.S., Al-Khayyal, F.A.: Enumeration approach for linear complementarity problems based on a reformulation-linearization technique. Journal of Optimization Theory and Applications 99, 481–507 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Tawarmalani, M.: Inclusion certificates and disjunctive programming. presented in Operations Research Seminar at GSIA, Carnegie Mellon University (2006)Google Scholar
  12. 12.
    Tawarmalani, M.: Inclusion certificates and simultaneous convexification of functions. Mathematical Programming (2010) (submitted)Google Scholar
  13. 13.
    Vandenbussche, D., Nemhauser, G.L.: A polyhedral study of nonconvex quadratic programs with box constraints. Mathematical Programming 102, 531–557 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Trang T. Nguyen
    • 1
  • Mohit Tawarmalani
    • 2
  • Jean-Philippe P. Richard
    • 1
  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaUSA
  2. 2.Krannert School of ManagementPurdue UniversityUSA

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