Journal of Global Optimization

, Volume 56, Issue 2, pp 727–736 | Cite as

Handelman rank of zero-diagonal quadratic programs over a hypercube and its applications

  • Myoung-Ju Park
  • Sung-Pil Hong


It has been observed that the Handelman’s certificate of positivity of a polynomial over a compact polyhedron offers a hierarchical relaxation scheme for polynomial programs. The Handelman hierarchy seems particularly suitable for a class of combinatorial optimizations that are formulated as a zero-diagonal quadratic program over a hypercube. In this paper, we present an error analysis of Handelman hierarchy applied to the special class of polynomial programs and its implications in the computation of the combinatorial optimization problems.


Polynomial optimization Handelman hierarchy The maximum cut problem The stable set problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Bollobás, B., Lovász, L.: Proving integrality gaps without knowing the linear program. In: FOCS ‘02: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pp. 313–322 (2002)Google Scholar
  2. 2.
    Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58(3), 295–324 (1993)CrossRefGoogle Scholar
  3. 3.
    Cheung K.K.H.: On Lovász-Schrijver lift-and-project procedures on the Dantzig–Fulkerson–Johnson relaxation of the TSP. SIAM J. Optim. 16(2), 380–399 (2005)CrossRefGoogle Scholar
  4. 4.
    Cheung K.K.H.: Computation of the Lasserre ranks of some polytopes. Math. Oper. Res. 32(1), 88–94 (2007)CrossRefGoogle Scholar
  5. 5.
    Cook W., Dash S.: On the matrix-cut rank of polyhedra. Math. Oper. Res. 26(1), 19–30 (2001)CrossRefGoogle Scholar
  6. 6.
    De Klerk E., Laurent M.: Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube. SIAM J. Optim. 20(6), 3104–3120 (2010)CrossRefGoogle Scholar
  7. 7.
    Handelman D.: Representing polynomials by positive linear functions on compact convex polyhedra. Pac. J. Math. 132(1), 35–62 (1988)CrossRefGoogle Scholar
  8. 8.
    Harant J.: Some news about the independence number of a graph. Discuss Math Graph Theory 20(1), 71–80 (2000)CrossRefGoogle Scholar
  9. 9.
    Hong S.-P., Tunçel L.: Unification of lower-bound analyses of the lift-and-project rank of combinatorial optimization polyhedra. Discret. Appl. Math. 156(1), 25–41 (2008)CrossRefGoogle Scholar
  10. 10.
    Lasserre J.B.: Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res. 27(2), 347–360 (2002)CrossRefGoogle Scholar
  11. 11.
    Laurent M.: A Comparison of the Sherali-Admas, Lovász-Schrijver and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)CrossRefGoogle Scholar
  12. 12.
    Lovász L., Schrijver A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Discret. Math. 1(2), 166–190 (1991)Google Scholar
  13. 13.
    Park M.-J., Hong S.-P.: Rank of handelman hierarchy for max-cut. Oper. Res. Lett. 39(5), 323–328 (2011)CrossRefGoogle Scholar
  14. 14.
    Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)CrossRefGoogle Scholar
  15. 15.
    Schmüdgen K.: The K-moment problem for compact semi-algebraic sets. Mathematische Annalen 289(2), 203–206 (1991)CrossRefGoogle Scholar
  16. 16.
    Sherali H.D., Adams W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero–one programmings. SIAM J. Discret. Math. 3(3), 411–430 (1990)CrossRefGoogle Scholar
  17. 17.
    Sherali H.D., Tuncbilek C.H.: A global optimization algorithm for polynomial programming problems using a reformulation linearization technique. J. Glob. Optim. 2(1), 101–112 (1992)CrossRefGoogle Scholar
  18. 18.
    Stephen T., Tunçel L.: On a representation of the matching polytope via semidefinite liftings. Math. Oper. Res. 24(1), 1–7 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Department of Industrial and Management Systems EngineeringKyung Hee UniversityYongin-si, Kyunggi-doRepublic of Korea
  2. 2.Department of Industrial EngineeringSeoul National UniversitySeoulRepublic of Korea

Personalised recommendations