Journal of Global Optimization

, Volume 52, Issue 3, pp 447–469 | Cite as

Reduced RLT representations for nonconvex polynomial programming problems

  • Hanif D. Sherali
  • Evrim DalkiranEmail author
  • Leo Liberti


This paper explores equivalent, reduced size Reformulation-Linearization Technique (RLT)-based formulations for polynomial programming problems. Utilizing a basis partitioning scheme for an embedded linear equality subsystem, we show that a strict subset of RLT defining equalities imply the remaining ones. Applying this result, we derive significantly reduced RLT representations and develop certain coherent associated branching rules that assure convergence to a global optimum, along with static as well as dynamic basis selection strategies to implement the proposed procedure. In addition, we enhance the RLT relaxations with v-semidefinite cuts, which are empirically shown to further improve the relative performance of the reduced RLT method over the usual RLT approach. We present computational results for randomly generated instances to test the different proposed reduction strategies and to demonstrate the improvement in overall computational effort when such reduced RLT mechanisms are employed.


Reformulation-Linearization Technique (RLT) Reduced basis techniques Polynomial programs Global optimization Semidefinite cuts BARON 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Belotti P., Lee J., Liberti L., Margot F., Wächter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)CrossRefGoogle Scholar
  2. 2.
    Cafieri, S., Hansen, P., Liberti, L., Letocart, L., Messine, F.: Tight and compact convex relaxations for polynomial programming problems. Manuscript, LIX, École Polytechnique, F-91128 Palaiseau, FranceGoogle Scholar
  3. 3.
    Caprara A., Locatelli M.: Global optimization problems and domain reduction strategies. Math. Program. 125(1), 123–137 (2010)CrossRefGoogle Scholar
  4. 4.
  5. 5.
    Dalkiran, E., Sherali, H. D.: Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality. Manuscript, Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA (2011)Google Scholar
  6. 6.
    Floudas C.A., Visweswaran V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217–270. Kluwer, Boston, MA (1995)Google Scholar
  7. 7.
    Gill P.E., Murray W., Saunders M.A.: An SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005)CrossRefGoogle Scholar
  8. 8.
    Liberti L.: Linearity embedded in nonconvex programs. J. Glob. Optim. 33, 157–196 (2005)CrossRefGoogle Scholar
  9. 9.
    Liberti, L.: Effective RLT tightening in continuous bilinear programs. Internal Report 2003.18, Politecnico di Milano, 20133 Milano, ItalyGoogle Scholar
  10. 10.
    Liberti L., Pantelides C.C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim. 36, 161–189 (2006)CrossRefGoogle Scholar
  11. 11.
    MATLAB: version 7.6.0 (R2008a). The MathWorks Inc., Natick, MA (2008)Google Scholar
  12. 12.
    Ryoo H.S., Sahinidis N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)CrossRefGoogle Scholar
  13. 13.
    Sahinidis, N.V., Tawarmalani, M.: BARON 9.0.6: Global optimization of mixed-integer nonlinear programs. User’s Manual (2010)Google Scholar
  14. 14.
    Sherali H.D., Adams W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Boston, MA (1999)Google Scholar
  15. 15.
    Sherali, H.D., Dalkiran, E.: Combined bound-grid-factor constraints for enhancing RLT relaxations for polynomial programs. J. Glob. Optim. (2011). doi: 10.1007/s10898-010-9639-0
  16. 16.
    Sherali, H.D., Dalkiran, E., Desai, J.: Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts. Comput. Optim. Appl. (accepted for publication)Google Scholar
  17. 17.
    Sherali H.D., Fraticelli B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Glob. Optim. 22, 233–261 (2002)CrossRefGoogle Scholar
  18. 18.
    Sherali H.D., Smith J.C., Adams W.P.: Reduced first-level representations via the reformulation-linearization technique: results, counterexamples, and computations. Discrete Appl. Math. 101, 247–267 (2000)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Sherali H.D., Tuncbilek C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7, 1–31 (1995)CrossRefGoogle Scholar
  21. 21.
    Sherali H.D., Tuncbilek C.H.: Comparison of two reformulation-linearization technique based linear programming relaxations for polynomial programming problems. J. Glob. Optim. 10, 381–390 (1997)CrossRefGoogle Scholar
  22. 22.
    Sherali H.D., Tuncbilek C.H.: New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Oper. Res. Lett. 21(1), 1–9 (1997)CrossRefGoogle Scholar
  23. 23.
    Shor N.Z.: Dual quadratic estimates in polynomial and Boolen programming. Ann. Oper. Res. 25, 163–168 (1990)CrossRefGoogle Scholar
  24. 24.
    Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Hanif D. Sherali
    • 1
  • Evrim Dalkiran
    • 1
    Email author
  • Leo Liberti
    • 2
  1. 1.Grado Department of Industrial and Systems EngineeringVirginia TechBlacksburgUSA
  2. 2.CNRS LIX, École PolytechniquePalaiseauFrance

Personalised recommendations