Skip to main content
SpringerLink
Log in

Reduced RLT representations for nonconvex polynomial programming problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  Google Scholar 

  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, France

  3. Caprara A., Locatelli M.: Global optimization problems and domain reduction strategies. Math. Program. 125(1), 123–137 (2010)

    Article  Google Scholar 

  4. Dalkiran, E.: http://filebox.vt.edu/users/dalkiran/website/#ProblemRRLT (2011)

  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)

  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. Gill P.E., Murray W., Saunders M.A.: An SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005)

    Article  Google Scholar 

  8. Liberti L.: Linearity embedded in nonconvex programs. J. Glob. Optim. 33, 157–196 (2005)

    Article  Google Scholar 

  9. Liberti, L.: Effective RLT tightening in continuous bilinear programs. Internal Report 2003.18, Politecnico di Milano, 20133 Milano, Italy

  10. Liberti L., Pantelides C.C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim. 36, 161–189 (2006)

    Article  Google Scholar 

  11. MATLAB: version 7.6.0 (R2008a). The MathWorks Inc., Natick, MA (2008)

  12. Ryoo H.S., Sahinidis N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996)

    Article  Google Scholar 

  13. Sahinidis, N.V., Tawarmalani, M.: BARON 9.0.6: Global optimization of mixed-integer nonlinear programs. User’s Manual (2010)

  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. 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. 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)

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  20. Sherali H.D., Tuncbilek C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7, 1–31 (1995)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  23. Shor N.Z.: Dual quadratic estimates in polynomial and Boolen programming. Ann. Oper. Res. 25, 163–168 (1990)

    Article  Google Scholar 

  24. Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    Hanif D. Sherali & Evrim Dalkiran

  2. CNRS LIX, École Polytechnique, 91128, Palaiseau, France

    Leo Liberti

Authors
  1. Hanif D. Sherali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evrim Dalkiran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Leo Liberti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evrim Dalkiran.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sherali, H.D., Dalkiran, E. & Liberti, L. Reduced RLT representations for nonconvex polynomial programming problems. J Glob Optim 52, 447–469 (2012). https://doi.org/10.1007/s10898-011-9757-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9757-3

Keywords

Search

Navigation