Journal of Global Optimization

, Volume 57, Issue 4, pp 1147–1172 | Cite as

Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality

  • Evrim DalkiranEmail author
  • Hanif D. Sherali


In this paper, we propose two sets of theoretically filtered bound-factor constraints for constructing reformulation-linearization technique (RLT)-based linear programming (LP) relaxations for solving polynomial programming problems. We establish related theoretical results for convergence to a global optimum for these reduced sized relaxations, and provide insights into their relative sizes and tightness. Extensive computational results are provided to demonstrate the relative effectiveness of the proposed theoretical filtering strategies in comparison to the standard RLT and a prior heuristic filtering technique using problems from the literature as well as randomly generated test cases.


Reformulation-linearization technique (RLT) Filtering strategies  Polynomial programming Branch-and-bound 



This research has been supported by the National Science Foundation under Grant No. CMMI-0969169. The authors also thank two anonymous referees for their constructive and insightful comments that have helped improve the substance and presentation in this paper


  1. 1.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)CrossRefGoogle Scholar
  2. 2.
    Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43(2–3), 471–484 (2009)CrossRefGoogle Scholar
  3. 3.
    Applegate, D., Dash, S., Cook, W., Espinoza, D.: QSopt_ex.
  4. 4.
    Bao, X., Sahinidis, N.V., Tawarmalani, M.: Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs. Optim. Methods Softw. 24(4–5), 485–504 (2009)CrossRefGoogle Scholar
  5. 5.
    Cafieri, S., Hansen, P., Létocart, L., Liberti, L., Messine, F.: Compact relaxations for polynomial programming problems. In: Klasing, R. (ed.) Experimental Algorithms, Lecture Notes in Computer Science, vol. 7276, pp. 75–86. Springer, Berlin (2012)Google Scholar
  6. 6.
    Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM Rev. 47(1), 99–131 (2005)CrossRefGoogle Scholar
  7. 7.
    Haverly, C.A.: Studies of the behavior of recursion for the pooling problem. SIGMAP Bull. 25, 19–28 (1978)CrossRefGoogle Scholar
  8. 8.
    Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1981)CrossRefGoogle Scholar
  9. 9.
    ILOG Cplex 12.3: Reference manual, 2012.
  10. 10.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)CrossRefGoogle Scholar
  11. 11.
    Lasserre, J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)CrossRefGoogle Scholar
  12. 12.
    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)CrossRefGoogle Scholar
  13. 13.
    Liberti, L.: Linearity embedded in nonconvex programs. J. Glob. Optim. 33, 157–196 (2005)CrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)CrossRefGoogle Scholar
  16. 16.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefGoogle Scholar
  17. 17.
    Nataraj, P.S.V., Arounassalame, M.: Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm. J. Glob. Optim. 49(2), 185–212 (2011)CrossRefGoogle Scholar
  18. 18.
    Rikun, A.D.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10(4), 425–437 (1997)CrossRefGoogle Scholar
  19. 19.
    Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Comput. Chem. Eng. 19(5), 551–566 (1995)Google Scholar
  20. 20.
    Sherali, H.D.: Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. ACTA Mathematica Vietnamica 22(1), 245–270 (1997)Google Scholar
  21. 21.
    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Boston (1999)CrossRefGoogle Scholar
  22. 22.
    Sherali, H.D., Dalkiran, E.: Combined bound-grid-factor constraints for enhancing RLT relaxations for polynomial programs. J. Glob. Optim. 51(3), 377–393 (2011)CrossRefGoogle Scholar
  23. 23.
    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. 52(2), 483–506 (2012)CrossRefGoogle Scholar
  24. 24.
    Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programs. J. Glob. Optim. 52(3), 447–469 (2012)CrossRefGoogle Scholar
  25. 25.
    Sherali, H.D., Fraticelli, B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Glob. Optim. 22(1–4), 233–261 (2002)CrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    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
  28. 28.
    Sherali, H.D., Wang, H.: Global optimization of nonconvex factorable programming problems. Math. Program. 89(3), 459–478 (2001)CrossRefGoogle Scholar
  29. 29.
    Shor, N.Z.: Dual quadratic estimates in polynomial and Boolean programming. Ann. Oper. Res. 25, 163–168 (1990)CrossRefGoogle Scholar
  30. 30.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program. 99, 563–591 (2004)CrossRefGoogle Scholar
  31. 31.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)CrossRefGoogle Scholar
  32. 32.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17(1), 218–242 (2006)CrossRefGoogle Scholar
  33. 33.
    Waki, H., Kim, S., Kojima, M., Muramatsu, M., Sugimoto, H.: SparsePOP—A sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Softw. 35(2), 15:1–13 (2008)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringWayne State UniversityDetroitUSA
  2. 2.Grado Department of Industrial and Systems EngineeringVirginia TechBlacksburgUSA

Personalised recommendations