Skip to main content

RLT-POS: Reformulation-Linearization Technique-based optimization software for solving polynomial programming problems

Abstract

In this paper, we introduce a Reformulation-Linearization Technique-based open-source optimization software for solving polynomial programming problems (RLT-POS). We present algorithms and mechanisms that form the backbone of RLT-POS, including constraint filtering techniques, reduced RLT representations, and semidefinite cuts. When implemented individually, each model enhancement has been shown in previous papers to significantly improve the performance of the standard RLT procedure. However, the coordination between different model enhancement techniques becomes critical for an improved overall performance since special structures in the original formulation that work in favor of a particular technique might be lost after implementing some other model enhancement. More specifically, we discuss the coordination between (1) constraint elimination via filtering techniques and reduced RLT representations, and (2) semidefinite cuts for sparse problems. We present computational results using instances from the literature as well as randomly generated problems to demonstrate the improvement over a standard RLT implementation and to compare the performances of the software packages BARON, COUENNE, and SparsePOP with RLT-POS.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: a global optimization method for general constrained nonconvex problems. J. Global Optim. 7, 337–363 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  2. Anstreicher, K.M.: Semidefinite programming versus the Reformulation-Linearization Technique for nonconvex quadratically constrained quadratic programming. J. Global Optim. 43(2–3), 471–484 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136(2), 233–251 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. Balas, E., Ceria, S., Cornuejols, G.: Mixed 0–1 programming by lift-and-project in a branch-and-cut framework (1996)

  5. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)

    Book  MATH  Google Scholar 

  6. 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)

    MathSciNet  Article  MATH  Google Scholar 

  7. 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 

  8. Dalkiran, E., Sherali, H.: Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality. J. Global Optim. 57(4), 1147–1172 (2013)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  10. Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin Heidelberg, New York (1981)

    Book  MATH  Google Scholar 

  11. Ibm, ILOG CPLEX Optimization Studio. http://www.ilog.com/products/cplex

  12. Lasserre, J.B.: Semidefinite programming vs. LP relaxations for polynomial programming. Math. Operations Res. 27(2), 347–360 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. Lasserre, J.B.: Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17(3), 822–843 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  14. Laurent, M., Rendl, F.: Semidefinite Programming and Integer Programming. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook on Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  17. MATLAB: version 7.12.0 (R2011a). The MathWorks Inc., Natick, Massachusetts (2011)

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

    MathSciNet  Article  MATH  Google Scholar 

  19. Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer Academic Publishers, Boston (1999)

    Book  MATH  Google Scholar 

  20. Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique (RLT) for semi-infinite and convex programs under mixed 0–1 and general discrete restrictions. Discrete Appl. Math. 157(6), 1319–1333 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  21. Sherali, H.D., Dalkiran, E.: Combined bound-grid-factor constraints for enhancing RLT relaxations for polynomial programs. J. Global Optim. 51(3), 377–393 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  22. 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)

    MathSciNet  Article  MATH  Google Scholar 

  23. Sherali, H.D., Dalkiran, E., Liberti, L.: Reduced RLT representations for nonconvex polynomial programs. J. Global Optim. 52(3), 447–469 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  24. Sherali, H.D., Fraticelli, B.M.P.: Enhancing RLT relaxations via a new class of semidefinite cuts. J. Global Optim. 22(1–4), 233–261 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  25. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique. J. Global Optim. 2(1), 101–112 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  26. Sherali, H.D., Tuncbilek, C.H.: New reformulation linearization/convexification relaxations for univariate and multivariate polynomial programming problems. Operations Res. Lett. 21(1), 1–9 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  27. Sherali, H.D., Wang, H.: Global optimization of nonconvex factorable programming problems. Math. Program. 89(3), 459–478 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  28. Sturm, J.F.: Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optimiz. Methods Softw. 11(1–4), 625–653 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  29. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  30. 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)

    MathSciNet  Article  MATH  Google Scholar 

  31. 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), 15:1–15:13 (2008)

    MathSciNet  Article  Google Scholar 

  32. Zorn, K., Sahinidis, N.V.: Global optimization of general nonconvex problems with intermediate polynomial structures. J. Global Optim. 59(2–3), 673–693 (2014)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors gratefully acknowledge Nick Sahinidis at Carnegie Mellon University for permitting the use of the BARON solver.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Evrim Dalkiran.

Additional information

This research has been supported by the National Science Foundation under Grant No. CMMI-0969169.

Appendix

Appendix

PP1 (Problem 119 in [10], where the parameter data is specified in (22) and in Table 13):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^2 + x_i + 1\right) \left( x_j^2 + x_j + 1\right) \\ \text {subject to}&\quad \nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1,\ldots , 8 \\&\quad 0 \le x_i \le 5, \quad \forall j=1, \ldots , 16. \end{aligned}$$
$$\begin{aligned} a_{ij} \!=\! 1, \forall (\!i, j\!)\!\in & {} \! \{ (\!1, 1), (\!1, 4), (\!1, 7\!), (\!1, 8), (\!1, 16), (\!2, 2), (\!2, 3), (\!2, 7), (\!2, 10), (\!3, 3), \nonumber \\&(\!3, 7), (\!3, 9), (\!3, 10), (3, 14), (4, 4), (4, 7),(4, 11), (4, 15), (5, 5),\nonumber \\&(5, 6), (5, 10), (5, 12), (5, 16), (6, 6), (6, 8), (6, 15), \, (7, 7), \, (7, 11),\nonumber \\&(7, 13), \, (8, 8),\, (8, 10), \, (8, 15), \, (9, 9), \, (9, 12), \, (9, 16), (10, 10), \nonumber \\&(\!10, 14\!), (\!11, 11\!), (\!11, 13\!), (\!12, 12\!), (\!12, 14\!), (\!13, 13\!), (\!13, 14\!), (\!14, 14\!), \nonumber \\&(\!15, 15\!), (16, 16) \} \end{aligned}$$
(22)
Table 13 Parameter data for Problems PP1, PP2, and PP3 (Problem 119 in [10])

PP2 (Problem 119 in [10]—as given by Problem PP1, but with a modified objective function):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^3 + x_i + 1\right) \left( x_j^2 + x_j + 1\right) \\ \text {subject to}&\nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1, \ldots , 8 \\&\quad 0 \le x_i \le 5,\quad \forall j=1, \ldots , 16. \end{aligned}$$

PP3 (Problem 119 in [10]—as given by Problem PP1, but with a modified objective function):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^3 + x_i^2 + 1\right) \left( x_j^3 + x_j^2 + 1\right) \\ \text {subject to}&\nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1, \ldots , 8 \\&\quad 0 \le x_i \le 5,\quad \forall j=1, \ldots , 16. \end{aligned}$$

PP4 (Problem 49 in [10]—, along with imposed variable bounds):

$$\begin{aligned} \text {Minimize}&\quad (x_1 - x_2)^2 + (x_3 -1)^2 + (x_4 -1)^4 + (x_5-1)^6 \\ \text {subject to}&\nonumber \\&\quad x_1 + x_2 + x_3 + 4x_4 = 7 \\&\quad x_3 + 5x_5 = 6 \\&\quad 0 \le x_j \le 5, \quad \forall j=1, \ldots , 5. \end{aligned}$$

PP5 (Problem 50 in [10]—, along with imposed variable bounds):

$$\begin{aligned} \text {Minimize}&\quad (x_1 - x_2)^2 + (x_2 - x_3)^2 + (x_3 -x_4)^4 + (x_4 -x_5)^2 \\ \text {subject to}&\nonumber \\&\quad x_1 + 2x_2 + 3x_3 = 6 \\&\quad x_2 + 2x_3 + 3x_4 = 6 \\&\quad x_3 + 2x_4 + 3x_5 = 6 \\&\quad 0 \le x_j \le 5,\quad \forall j=1, \ldots , 5. \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dalkiran, E., Sherali, H.D. RLT-POS: Reformulation-Linearization Technique-based optimization software for solving polynomial programming problems. Math. Prog. Comp. 8, 337–375 (2016). https://doi.org/10.1007/s12532-016-0099-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-016-0099-5

Keywords

  • Reformulation-Linearization Technique (RLT)
  • Open-source code
  • Constraint filtering strategies
  • Valid inequalities
  • Reduced RLT representations
  • Polynomial programming

Mathematics Subject Classification

  • 65K05
  • 90-08
  • 90C26
  • 90C57