[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

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

Abstract

In this work, we develop an adaptive, multivariate partitioning algorithm for solving nonconvex, Mixed-Integer Nonlinear Programs (MINLPs) with polynomial functions to global optimality. In particular, we present an iterative algorithm that exploits piecewise, convex relaxation approaches via disjunctive formulations to solve MINLPs that is different than conventional spatial branch-and-bound approaches. The algorithm partitions the domains of variables in an adaptive and non-uniform manner at every iteration to focus on productive areas of the search space. Furthermore, domain reduction techniques based on sequential, optimization-based bound-tightening and piecewise relaxation techniques, as a part of a presolve step, are integrated into the main algorithm. Finally, we demonstrate the effectiveness of the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare our algorithm with state-of-the-art global optimization solvers. With our novel approach, we solve several large-scale instances, some of which are not solvable by state-of-the-art solvers. We also succeed in reducing the best known optimality gap for a hard, generalized pooling problem instance.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. In the case of a higher order univariate monomial, i.e., \(x_i^5\), apply a reduction of the form \(x_i^2x_i^2x_i \Rightarrow \tilde{x}_i^2x_i \Rightarrow \tilde{\tilde{x}}_ix_i\).

  2. Global optimum is defined numerically by a tolerance, \(\epsilon \).

  3. To keep the algorithm notation simple, this detail is omitted from Algorithm 2.

  4. Exhaustiveness of the partitioning scheme implies AMP will eventually partition all other domains small enough such that AMP will pick an active partition with the global optimal whose length is \(\le \epsilon ^l_i+\epsilon ^u_i\).

  5. See [8] for more details on strategies for choosing the variables for partitioning.

  6. meyer15 is a generalized pooling problem instance. These problems are typically considered hard (bilinear) MINLP for global optimization [9, 36].

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Progr. Comput. 1(1), 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belotti, P.: Bound reduction using pairs of linear inequalities. J. Glob. Optim. 56(3), 787–819 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belotti, P., Cafieri, S., Lee, J., Liberti, L.: On feasibility based bounds tightening (2012). https://hal.archives-ouvertes.fr/file/index/docid/935464/filename/377.pdf

  5. 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  MathSciNet  MATH  Google Scholar 

  6. Bent, R., Nagarajan, H., Sundar, K., Wang, S., Hijazi, H.: A polyhedral outer-approximation, dynamic-discretization optimization solver, 1.x. Tech. rep., Los Alamos National Laboratory, Los Alamos, NM, USA (2017). https://github.com/lanl-ansi/POD.jl

  7. Bergamini, M.L., Grossmann, I., Scenna, N., Aguirre, P.: An improved piecewise outer-approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms. Comput. Chem. Eng. 32(3), 477–493 (2008)

    Article  Google Scholar 

  8. Boukouvala, F., Misener, R., Floudas, C.A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization, CDFO. Eur. J. Oper. Res. 252(3), 701–727 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib–a collection of test models for mixed-integer nonlinear programming. Inf. J. Comput. 15(1), 114–119 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cafieri, S., Lee, J., Liberti, L.: On convex relaxations of quadrilinear terms. J. Glob. Optim. 47(4), 661–685 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro, P.M.: Normalized multiparametric disaggregation: An efficient relaxation for mixed-integer bilinear problems. J. Glob. Optim. 64(4), 765–784 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Castro, P.M.: Tightening piecewise McCormick relaxations for bilinear problems. Comput. Chem. Eng. 72, 300–311 (2015)

    Article  Google Scholar 

  13. Coffrin, C., Hijazi, H.L., Van Hentenryck, P.: Strengthening convex relaxations with bound tightening for power network optimization. In: Principles and Practice of Constraint Programming, pp. 39–57. Springer, Berlin (2015)

  14. Dunning, I., Huchette, J., Lubin, M.: JuMP: A modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. D’Ambrosio, C., Lodi, A., Martello, S.: Piecewise linear approximation of functions of two variables in milp models. Oper. Res. Lett. 38(1), 39–46 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Faria, D.C., Bagajewicz, M.J.: Novel bound contraction procedure for global optimization of bilinear MINLP problems with applications to water management problems. Comput. Chem. Eng. 35(3), 446–455 (2011)

    Article  Google Scholar 

  17. Faria, D.C., Bagajewicz, M.J.: A new approach for global optimization of a class of MINLP problems with applications to water management and pooling problems. AIChE J. 58(8), 2320–2335 (2012)

    Article  Google Scholar 

  18. Grossmann, I.E., Trespalacios, F.: Systematic modeling of discrete-continuous optimization models through generalized disjunctive programming. AIChE J. 59(9), 3276–3295 (2013)

    Article  Google Scholar 

  19. Hasan, M., Karimi, I.: Piecewise linear relaxation of bilinear programs using bivariate partitioning. AIChE J. 56(7), 1880–1893 (2010)

    Article  Google Scholar 

  20. Hijazi, H., Coffrin, C., Van Hentenryck, P.: Convex quadratic relaxations for mixed-integer nonlinear programs in power systems. Math. Progr. Comput. 9(3), 321–367 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes. J. Optim. Theory Appl. 30(1), 127–129 (1980)

    Article  MATH  Google Scholar 

  22. Horst, R., Pardalos, P.M.: Handbook of global optimization, vol. 2. Springer, Berlin (2013)

    MATH  Google Scholar 

  23. Horst, R., Tuy, H.: Global optimization: deterministic approaches. Springer, Berlin (2013)

    MATH  Google Scholar 

  24. Karuppiah, R., Grossmann, I.E.: Global optimization for the synthesis of integrated water systems in chemical processes. Comput. Chem. Eng. 30(4), 650–673 (2006)

    Article  Google Scholar 

  25. Kocuk, B., Dey, S.S., Sun, X.A.: Strong SOCP relaxations for the optimal power flow problem. Oper. Res. 64(6), 1177–1196 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kolodziej, S.P., Grossmann, I.E., Furman, K.C., Sawaya, N.W.: A discretization-based approach for the optimization of the multiperiod blend scheduling problem. Comput. Chem. Eng. 53, 122–142 (2013)

    Article  Google Scholar 

  27. Li, H.L., Huang, Y.H., Fang, S.C.: A logarithmic method for reducing binary variables and inequality constraints in solving task assignment problems. Inf. J. Comput. 25(4), 643–653 (2012)

    Article  MathSciNet  Google Scholar 

  28. Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15(1), 1–17 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lu, M., Nagarajan, H., Bent, R., Eksioglu, S., Mason, S.: Tight piecewise convex relaxations for global optimization of optimal power flow. In: Power Systems Computation Conference (PSCC), pp. 1–7. IEEE (2018)

  30. Lu, M., Nagarajan, H., Yamangil, E., Bent, R., Backhaus, S., Barnes, A.: Optimal transmission line switching under geomagnetic disturbances. IEEE Trans. Power Syst. 33(3), 2539–2550 (2018). https://doi.org/10.1109/TPWRS.2017.2761178

    Article  Google Scholar 

  31. Luedtke, J., Namazifar, M., Linderoth, J.: Some results on the strength of relaxations of multilinear functions. Math. Progr. 136(2), 325–351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part i–convex underestimating problems. Math. Progr. 10(1), 147–175 (1976)

    Article  MATH  Google Scholar 

  33. Meyer, C.A., Floudas, C.A.: Global optimization of a combinatorially complex generalized pooling problem. AIChE J. 52(3), 1027–1037 (2006)

    Article  Google Scholar 

  34. Misener, R., Floudas, C.: Generalized pooling problem (2011). Available from Cyber-Infrastructure for MINLP [www.minlp.org, a collaboration of Carnegie Mellon University and IBM Research] at: www.minlp.org/library/problem/index.php?i=123

  35. Misener, R., Floudas, C.A.: GloMIQO: Global mixed-integer quadratic optimizer. J. Glob. Optim. 57(1), 3–50 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Misener, R., Thompson, J.P., Floudas, C.A.: APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput. Chem. Eng. 35(5), 876–892 (2011)

    Article  Google Scholar 

  37. Mouret, S., Grossmann, I.E., Pestiaux, P.: Tightening the linear relaxation of a mixed integer nonlinear program using constraint programming. In: Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, pp. 208–222. Springer, Berlin (2009)

  38. Nagarajan, H., Lu, M., Yamangil, E., Bent, R.: Tightening McCormick relaxations for nonlinear programs via dynamic multivariate partitioning. In: International Conference on Principles and Practice of Constraint Programming, pp. 369–387. Springer, Berlin (2016)

  39. Nagarajan, H., Pagilla, P., Darbha, S., Bent, R., Khargonekar, P.: Optimal configurations to minimize disturbance propagation in manufacturing networks. In: American Control Conference (ACC), 2017, pp. 2213–2218. IEEE (2017)

  40. Nagarajan, H., Sundar, K., Hijazi, H., Bent, R.: Convex hull formulations for mixed-integer multilinear functions. In: Proceedings of the XIV International Global Optimization Workshop (LEGO 18) (2018)

  41. Nagarajan, H., Yamangil, E., Bent, R., Van Hentenryck, P., Backhaus, S.: Optimal resilient transmission grid design. In: Power Systems Computation Conference (PSCC), 2016, pp. 1–7. IEEE (2016)

  42. Puranik, Y., Sahinidis, N.V.: Domain reduction techniques for global NLP and MINLP optimization. Constraints 22(3), 338–376 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rikun, A.D.: A convex envelope formula for multilinear functions. J. Glob. Optim. 10, 425–437 (1997). https://doi.org/10.1023/A:1008217604285

    Article  MathSciNet  MATH  Google Scholar 

  44. Ruiz, J.P., Grossmann, I.E.: Global optimization of non-convex generalized disjunctive programs: a review on reformulations and relaxation techniques. J. Glob. Optim. 67(1–2), 43–58 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19(4), 403–424 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sahinidis, N.V.: Baron: A general purpose global optimization software package. J. Glob. Optim. 8(2), 201–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. Speakman, E.E.: Volumetric Guidance for Handling Triple Products in Spatial Branch-and-Bound by. Ph.D. thesis, University of Michigan (2017)

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Teles, J.P., Castro, P.M., Matos, H.A.: Univariate parameterization for global optimization of mixed-integer polynomial problems. Eur. J. Oper. Res. 229(3), 613–625 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Trespalacios, F., Grossmann, I.E.: Cutting plane algorithm for convex generalized disjunctive programs. Inf. J. Comput. 28(2), 209–222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Progr. 128(1), 49–72 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wicaksono, D.S., Karimi, I.: Piecewise MILP under-and overestimators for global optimization of bilinear programs. AIChE J. 54(4), 991–1008 (2008)

    Article  Google Scholar 

  54. Wu, F., Nagarajan, H., Zlotnik, A., Sioshansi, R., Rudkevich, A.: Adaptive convex relaxations for gas pipeline network optimization. In: American Control Conference (ACC), 2017, pp. 4710–4716. IEEE (2017)

Download references

Acknowledgements

The work was funded by the Center for Nonlinear Studies (CNLS) at LANL and the LANL’s directed research and development project “POD: A Polyhedral Outer-approximation, Dynamic-discretization optimization solver”. Work was carried out under the auspices of the U.S. DOE under Contract No. DE-AC52-06NA25396.

Author information

Authors and Affiliations

  1. Center for Nonlinear Studies, Los Alamos National Laboratory, New Mexico, USA

    Harsha Nagarajan, Russell Bent & Kaarthik Sundar

  2. Department of Industrial Engineering, Clemson University, Clemson, USA

    Mowen Lu & Site Wang

Authors
  1. Harsha Nagarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mowen Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Site Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Russell Bent
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kaarthik Sundar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harsha Nagarajan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A: Appendix

A: Appendix

1.1 A.1: Sensitivity analysis of \(\Delta \)

One of the important details of MINLP algorithms and approaches is their parameterization. As seen in the earlier sections, AMP is no different. The quality of the solutions depend heavily on the choice of \(\Delta \). However, in spite of this problem specific dependence, it is often interesting to identify reasonable default values. Table 8 presents computational results on all instances for different choices of \(\Delta \). From these results, AMP is most effective when \(\Delta \) is between 4 and 10.

Table 8 This table shows a sensitivity analysis of AMP’s performance to the choice of \(\Delta \)
Full size table

1.2 A.2: Logarithmic and linear encodings of partition variables

In Sect. 2, the discussion on piecewise convex relaxations described formulations that encoded the partition variables with a linear number of variables and a logarithmic number of variables [52]. Table 9 compares the performance of AMP using both formulations. Despite fewer variables in the logarithmic formulation, this encoding is only effective on a few problems, generally on problems that require a significant number of partitions. These results suggest that when the logarithmic encoding has nearly the same number of partition variables as the linear encoding, the linear encoding is more effective.

Table 9 This table compares the logarithmic formulation of partition variables with the linear representation
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nagarajan, H., Lu, M., Wang, S. et al. An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs. J Glob Optim 74, 639–675 (2019). https://doi.org/10.1007/s10898-018-00734-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-018-00734-1

Keywords

Search

Navigation