Skip to main content
SpringerLink
Log in

Extended reverse-convex programming: an approximate enumeration approach to global optimization

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

Abstract

A new approach to solving a large class of factorable nonlinear programming (NLP) problems to global optimality is presented in this paper. Unlike the traditional strategy of partitioning the decision-variable space employed in many branch-and-bound methods, the proposed approach approximates the NLP problem by a reverse-convex programming (RCP) problem to a controlled precision, with the latter then solved by an enumerative search. To establish the theoretical guarantees of the method, the notion of “RCP regularity” is introduced and it is proven that enumeration is guaranteed to yield a global optimum when the RCP problem is regular. An extended RCP algorithmic framework is then presented and its performance is examined for a small set of test problems.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)

    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. Bunin, G. A.: Extended reverse convex programming: An active-set approach to global optimization, pp 1–48. arXiv [math.OC] 1308.2828v1 (2013)

  4. Bunin, G. A.: On the piecewise-concave approximations of functions, pp 1-4. arXiv [math.OC] 1403.3882 (2014)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. CVX Research, Inc.: CVX: MATLAB software for disciplined convex programming, version 2.0 beta. http://cvxr.com/cvx (2012). Accessed September 2012

  7. Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Manag. Sci. 15(9), 550–569 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999)

    Book  MATH  Google Scholar 

  9. Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.: Global optimization in the 21st century: Advances and challenges. Comput. Chem. Eng. 29, 1185–1202 (2005)

    Article  Google Scholar 

  10. Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Inform. Process. Lett. 1, 132–133 (1972)

    Article  MATH  Google Scholar 

  11. Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs, recent advances in learning and control (a tribute to M. Vidyasagar). In: Blondel, V., Boyd, S., Kimura, H. (eds.) Lecture Notes in Control and Information Sciences, pp. 95-110, Springer (2008)

  12. Hillestad, R.J., Jacobsen, S.E.: Reverse convex programming. Appl. Math. Optim. 6, 63–78 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. Horst, R., Pardalos, P.M., Thoai, N.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)

    MATH  Google Scholar 

  14. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I - convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. McKeown, P.: Extreme point ranking algorithms: A computational survey. In: White, W.W. (ed.) Computers and Mathematical Programming: Proceedings of the Bicentennial Conference on Mathematical Programming, pp. 216–222. U.S. Government Printing Office, Washington (1976)

  16. Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13(1), 271–369 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Neumaier, A., Shcherbina, O., Huyer, W., Vinkó, T.: A comparison of complete global optimization solvers. Math. Program. 103(2), 335–356 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nowak, I.: Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming. Birkhäuser Verlag, Basel (2005)

    MATH  Google Scholar 

  19. Pardalos, P.M., Rosen, J.: Methods for global concave minimization: A bibliographic survey. SIAM Rev. 28(3), 367–379 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rozvany, G.I.N.: A new calculus for optimum design. Int. J. Mech. Sci. 9, 885–886 (1967)

    Article  MATH  Google Scholar 

  21. Rozvany, G.I.N.: Concave programming in structural optimization. Int. J. Mech. Sci. 12, 131–142 (1970)

    Article  MATH  Google Scholar 

  22. Rozvany, G.I.N.: Concave programming and piece-wise linear programming. Int. J. Numer. Methods Eng. 3, 131–144 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program. 99(3), 563–591 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ueing, U.: A combinatorial method to compute a global solution of certain non-convex optimization problems. In: Lootsma, F.A. (ed.) Numerical Methods for Non-Linear Optimization, pp. 223–230. Academic Press, San Diego (1972)

    Google Scholar 

  28. Zamora, J.M., Grossmann, I.E.: A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms. J. Glob. Optim. 14(3), 217–249 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zangwill, W.I.: The piecewise concave function. Manag. Sci. 13(11), 900–912 (1967)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to extend his most profound thanks to the anonymous reviewer, whose numerous suggestions have greatly contributed to the improvement of the present document.

Author information

Author notes

  1. Gene A. Bunin

    Present address: Xinjiang Arts Institute, 734 Tuanjie Road, Urumqi, 830000, Xinjiang Uyghur Autonomous Region, People’s Republic of China

Authors and Affiliations

    Authors
    1. Gene A. Bunin
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Gene A. Bunin.

    Rights and permissions

    Reprints and permissions

    About this article

    Check for updates. Verify currency and authenticity via CrossMark

    Cite this article

    Bunin, G.A. Extended reverse-convex programming: an approximate enumeration approach to global optimization. J Glob Optim 65, 191–229 (2016). https://doi.org/10.1007/s10898-015-0352-x

    Download citation

    • Received:

    • Accepted:

    • Published:

    • Issue Date:

    • DOI: https://doi.org/10.1007/s10898-015-0352-x

    Keywords

    Search

    Navigation