Skip to main content
SpringerLink
Log in

An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems

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

Abstract

We present a new relaxation method for the deterministic global optimization of general nonconvex and \({\mathscr {C}}^2\)-continuous problems. Instead of using a convex underestimator, the method uses an edge-concave (componentwise concave) underestimator to relax a nonconvex function. The underestimator is constructed by subtracting a positive quadratic expression such that all nonedge-concavities in the original function is overpowered by the added expression. While the edge-concave underestimator is nonlinear, the linear facets of its vertex polyhedral convex envelope leads to a linear programming (LP)-based relaxation of the original nonconvex problem. We present some theoretical results on this new class of underestimators and compare the performance of the LP relaxation with relaxations obtained by convex underestimators such as \(\alpha \hbox {BB}\) and its variants for several test problems. We also discuss the potential of a hybrid relaxation, relying on the dynamic selection of convex and edge-concave underestimators using criteria such as maximum separation distance.

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

Similar content being viewed by others

References

  1. Floudas, C.A., Pardalos, P.M.: State-of-the-art in global optimization—computational methods and applications—preface. J. Glob. Optim. 7(2), 113 (1995)

    Article  Google Scholar 

  2. Sherali, H.D., Adams, W.P.: Reformulation-Linearization Techniques in Discrete, Continuous Optimization. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Floudas, C.A.: Deterministic Global Optimization: Theory, Methods, Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  5. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, Second edn. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  6. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Applications, Software, and Applications. Kluwer Academic Publishers, Norwell (2002)

    Book  MATH  Google Scholar 

  7. Floudas, C.A., Pardalos, P.M.: Frontiers in Global Optimization. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  8. Floudas, C.A.: Research challenges opportunities, synergism in systems engineering, computational biology. AIChE J. 51, 1872–1884 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  10. Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part 1-convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Meyer, C.A., Floudas, C.A.: Trilinear monomials with positive or negative domains: facets of the convex, concave envelopes. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 327–352. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  14. Meyer, C.A., Floudas, C.A.: Trilinear monomials with mixed sign domains: facets of the convex, concave envelopes. J. Glob. Optim. 29(2), 125–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143–182 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20(2), 133–154 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tawarmalani, M., Sahinidis, N.V.: Convex extensions, envelopes of lower semi-continuous functions. Math. Program. 247–263, 93 (2002)

    MathSciNet  MATH  Google Scholar 

  19. Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25(2), 157–168 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103(2), 207–224 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tardella, F.: On a class of functions attaining their maximum at the vertices of a polyhedron. Discrete Appl. Math. 22, 191–195 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tardella, F.: On the existence of polyhedral convex envelopes. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 563–573. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  23. Tardella, F.: Existence, sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Caratzoulas, S., Floudas, C.A.: Trigonometric convex underestimator for the base functions in Fourier space. J. Optim. Theory Appl. 124, 339–362 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maranas, C.D., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4, 135–170 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Adjiman, C.S., Floudas, C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9, 23–40 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A Global Optimization method, \(\alpha \text{ BB }\), for general twice differentiable NLPs-I. Theoretical advances. Comput. Chem. Eng. 22, 1137–1158 (1998)

    Article  Google Scholar 

  29. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, \(\alpha \text{ BB }\), for general twice differentiable NLPs-II. Implementation, computional results. Comput. Chem. Eng. 22, 1159–1179 (1998)

    Article  Google Scholar 

  30. Akrotirianakis, I.G., Floudas, C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30, 367–390 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Akrotirianakis, I.G., Floudas, C.A.: Computational experience with a new class of convex underestimators: box-constrained NLP problems. J. Glob. Optim. 29, 249–264 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Floudas, C.A., Kreinovich, V.: Towards optimal techniques for solving global optimization problems: symmetry-based approach. In: Torn, A., Zilinskas, J. (eds.) Models, Algorithms for Global Optimization, pp. 21–42. Springer, Berlin (2006)

    Google Scholar 

  33. Floudas, C.A., Kreinovich, V.: On the functional form of convex underestimators for twice continuously differentiable functions. Optim. Lett. 1(2), 187–192 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Meyer, C.A., Floudas, C.A.: Convex underestimation of twice continuously differentiable functions by piecewise quadratic perturbation: spline \(\alpha \text{ BB }\) underestimators. J. Glob. Optim. 32, 221–258 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \(C^2\)-continuous problems: I. Univariate functions. J. Glob. Optim. 42(1), 51–67 (2008)

    Article  MATH  Google Scholar 

  36. Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \(C^2\)-continuous problems: II. Multivariate functions. J. Glob. Optim. 42(1), 69–89 (2008)

    Article  MATH  Google Scholar 

  37. Misener, R., Gounaris, C.E., Floudas, C.A.: Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints. Comput. Chem. Eng. 34(9), 1432–1456 (2010)

    Article  Google Scholar 

  38. Misener, R., Floudas, C.A.: Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear, edge-concave relaxations. Math. Program. 136(1), 155–182 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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 

  42. Hertz, D., Adjiman, C.S., Floudas, C.A.: Two results on bounding the roots of interval polynomials. Comput. Chem. Eng. 23, 1333–1339 (1999)

    Article  Google Scholar 

  43. Skjäl, A., Westerlund, T., Misener, R., Floudas, C.A.: A generalization of the classical \(\alpha \text{ BB }\) convex underestimation via diagonal and nondiagonal quadratic terms. J. Optim. Theory Appl. 154(2), 462–490 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Skjäl, A., Westerlund, T.: New methods for calculating \(\alpha \text{ BB }\)-type underestimators. J. Glob. Optim. 58(3), 411–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Guzman, Y.A., Hasan, M.M.F., Floudas, C.A.: Performance of convex underestimators in a branch-and-bound global optimization framework. Optim. Lett. 10(2), 283–308 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Berna, T., Locke, M., Westerberg, A.W.: A new approach to optimization of chemical processes. AIChE J. 26(1), 37–43 (1980)

    Article  Google Scholar 

Download references

Acknowledgements

Financial support from the U.S. National Science Foundation (Award Number CBET-1606027) is gratefully acknowledged. M.M.F.H. likes to thank Dr. Yannis Guzman and Dr. Eric First for their help in comparing results with previous methods.

Author information

Authors and Affiliations

  1. Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, TX, 77843, USA

    M. M. Faruque Hasan

Authors
  1. M. M. Faruque Hasan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Faruque Hasan.

Additional information

This article is dedicated to the memory of our dear mentor Professor Christodoulos A. Floudas, who died on 14 August 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hasan, M.M.F. An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems. J Glob Optim 71, 735–752 (2018). https://doi.org/10.1007/s10898-018-0646-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-018-0646-x

Keywords

Search

Navigation