Advertisement

Journal of Global Optimization

, Volume 71, Issue 4, pp 735–752 | Cite as

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

  • M. M. Faruque Hasan
Article

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.

Keywords

Global optimization Edge-concave underestimator Relaxation Nonconvex NLP 

Notes

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.

References

  1. 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)CrossRefGoogle Scholar
  2. 2.
    Sherali, H.D., Adams, W.P.: Reformulation-Linearization Techniques in Discrete, Continuous Optimization. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  3. 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)CrossRefzbMATHGoogle Scholar
  4. 4.
    Floudas, C.A.: Deterministic Global Optimization: Theory, Methods, Applications. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  5. 5.
    Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, Second edn. Kluwer Academic Publishers, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  6. 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)CrossRefzbMATHGoogle Scholar
  7. 7.
    Floudas, C.A., Pardalos, P.M.: Frontiers in Global Optimization. Kluwer Academic Publishers, Dordrecht (2003)zbMATHGoogle Scholar
  8. 8.
    Floudas, C.A.: Research challenges opportunities, synergism in systems engineering, computational biology. AIChE J. 51, 1872–1884 (2005)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part 1-convex underestimating problems. Math. Program. 10(1), 147–175 (1976)CrossRefzbMATHGoogle Scholar
  12. 12.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ryoo, H.S., Sahinidis, N.V.: Analysis of bounds for multilinear functions. J. Glob. Optim. 19, 403–424 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7(2), 143–182 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tawarmalani, M., Sahinidis, N.V.: Semidefinite relaxations of fractional programs via novel convexification techniques. J. Glob. Optim. 20(2), 133–154 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tawarmalani, M., Sahinidis, N.V.: Convex extensions, envelopes of lower semi-continuous functions. Math. Program. 247–263, 93 (2002)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liberti, L., Pantelides, C.C.: Convex envelopes of monomials of odd degree. J. Glob. Optim. 25(2), 157–168 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meyer, C.A., Floudas, C.A.: Convex envelopes for edge-concave functions. Math. Program. 103(2), 207–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tardella, F.: On a class of functions attaining their maximum at the vertices of a polyhedron. Discrete Appl. Math. 22, 191–195 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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. 23.
    Tardella, F.: Existence, sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2, 363–375 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Caratzoulas, S., Floudas, C.A.: Trigonometric convex underestimator for the base functions in Fourier space. J. Optim. Theory Appl. 124, 339–362 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Maranas, C.D., Floudas, C.A.: Global minimum potential energy conformations of small molecules. J. Glob. Optim. 4, 135–170 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Adjiman, C.S., Floudas, C.A.: Rigorous convex underestimators for general twice-differentiable problems. J. Glob. Optim. 9, 23–40 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)CrossRefzbMATHGoogle Scholar
  36. 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)CrossRefzbMATHGoogle Scholar
  37. 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)CrossRefGoogle Scholar
  38. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. 57(1), 3–50 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Skjäl, A., Westerlund, T.: New methods for calculating \(\alpha \text{ BB }\)-type underestimators. J. Glob. Optim. 58(3), 411–427 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Berna, T., Locke, M., Westerberg, A.W.: A new approach to optimization of chemical processes. AIChE J. 26(1), 37–43 (1980)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Artie McFerrin Department of Chemical EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations