Abstract
We consider convex underestimators that are used in the global optimization αBB method and its variants. The method is based on augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators. The purpose of this paper is also to turn attention of researchers to the possibility of symbolic differentiation (and its combination with automatic differentiation) and investigation of the most convenient way for interval evaluation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, αBB, for general twice-differentiabe constrained NLPs – II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159–1179 (1998)
Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, αBB, for general twice-differentiable constrained NLPs – I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)
Androulakis, I.P., Maranas, C.D., Floudas, C.A.: αBB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995)
Beaumont, O.: Solving interval linear systems with linear programming techniques. Linear Algebra Appl. 281(1–3), 293–309 (1998)
Floudas, C.A.: Deterministic global optimization. Theory, methods and applications. In: Nonconvex Optimization and its Applications, vol. 37. Kluwer, Dordrecht (2000)
Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3–38 (2009)
Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization, 2nd edn. Springer, New York (2009)
Floudas, C., Akrotirianakis, I., Caratzoulas, S., Meyer, C., Kallrath, J.: Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29(6), 1185–1202 (2005)
Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for \(\mathcal{C}^{2}\)-continuous problems. II: multivariate functions. J. Glob. Optim. 42(1), 69–89 (2008)
Hansen, E.R.: Sharpness in interval computations. Reliab. Comput. 3(1), 17–29 (1997)
Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2004)
Hladík, M.: Bounds on eigenvalues of real and complex interval matrices. Appl. Math. Comput. 219(10), 5584–5591 (2013)
Hladík, M.: On the efficient Gerschgorin inclusion usage in the global optimization αBB method. J. Glob. Optim. 61(2), 235–253 (2015)
Hladík, M., Daney, D., Tsigaridas, E.: Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl. 31(4), 2116–2129 (2010)
Hladík, M., Daney, D., Tsigaridas, E.P.: A filtering method for the interval eigenvalue problem. Appl. Math. Comput. 217(12), 5236–5242 (2011)
Li, J., Misener, R., Floudas, C.: Continuous-time modeling and global optimization approach for scheduling of crude oil operations. AIChE J. 58(1), 205–226 (2012)
Miró, A., Pozo, C., Guillén-Gosálbez, G., Egea, J., Jiménez, L.: Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems. BMC Bioinf. 13(1), 90 (2012)
Mönnigmann, M.: Fast calculation of spectral bounds for hessian matrices on hyperrectangles. SIAM J. Matrix Anal. Appl. 32(4), 1351–1366 (2011)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)
Rump, S.M.: INTLAB – INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/
Shiu, T.J., Wu, S.Y.: Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems. Comput. Optim. Appl. 53(1), 91–113 (2012)
Skjäl, A., Westerlund, T.: New methods for calculating αBB-type underestimators. J. Glob. Optim. 58(3), 411–427 (2014)
Skjäl, A., Westerlund, T., Misener, R., Floudas, C.A.: A generalization of the classical αBB convex underestimation via diagonal and nondiagonal quadratic terms. J. Optim. Theory Appl. 154(2), 462–490 (2012)
Stein, O.: How to solve a semi-infinite optimization problem. Eur. J. Oper. Res. 223(2), 312–320 (2012)
Acknowledgements
The author was supported by the Czech Science Foundation Grant P402-13-10660S.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Hladík, M. (2017). The Effect of Hessian Evaluations in the Global Optimization αBB Method. In: Bock, H., Phu, H., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes HPSC 2015 . Springer, Cham. https://doi.org/10.1007/978-3-319-67168-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-67168-0_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67167-3
Online ISBN: 978-3-319-67168-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)