Abstract
In order to generate valid convex lower bounding problems for nonconvex twice-differentiable optimization problems, a method that is based on second-order information of general twice-differentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the αBB, a branch and bound algorithm which relies on this underestimation procedure [3].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. A. Adjiman, I. P. Androulakis, C. D. Maranas and C. A. Floudas, A Global Optimisation Method, αBB, for Process Design, Proc. of the Sixth European Symposium on Computer Aided Process Engineering, To be published, May 26–29 1996.
F. A., Al-Khayyal and J.E., Falk, Jointly Constrained Biconvex Programming, Maths Ops Res., 8:273–286, 1983.
I. P., Androulakis, C. D., Maranas and C. A., Floudas, αBB: A Global Optimization Method for General Constrained Nonconvex Problems, J. Global. Opt., 7:337–363, 1995.
A. Brooke, D. Kendrick and A. Meeraus, GAMS — Release 2.25. A User's Guide, boyd & fraser publishing company, 1992.
L. Dixon and G. P. Szegö, Towards Global Optimization, Proc. of a Workshop at the University of Cagliari, Italy, 1990.
C. A. Floudas and P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, 1990.
C. A, Floudas and V., Visweswaran, A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs: I. Theory, Comp. Chem. Engng., 14:1397–1417, 1990.
C. A, Floudas and V., Visweswaran, A Primal-Relaxed Dual Global Optimization Approach, J. Opt. Theory and App., 78:187–225, 1993.
E., Hansen, Global Optimization Using Interval Analysis, Marcel Dekkar, New York, 1992.
D., Hertz, The Extreme Eigenvalues and Stability of Real Symmetric Interval Matrices, IEEE Transactions on Automatic Control, 37:532–535, 1992.
C. V. Hollot and A. C. Bartlett, On the Eigenvalues of Interval Matrices, Proc. 1987 Conf. Decision Contr., 794–799, 1987.
V. L., Kharitonov, Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations, Differential Equations, 78:1483–1485, 1979.
C. D., Maranas and C. A., Floudas, A Global Optimization Approach for Lennard-Jones Microclusters, J. Chem. Phys., 97:7667–7677, 1992.
C. D., Maranas and C. A., Floudas, Global Optimization for Molecular Conformation Problems, Annals of Operations Research, 42:85–117, 1993.
C. D., Maranas and C. A., Floudas, Global Minimum Potential Energy Conformations of Small Molecules, J. Global. Opt., 4:135–170, 1994.
C. D., Maranas and C. A., Floudas, Finding All Solutions of Nonlinearly Constrained Systems of Equations, J. Global. Opt., 7:143–182, 1995
B. A., Murtagh and M. A., Saunders, MINOS 5.0 User's Guide, Systems Optimization Laboratory, Dept. of Operations Research, Stanford University, CA., 1988.
A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, 1990.
C. Reinsch and J. H. Wilkinson, Linear Algebra, Handbook for Automatic Computation, Vol. 2, Springer-Verlag, 1971.
H. S., Ryoo and N. V., Sahinidis, Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design, Comp. Chem. Engng., 19:551–566, 1995.
S.A. Vavasis, Nonlinear Optimization — Complexity Issues, Oxford Science Publications, 1991.
V. Visweswaran and C. A. Floudas, New Formulations and Branching Strategies for the GOP Algorithm, Global Optimization in Chemical Engineering, I.E. Grossmann, Editor, Kluwer Academic Publishers, Chapter 3, 75–110, 1996.
V. Visweswaran and C. A. Floudas, Computational Results for an Efficient Implementation of the GOP Algorithm and its Variants, Global Optimization in Chemical Engineering, I.E. Grossmann, Editor, Kluwer Academic Publishers, Chapter 4, 111–154, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Adjiman, C.S., Floudas, C.A. Rigorous convex underestimators for general twice-differentiable problems. J Glob Optim 9, 23–40 (1996). https://doi.org/10.1007/BF00121749
Issue Date:
DOI: https://doi.org/10.1007/BF00121749