Abstract
This paper describes the construction of convex underestimators for twice continuously differentiable functions over box domains through piecewise quadratic perturbation functions. A refinement of the classical α BB convex underestimator, the underestimators derived through this approach may be significantly tighter than the classical αBB underestimator. The convex underestimator is the difference of the nonconvex function f and a smooth, piecewise quadratic, perturbation function, q. The convexity of the underestimator is guaranteed through an analysis of the eigenvalues of the Hessian of f over all subdomains of a partition of the original box domain. Smoothness properties of the piecewise quadratic perturbation function are derived in a manner analogous to that of spline construction.
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References
C.S. Adjiman I.P. Androulakis C.A. Floudas (1998) ArticleTitleA global optimization method,α BB, for general twice differentiable NLPs-II. Implementation and computional results Computers & Chemical Engineering 22 1159–1179
C.S. Adjiman I.P. Androulakis C.D. Maranas C.A. Floudas (1996) ArticleTitleA global optimization method,α BB, for process design Computers & Chemical Engineering Supplement 20 S419–S424
C.S. Adjiman S Dallwig C.A. Floudas A. Neumaier (1998a) ArticleTitleA global optimization method,α BB, for general twice differentiable NLPs-I Theoretical advances. Computers & Chemical Engineering 22 1137–1158
A.S. Deif (1991) ArticleTitleThe interval eigenvalue problem Zeitschrift fur Angewandte Mathematik and Mechanik 71 IssueID1 61–64
Dixon L., Szegö G.P. (ed.) (1975). Towards Global Optimization. In: Proceedings of a Workshop at the University of Cagliari, Italy. North Holland
Floudas, C.A. (2000). Deterministic Global Optimization: Theory, Alogorithms and Applications. Kluwer Academic Publishers
S. Gerschgorin (1931) ArticleTitleÜber die Abgrenzung der Eigenwerte einer Matrix, Izvestiya Akademii Nauk Azerbaidzhanskoi SSSR Seriya Fiziko-tekhnicheskikh i Matematicheskikh Nauk 6 749–754
D. Hertz (1992) ArticleTitleThe extreme eigenvalues and stability of real symmetric interval matrices IEEE Transactions on Automatic Control 37 IssueID4 532–535 Occurrence Handle10.1109/9.126593
Hiriart-Urruty, J., Lemaréchal, C. (1993). Convex analysis and minimization algorithms I. In: Grundlehren der Mathematischen Wissenschaften, Vol. 305. Springer Verlag
V. Kharitonov (1979) ArticleTitleAsymptotic stability of an equilibrium position of a family of systems of linear differential equations Differential Equations 78 1483–1485
Krämer W., Geulig I. (2001). Interval Calculus in Maple. Wissenschaftliches Rechnen. Bergische Universität, GH Wuppertal
C.D. Maranas C. Floudas (1994) ArticleTitleGlobal minimum potential energy conformations of small molecules Journal of Global Optimization 4 135–170 Occurrence Handle10.1007/BF01096720
G.P. McCormick (1976) ArticleTitleComputability of global solutions to factorable non-convex programs: Part I – Convex underestimating problems Mathematical Programming 10 147–175 Occurrence Handle10.1007/BF01580665
R.E. Moore (1966) Interval Analysis Prentice-Hall Englewood Cliffs, NJ
Mori T., Kokame H. (1994). Eigenvalue bounds for a certain class of interval matrices, IEICE Transactions on Fundamentals E77-A(10). 1707–1709
Neumaier, A. (1990). Interval methods for systems of equations. In: Encyclopedia of Mathematics and its Applications. Cambridge University Press
A. Neumaier (1992) ArticleTitleAn optimality criterion for global quadratic optimization Journal of Global Optimization 2 201–208 Occurrence Handle10.1007/BF00122055
Ratschek, H., Rokne, J. (1984). Computer methods for the range of functions. In: Ellis Horwood Series in Mathematics and its Applications. Halsted Press
Rockafellar, R.T. (1970). Convex Analysis. Princeton University Press
Rohn, J. (1996). Bounds on Eigenvalues of Interval Matrices, Technical Report no. 688, Institute of Computer Science, Academy of Sciences, Prague
Stephens, C. (1997). Interval and bounding Hessians. In: Bomze, I.M. (ed.). Developements in Global Optimization. Kluwer Academic Publishers, pp. 109–199
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Meyer, C.A., Floudas, C.A. Convex Underestimation of Twice Continuously Differentiable Functions by Piecewise Quadratic Perturbation: Spline αBB Underestimators. J Glob Optim 32, 221–258 (2005). https://doi.org/10.1007/s10898-004-2704-9
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DOI: https://doi.org/10.1007/s10898-004-2704-9