A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

Type
Article

Authors
Fowkes, Jaroslav M.
Gould, Nicholas I. M.
Farmer, Chris L.

KAUST Grant Number
KUK-C1-013-04

Online Publication Date
2012-06-21

Print Publication Date
2013-08

Date
2012-06-21

Abstract
We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other branch and bound algorithms, lower bounds are obtained via nonconvex underestimators of the function. For a numerical example, we apply the proposed branch and bound algorithm to radial basis function approximations. © 2012 Springer Science+Business Media, LLC.

Citation
Fowkes JM, Gould NIM, Farmer CL (2012) A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions. J Glob Optim 56: 1791–1815. Available: http://dx.doi.org/10.1007/s10898-012-9937-9.

Acknowledgements
We would like to thank Coralia Cartis for helpful suggestions on an early draft of thispaper. We would also like to thank an anonymous referee for positive comments and revisions which havehelped improve the paper. This research was supported through an EPSRC Industrial CASE studentship inconjunction with Schlumberger. The work of Nick Gould was supported by the EPSRC grants EP/E053351/1,EP/F005369/1 and EP/I013067/1. This publication was also based on work supported in part by Award NoKUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) (CLF).

Publisher
Springer Nature

Journal
Journal of Global Optimization

DOI
10.1007/s10898-012-9937-9

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