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

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

Type

Article

Authors

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

KAUST Grant Number

KUK-C1-013-04

Date

2012-06-21

Online Publication Date

2012-06-21

Print Publication Date

2013-08

Permanent link to this record

http://hdl.handle.net/10754/597224

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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.

Sponsors

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).