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

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. 1 School of Mathematics, The King’s Buildings, University of Edinburgh, EH9 3JZ, Scotland, EU. Email: jaroslav.fowkes@ed.ac.uk . Current reports available from “http://www.maths.ed.ac.uk/ERGO/preprints.html”. 2 This research was supported through an EPSRC Industrial CASE studentship at the University of Oxford in conjunction with Schlumberger. 3 Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU. Email: nick.gould@stfc.ac.uk . Current reports available from “http://www.numerical.rl.ac.uk/reports/reports.shtml”. 4 This work was supported by the EPSRC grants EP/E053351/1, EP/F005369/1 and EP/I013067/1. 5 Mathematical Institute, 24-29 St Giles’, University of Oxford, OX1 3LB, England, EU. Email: Chris.Farmer@maths.ox.ac.uk . Computational Science and Engineering Department Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX 1st November 2011 Branch and Bound Global Optimization of Hessian Lipschitz Continuous Functions 1

DOI: 10.1007/s10898-012-9937-9

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Cite this paper

@article{Fowkes2013ABA, title={A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions}, author={Jaroslav M. Fowkes and Nicholas I. M. Gould and Chris L. Farmer}, journal={J. Global Optimization}, year={2013}, volume={56}, pages={1791-1815} }