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

Authors:
Jaroslav M. Fowkes;Nicholas I. Gould;Chris L. Farmer
Affiliations:
School of Mathematics, University of Edinburgh, Edinburgh, UK EH9 3JZ;Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, UK OX11 0QX;Mathematical Institute, University of Oxford, Oxford, UK OX1 3LB
Venue:
Journal of Global Optimization
Year:
2013

Citing 14
Cited 0

A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization

Journal of Optimization Theory and Applications
Lipschitzian optimization without the Lipschitz constant

Journal of Optimization Theory and Applications
Neural networks for pattern recognition

Neural networks for pattern recognition
Trust-region methods

Trust-region methods
A Taxonomy of Global Optimization Methods Based on Response Surfaces

Journal of Global Optimization
Introduction to Stochastic Search and Optimization

Introduction to Stochastic Search and Optimization
Convex Optimization

Convex Optimization
Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem

Journal of Global Optimization
Cubic regularization of Newton method and its global performance

Mathematical Programming: Series A and B
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)

Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Introduction to Global Optimization (Nonconvex Optimization and Its Applications)

Introduction to Global Optimization (Nonconvex Optimization and Its Applications)
Hierarchical Nonlinear Approximation for Experimental Design and Statistical Data Fitting

SIAM Journal on Scientific Computing
SNOBFIT -- Stable Noisy Optimization by Branch and Fit

ACM Transactions on Mathematical Software (TOMS)
Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results

Mathematical Programming: Series A and B

Quantified Score

Hi-index	0.00

Visualization

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.