Tighter αBB relaxations through a refi nement scheme for the scaled Gerschgorin theorem
File(s)
Accepted version
Author(s)
Nerantzis, Dimitrios
Adjiman, Claire
Type
Journal Article
Abstract
Of central importance to the αBB algorithm is the calculation of the α values that guarantee the convexity of the underestimator. Improvement (reduction) of these values can result in tighter underestimators and thus increase the performance of the algorithm. For instance, it was shown by Wechsung et al. (J Glob Optim 58(3):429-438, 2014) that the emergence of the cluster effect can depend on the magnitude of the α values. Motivated by this, we present a refinement method that can improve (reduce) the magnitude of α values given by the scaled Gerschgorin method and thus create tighter convex underestimators for the αBB algorithm. We apply the new method and compare it with the scaled Gerschgorin on randomly generated interval symmetric matrices as well as interval Hessians taken from test functions. As a measure of comparison, we use the maximal separation distance between the original function and the underestimator. Based on the results obtained, we conclude that the proposed refinement method can significantly reduce the maximal separation distance when compared to the scaled Gerschgorin method. This approach therefore has the potential to improve the performance of the αBB algorithm.
Date Issued
2019-03-01
Date Acceptance
2018-09-09
Citation
Journal of Global Optimization, 2019, 73 (3), pp.467-483
ISSN
0925-5001
Publisher
Springer Verlag
Start Page
467
End Page
483
Journal / Book Title
Journal of Global Optimization
Volume
73
Issue
3
Copyright Statement
© 2019 Springer-Verlag. The final publication is available at Springer via https://doi.org/10.1007/s10898-018-0718-y.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/J003840/1
Subjects
Science & Technology
Technology
Physical Sciences
Operations Research & Management Science
Mathematics, Applied
Mathematics
Global optimization
Branch-and-bound
Convex underestimators
Interval matrix
GLOBAL OPTIMIZATION METHOD
DIFFERENTIABLE CONSTRAINED NLPS
CONVEX UNDERESTIMATORS
CLUSTER PROBLEM
EIGENVALUES
Operations Research
0102 Applied Mathematics
0103 Numerical and Computational Mathematics
0802 Computation Theory and Mathematics
Publication Status
Published
Date Publish Online
2019-01-11