New methods for calculating α BB-type underestimators

A1 Journal article (refereed)

Internal Authors/Editors

Publication Details

List of Authors: Anders Skjäl, Tapio Westerlund
Publisher: SPRINGER
Publication year: 2014
Journal: Journal of Global Optimization
Journal acronym: J GLOBAL OPTIM
Volume number: 58
Issue number: 3
Start page: 411
End page: 427
Number of pages: 17
ISSN: 0925-5001
eISSN: 1573-2916


Most branch-and-bound algorithms in global optimization depend on convex underestimators to calculate lower bounds of a minimization objective function. The BB methodology produces such underestimators for sufficiently smooth functions by analyzing interval Hessian approximations. Several methods to rigorously determine the BB parameters have been proposed, varying in tightness and computational complexity. We present new polynomial-time methods and compare their properties to existing approaches. The new methods are based on classical eigenvalue bounds from linear algebra and a more recent result on interval matrices. We show how parameters can be optimized with respect to the average underestimation error, in addition to the maximum error commonly used in BB methods. Numerical comparisons are made, based on test functions and a set of randomly generated interval Hessians. The paper shows the relative strengths of the methods, and proves exact results where one method dominates another.


alpha BB, Convex relaxation, Global optimization, Nonconvex optimization

Last updated on 2020-27-01 at 03:39