# An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices

## Author

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() (Charles University)

## Abstract

Abstract The classical $$\alpha \hbox {BB}$$ α BB method is a global optimization method the important step of which is to determine a convex underestimator of an objective function on an interval domain. Its particular point is to enclose the range of a Hessian matrix in an interval matrix. To have a tighter estimation of the Hessian matrices, we investigate a linear parametric form enclosure in this paper. One way to obtain this form is by using a slope extension of the Hessian entries. Numerical examples indicate that our approach can sometimes significantly reduce overestimation on the objective function. However, the slope extensions highly depend on a choice of the center of linearization. We compare some naive choices and also propose a heuristic one, which performs well in executed examples, but it seems there is no one global winner.

## Suggested Citation

• Milan Hladík, 2016. "An extension of the $$\alpha \hbox {BB}$$ α BB -type underestimation to linear parametric Hessian matrices," Journal of Global Optimization, Springer, vol. 64(2), pages 217-231, February.
• Handle: RePEc:spr:jglopt:v:64:y:2016:i:2:d:10.1007_s10898-015-0304-5
DOI: 10.1007/s10898-015-0304-5
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## References listed on IDEAS

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1. Anders Skjäl & Tapio Westerlund, 2014. "New methods for calculating $$\alpha$$ BB-type underestimators," Journal of Global Optimization, Springer, vol. 58(3), pages 411-427, March.
2. Ferenc Domes & Arnold Neumaier, 2012. "Rigorous filtering using linear relaxations," Journal of Global Optimization, Springer, vol. 53(3), pages 441-473, July.
3. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
4. Jordan Ninin & Frédéric Messine, 2011. "A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms," Journal of Global Optimization, Springer, vol. 50(4), pages 629-644, August.
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### Keywords

Global optimization; Interval computation; Convex relaxation;

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