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An extension of the \(\alpha \hbox {BB}\)-type underestimation to linear parametric Hessian matrices

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Abstract

The classical \(\alpha \hbox {BB}\) 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.

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Acknowledgments

The author was supported by the Czech Science Foundation Grant P402-13-10660S.

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Hladík, M. An extension of the \(\alpha \hbox {BB}\)-type underestimation to linear parametric Hessian matrices. J Glob Optim 64, 217–231 (2016). https://doi.org/10.1007/s10898-015-0304-5

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  • DOI: https://doi.org/10.1007/s10898-015-0304-5

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