Milan Hladík. An extension of the αBB-type underestimation to linear parametric Hessian matrices. J. Glob. Optim., 64(2):217–231, 2016.
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The classical alpha-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.
@article{Hla2016a,
author = "Milan Hlad\'{\i}k",
title = "An extension of the $\alpha${BB}-type underestimation to linear parametric {Hessian} matrices",
webtitle = "An extension of the αBB-type underestimation to linear parametric {Hessian} matrices",
journal = "J. Glob. Optim.",
fjournal = "Journal of Global Optimization",
volume = "64",
number = "2",
pages = "217-231",
year = "2016",
doi = "10.1007/s10898-015-0304-5",
issn = "0925-5001",
url = "https://doi.org/10.1007/s10898-015-0304-5",
bib2html_dl_html = "https://link.springer.com/article/10.1007%2Fs10898-015-0304-5",
bib2html_dl_pdf = "https://rdcu.be/cnoZG",
abstract = "The classical alpha-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.",
keywords = "Global optimization; Interval computation; Convex relaxation",
}
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