Milan Hladík. The effect of Hessian evaluations in the global optimization αBB method. In H. G. Bock and others, editors, Modeling, Simulation and Optimization of Complex Processes HPSC 2015, pp. 67–79, Springer, Cham, 2017.
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We consider convex underestimators that are used in the global optimization $\alpha$BB method and its variants. The method is based on augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators. The purpose of this paper is also to turn attention of researchers to the possibility of symbolic differentiation (and its combination with automatic differentiation) and investigation of the most convenient way for interval evaluation.
@inCollection{Hla2017f,
author = "Milan Hlad\'{\i}k",
title = "The effect of {Hessian} evaluations in the global optimization $\alpha${BB} method",
webtitle = "The effect of {Hessian} evaluations in the global optimization α{BB} method",
editor = "H. G. Bock and others",
feditor = "Bock, Hans Georg and Phu, Hoang Xuan and Rannacher, Rolf and Schl{\"o}der, Johannes P.",
booktitle = "Modeling, Simulation and Optimization of Complex Processes {HPSC} 2015",
fbooktitle = "Modeling, Simulation and Optimization of Complex Processes {HPSC} 2015: {Proceedings} of the Sixth International Conference on High Performance Scientific Computing, March 16-20, 2015, Hanoi, Vietnam",
publisher = "Springer",
address = "Cham",
pages = "67-79",
year = "2017",
doi = "10.1007/978-3-319-67168-0_6",
isbn = "978-3-319-67168-0",
url = "https://link.springer.com/chapter/10.1007/978-3-319-67168-0_6",
bib2html_dl_html = "https://doi.org/10.1007/978-3-319-67168-0_6",
abstract = "We consider convex underestimators that are used in the global optimization $\alpha$BB method and its variants. The method is based on augmenting the original nonconvex function by a relaxation term that is derived from an interval enclosure of the Hessian matrix. In this paper, we discuss the advantages of symbolic computation of the Hessian matrix. Symbolic computation often allows simplifications of the resulting expressions, which in turn means less conservative underestimators. We show by examples that even a small manipulation with the symbolic expressions, which can be processed automatically by computers, can have a large effect on the quality of underestimators. The purpose of this paper is also to turn attention of researchers to the possibility of symbolic differentiation (and its combination with automatic differentiation) and investigation of the most convenient way for interval evaluation.",
keywords = "Global optimzation; Convex underestimator; Interval computation",
}
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