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On the efficient Gerschgorin inclusion usage in the global optimization \(\alpha \hbox {BB}\) method

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Abstract

In this paper, we revisit the \(\alpha \)BB method for solving global optimization problems. We investigate optimality of the scaling vector used in Gerschgorin’s inclusion theorem to calculate bounds on the eigenvalues of the Hessian matrix. We propose two heuristics to compute a good scaling vector \(d\), and state three necessary optimality conditions for an optimal scaling vector. Since the scaling vectors calculated by the presented methods satisfy all three optimality conditions, they serve as cheap but efficient solutions. A small numerical study shows that they are practically always optimal.

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Acknowledgments

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

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  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00 , Prague, Czech Republic

    Milan Hladík

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  1. Milan Hladík
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Correspondence to Milan Hladík.

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Hladík, M. On the efficient Gerschgorin inclusion usage in the global optimization \(\alpha \hbox {BB}\) method. J Glob Optim 61, 235–253 (2015). https://doi.org/10.1007/s10898-014-0161-7

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  • DOI: https://doi.org/10.1007/s10898-014-0161-7

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