Milan Hladík, Lubomir V. Kolev, and Iwona Skalna. Linear interval parametric approach to testing pseudoconvexity. J. Glob. Optim., 79(2):351–368, 2021.
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The recent paper (DOI: 10.1007/s10898-017-0537-6) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative approach which is based on the use of more accurate affine form enclosures and affine arithmetic. We modify the tests to work with linear interval parametric enclosures of the gradients and the Hessians. We also present computational complexity results, showing that performing some tests exactly is NP-hard. It is shown by numerical experiments on random and benchmark data that the new approach results in more efficient tests for checking pseudoconvexity, however, at the expense of higher computation time.
@article{HlaSka2021a,
author = "Milan Hlad\'{\i}k and Lubomir V. Kolev and Iwona Skalna",
title = "Linear interval parametric approach to testing pseudoconvexity",
journal = "J. Glob. Optim.",
fjournal = "Journal of Global Optimization",
volume = "79",
number = "2",
pages = "351-368",
year = "2021",
doi = "10.1007/s10898-020-00924-w",
issn = "1573-2916",
bib2html_dl_html = "https://doi.org/10.1007/s10898-020-00924-w",
bib2html_dl_pdf = "https://rdcu.be/b5S9u",
abstract = "The recent paper (DOI: 10.1007/s10898-017-0537-6) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative approach which is based on the use of more accurate affine form enclosures and affine arithmetic. We modify the tests to work with linear interval parametric enclosures of the gradients and the Hessians. We also present computational complexity results, showing that performing some tests exactly is NP-hard. It is shown by numerical experiments on random and benchmark data that the new approach results in more efficient tests for checking pseudoconvexity, however, at the expense of higher computation time.",
keywords = "Global optimization; Pseudoconvexity; Interval computation; Linear interval parametric enclosures",
}
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