Milan Hladík. Positive semidefiniteness and positive definiteness of a linear parametric interval matrix. In Martine Ceberio and Vladik Kreinovich, editors, Constraint Programming and Decision Making: Theory and Applications, Studies in Systems, Decision and Control, pp. 77–88, Springer, Cham, 2018.
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We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient / necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques.
@inCollection{Hla2018a,
author = "Milan Hlad\'{\i}k",
title = "Positive semidefiniteness and positive definiteness of a linear parametric interval matrix",
editor = "Ceberio, Martine and Kreinovich, Vladik",
booktitle = "Constraint Programming and Decision Making: {Theory} and Applications",
publisher = "Springer",
address = "Cham",
series = "Studies in Systems, Decision and Control",
volume = "100",
pages = "77-88",
year = "2018",
doi = "10.1007/978-3-319-61753-4_11",
isbn = "978-3-319-61753-4",
url = "https://link.springer.com/chapter/10.1007/978-3-319-61753-4_11",
bib2html_dl_html = "https://doi.org/10.1007/978-3-319-61753-4_11",
abstract = "We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters, the matrix is positive definite (or positive semidefinite). We state a characterization in the form of equivalent conditions, and also propose some computationally cheap sufficient / necessary conditions. Our results extend the classical results on positive (semi-)definiteness of interval matrices. They may be useful for checking convexity or non-convexity in global optimization methods based on branch and bound framework and using interval techniques.",
keywords = "Interval computation; Interval matrix; Parametric matrix; Positive semidefiniteness; Positive definiteness; Global optimization",
}
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