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Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

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Constraint Programming and Decision Making: Theory and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 100))

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Applied Mathematics, 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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Editors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, El Paso, Texas, USA

    Martine Ceberio

  2. Department of Computer Science, University of Texas at El Paso, El Paso, Texas, USA

    Vladik Kreinovich

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Hladík, M. (2018). Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix . In: Ceberio, M., Kreinovich, V. (eds) Constraint Programming and Decision Making: Theory and Applications. Studies in Systems, Decision and Control, vol 100. Springer, Cham. https://doi.org/10.1007/978-3-319-61753-4_11

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  • DOI: https://doi.org/10.1007/978-3-319-61753-4_11

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