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Outer enclosures to the parametric AE solution set

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We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer–Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods.

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Acknowledgments

This work was inspired by the discussions held during the Dagstuhl seminar 11371 in Dagstuhl, Germany, September 2011. E. Popova was partially supported by the Bulgarian National Science Fund under grant No. DO 02-359/2008. M. Hladík was partially supported by the Czech Science Foundation Grant P403/12/1947. The authors thank the anonymous reviewers for the numerous remarks improving readability of the paper.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113, Sofia, Bulgaria

    Evgenija D. Popova

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 11800, Prague, Czech Republic

    Milan Hladík

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  1. Evgenija D. Popova
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  2. Milan Hladík
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Correspondence to Evgenija D. Popova.

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Communicated by V. Kreinovich.

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Popova, E.D., Hladík, M. Outer enclosures to the parametric AE solution set. Soft Comput 17, 1403–1414 (2013). https://doi.org/10.1007/s00500-013-1011-0

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