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Solving the linear interval tolerance problem

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  • Shary, Sergey P.

Abstract

For the interval linear system Ax = b, the linear tolerance problem is considered that requires inner evaluation of the tolerable solution setΣ∀∃(A, b) = {x ∈ Rn ∣ (∀A ∈ A)(Ax ∈ b)} formed by all point vectors x such that the product Ax remains within b for all possible A ∈ A. Along with the simple incompatibility criterion, we develop comprehensive solvability theory for the linear tolerance problem that not only settles whether Σ∀∃ is empty or not, but also enables modification of the problem to ensure its desired properties. To conclude, we advance several numerical methods of various accuracy and complexity for construction of an interval solution to the linear tolerance problem around a given center.

Suggested Citation

  • Shary, Sergey P., 1995. "Solving the linear interval tolerance problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 39(1), pages 53-85.
    • Handle: RePEc:eee:matcom:v:39:y:1995:i:1:p:53-85
    DOI: 10.1016/0378-4754(95)00135-K
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    References listed on IDEAS

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    1. Walter, Eric & Piet-Lahanier, Hélène, 1990. "Estimation of parameter bounds from bounded-error data: a survey," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 32(5), pages 449-468.
    2. Rohn, Jiri, 1980. "Input-Output Model with Interval Data," Econometrica, Econometric Society, vol. 48(3), pages 767-769, April.
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    Cited by:

    1. Ignacio Araya & Gilles Trombettoni & Bertrand Neveu & Gilles Chabert, 2014. "Upper bounding in inner regions for global optimization under inequality constraints," Journal of Global Optimization, Springer, vol. 60(2), pages 145-164, October.
    2. Leela-apiradee, Worrawate & Gorka, Artur & Burimas, Kanokwan & Thipwiwatpotjana, Phantipa, 2022. "Tolerance-localized and control-localized solutions of interval linear equations system and their application to course assignment problem," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    3. Ivan Contreras & Remei Calm & Miguel A. Sainz & Pau Herrero & Josep Vehi, 2021. "Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty," Mathematics, MDPI, vol. 9(6), pages 1-20, March.
    4. Stefania Corsaro & Marina Marino, 2006. "Interval linear systems: the state of the art," Computational Statistics, Springer, vol. 21(2), pages 365-384, June.

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