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On the unbounded parametric tolerable solution set

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Abstract

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters p,q varying within given intervals p, q. The so-called parametric tolerable solution set Σ t o l (A(p),b(q),p,q):={x∈ℝ n∣(∀p∈p)(∃q∈q)(A(p)x=b(q))} is studied for unboundedness. Basing on a characterization of the parametric tolerable solution set as a convex polyhedron, we present necessary and sufficient conditions (in both general and computable forms) for a nonempty parametric tolerable solution set to be unbounded. Every parametric tolerable solution set is represented as a sum of a linear subspace and a bounded convex polyhedron. The latter implies better estimations (outer and inner) for the unbounded parametric tolerable solution sets. Numerical examples illustrate the discussed methodology and the solution sets.

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Authors and Affiliations

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

    Evgenija D. Popova

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  1. Evgenija D. Popova
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Correspondence to Evgenija D. Popova.

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This work is partly supported by the grant BG051PO001/3.3-05-001 of the Bulgarian Ministry of Science and Education.

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Popova, E.D. On the unbounded parametric tolerable solution set. Numer Algor 69, 169–182 (2015). https://doi.org/10.1007/s11075-014-9888-y

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  • DOI: https://doi.org/10.1007/s11075-014-9888-y

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