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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Beaumont, O., Philippe, B.: Linear interval tolerance problem and linear programming techniques. Reliab. Comput. 7, 433–447 (2001)
Dymowa, L.: Soft Computing in Economics and Finance Intelligent Systems Reference Library, Vol. 6. Springer, Heidelberg (2011)
Goldsztejn, A.: A right-preconditioning process for the formal-algebraic approach to inner and outer estimation of AE-solution sets. Reliab. Comput. 11, 443–478 (2005)
Golub, G. H., Van Loan, C. F., 3rd ed.: Matrix computations. Johns Hopkins, Baltimore (1996)
Kelling, B., Oelschlägel, D.: Zur Lösung von linearen Toleranzproblemen, Wiss. Zeitschrift TH Leuna-Merseburg 33 (1), 121–131 (1991)
Neumaier, A.: Tolerance analysis with interval arithmetic. In: Freiburger Intervall-Berichte 86/9, pp 5–19. Albert-Ludwigs Univ., Freiburg (1986)
Nüding, E., Wilhelm, W.: Über Gleichungen und über Lösungen. ZAMM 52, T188–T190 (1972)
Popova, E. D.: Improved solution enclosures for over- and underdetermined interval linear systems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) , Vol. 3743, pp 305–312. Springer, Heidelberg (2006)
Popova, E. D.: Explicit description of AE solution sets for parametric linear systems. SIAM J. Matrix Anal. Appl. 33, 1172–1189 (2012)
Popova, E. D.: Inner estimation of the parametric tolerable solution set. Comput. Math. Appl. 66, 1655–1665 (2013)
Popova, E. D., Hladík, M.: Outer enclosures to the parametric AE solution set. Soft. Comput. 17, 1403–1414 (2013)
Rohn, J.: Input–output model with interval data. Econometrica 48, 767–769 (1980)
Rohn, J.: Inner solutions of linear interval systems. In: Nickel, K. (ed.) Interval Mathematics 1985, Lecture Notes on Computer Science, Vol. 212, pp 157–158. Springer Verlag, New York (1986)
Shary, S. P.: Solving the linear interval tolerance problem. Math. Comput. Simul. 39, 53–85 (1995)
Shary, S. P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput. 8, 321–418 (2002)
Sharaya, I. A.: Structure of the tolerable solution set of an interval linear system. Comput. Technol. 10, 103–119 (2005). in Russian
Sharaya, I. A.: On unbounded tolerable solution sets. Reliab. Comput. 11, 425–432 (2005)
Sharaya, I. A., Shary, S. P.: Tolerable solution set for interval linear systems with constraints on coefficients. Reliab. Comput. 15, 345–357 (2011)
Sokolova, S. P., Kuzmina, E. A.: Dynamic properties of interval systems, pp 215–221. SPIIRAS Proceedings, Nauka (2008). in Russian
Wang, Y.: Semantic tolerance modeling with generalized intervals. ASME J. Mech. Des. 130 (8), 081701 (2008)
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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