Abstract
The problem of finding the smallest box enclosing the united solution set of a linear interval system, also known as the “interval hull” problem, was proven to be NP-hard. However, Hansen, Bliek, and others subsequently, have provided a polynomial-time solution in the case of systems preconditioned by the midpoint inverse matrix.
Based upon a similar approach, this paper deals with the interval hull problem in the context of AE-solution sets, where parameters may be given different quantifiers. A polynomial-time algorithm is proposed for computing the hull of AE-solution sets where parameters involved in the matrix are constrained to be existentially quantified. Such AE-solution sets are called right-quantified solution sets. They have recently been shown to be of practical interest.
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References
Aberth O. (1997) The Solution of Linear Interval Equations by a Linear Programming Method. Linear Algebra and Its Applications 259: 271–279
Bliek, C.: Computer Methods for Design Automation, PhD Thesis, Massachusetts Institute of Technology, 1992.
Dantzig, G. B.: Linear Programming and Extensions, Princeton University Press, 1963.
Goldsztejn, A.: A Branch and Prune Algorithm for the Approximation of Non-Linear AE-Solution Sets, in: ACM SAC, 2006, pp. 1650–1654.
Goldsztejn A. (2005) A Right-Preconditioning Process for the Formal-Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets. Reliable Computing 11(6): 443–478
Goldsztejn, A.: Définition et Applications des Extensions des Fonctions Réelles aux Intervalles Généralisés, PhDThesis, Université de Nice-Sophia Antipolis, 2005.
Hansen E.R. (1992) Bounding the Solution of Interval Linear Equations. SIAM J. Numer. Anal. 29(5): 1493–1503
Heindl G., Kreinovich V., Lakeyev A.V. (1998) Solving Linear Interval Systems Is NP-Hard Even If We Exclude Overflow and Underflow. Reliable Computing 4(4): 377–381
Markov, S., Popova, E., and Ullrich,Ch.: On the Solution of Linear Algebraic Equations Involving Interval Coefficients, in: Margenov, S. and Vassilevski, P. (eds), Iterative Methods in Linear Algebra, II, IMACS Series in Computational and Applied Mathematics 3 (1996), pp. 216–225.
Moore, R.: Interval Analysis, Prentice Hall, 1966.
Neumaier A. (1999) A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations. Reliable Computing 5(2): 131–136
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.
Ning S., Kearfott R.B. (1997) A Comparison of Some Methods for Solving Linear Interval Equations. SIAM J. Numer. Anal. 34(1): 1289–1305
Oettli W. (1965) On the Solution Set of a Linear System with Inaccurate Coefficients, SIAM J. Numer. Anal. 2(1): 115–118
Rohn J. (1993) Cheap and Tight Bounds: The Recent Result by E.Hansen Can Be Made More Efficient. Interval Computations 1(4): 13–21
Shary S.P. (2002) A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity. Reliable Computing 8(5): 321–418
Shary, S. P.: Algebraic Solutions to Interval Linear Equations and Their Application, in: IMACS— GAMM International Symposium on Numerical Methods and Error Bounds, 1996.
Shary S.P. (2001) Interval Gauss-Seidel Method for Generalized Solution Sets to Interval Linear Systems. Reliable Computing 7(2): 141–155
Shary S.P. (1999) Outer Estimation of Generalized Solution Sets to Interval Linear Systems. Reliable Computing 5(3): 323–335
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Chabert, G., Goldsztejn, A. Extension of the Hansen-Bliek Method to Right-Quantified Linear Systems. Reliable Comput 13, 325–349 (2007). https://doi.org/10.1007/s11155-007-9037-6
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DOI: https://doi.org/10.1007/s11155-007-9037-6