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Maximal inner boxes in parametric AE-solution sets with linear shape

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  • Hladík, Milan
  • Popova, Evgenija D.

Abstract

We consider linear systems of equations A(p)x=b(p), where the parameters p are linearly dependent and come from prescribed boxes, and the sets of solutions (defined in various ways) which have linear boundary. One fundamental problem is to compute a box being inside a parametric solution set. We first consider parametric tolerable solution sets (being convex polyhedrons). For such solution sets we prove that finding a maximal inner box is an NP-hard problem. This justifies our exponential linear programming methods for computing maximal inner boxes. We also propose a polynomial heuristic that yields a large, but not necessarily the maximal, inner box. Next, we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric AE-solution sets with linear shape. Numerical examples illustrate the properties of the methods and their application.

Suggested Citation

  • Hladík, Milan & Popova, Evgenija D., 2015. "Maximal inner boxes in parametric AE-solution sets with linear shape," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 606-619.
    • Handle: RePEc:eee:apmaco:v:270:y:2015:i:c:p:606-619
    DOI: 10.1016/j.amc.2015.08.003
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    References listed on IDEAS

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    1. Filippi, Carlo, 2005. "A fresh view on the tolerance approach to sensitivity analysis in linear programming," European Journal of Operational Research, Elsevier, vol. 167(1), pages 1-19, November.
    2. Richard E. Wendell, 1985. "The Tolerance Approach to Sensitivity Analysis in Linear Programming," Management Science, INFORMS, vol. 31(5), pages 564-578, May.
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    Cited by:

    1. Popova, Evgenija D., 2017. "Parameterized outer estimation of AE-solution sets to parametric interval linear systems," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 353-360.
    2. Jianzhe Zhen & Dick Hertog, 2017. "Centered solutions for uncertain linear equations," Computational Management Science, Springer, vol. 14(4), pages 585-610, October.
    3. Alexandre dit Sandretto, Julien & Hladík, Milan, 2017. "Solving over-constrained systems of non-linear interval equations – And its robotic application," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 180-195.

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