Milan Hladík and Evgenija D. Popova. Maximal inner boxes in parametric AE-solution sets with linear shape. Appl. Math. Comput., 270:606–619, 2015.
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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.
@article{HlaPop2015a,
author = "Milan Hlad\'{\i}k and Evgenija D. Popova",
title = "Maximal inner boxes in parametric {AE}-solution sets with linear shape",
journal = "Appl. Math. Comput.",
fjournal = "Applied Mathematics and Computation",
volume = "270",
pages = "606-619",
year = "2015",
doi = "10.1016/j.amc.2015.08.003",
issn = "0096-3003",
bib2html_dl_html = "http://dx.doi.org/10.1016/j.cam.2015.07.034",
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.",
keywords = "Linear equations; Dependent interval parameters; Tolerable solution set; AE-solution set; Inner estimation",
}
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