Milan Hladík. Robust Slater's condition in an uncertain environment. In 39th International Conference on Mathematical Methods in Economics 2021. Conference Proceedings, pp. 161–166, Czech University of Life Sciences Prague, 2021.
Slater's condition is, no doubt, an important regularity condition used in nonlinear programming. It states that the feasible set must contain an interior point. We analyse this condition in an uncertain environment. We assume that uncertainty of the input data has the form of intervals covering the true values; we assume no other information about the uncertainty is known. Then Slater's condition holds robustly if it is satisfied for each possible realization of the interval values. In particular, we investigate interval systems of linear equations and inequalities. Therein, Slater's condition has the form of strong solvability with strict inequalities. We present a finite characterization of this property and inspect its computational complexity – in some cases it is polynomial, but in some cases it is NP-hard. As an illustration, we apply our results in interval linear programming in the problem of testing boundedness of the optimal solution set.
@InProceedings{Hla2021c,
author = "Milan Hlad\'{\i}k",
title = "Robust {Slater}'s condition in an uncertain environment",
editor = "Robert Hlavat\'{y}",
booktitle = "39th International Conference on Mathematical Methods in Economics 2021. Conference Proceedings",
pages = "161-166",
year = "2021",
publisher = "Czech University of Life Sciences Prague",
isbn = "978-80-213-3126-6",
url = "https://mme2021.v2.czu.cz/dl/99363?lang=en",
bib2html_dl_pdf = "https://mme2021.v2.czu.cz/dl/99363?lang=en",
abstract = "Slater's condition is, no doubt, an important regularity condition used in nonlinear programming. It states that the feasible set must contain an interior point. We analyse this condition in an uncertain environment. We assume that uncertainty of the input data has the form of intervals covering the true values; we assume no other information about the uncertainty is known. Then Slater's condition holds robustly if it is satisfied for each possible realization of the interval values. In particular, we investigate interval systems of linear equations and inequalities. Therein, Slater's condition has the form of strong solvability with strict inequalities. We present a finite characterization of this property and inspect its computational complexity – in some cases it is polynomial, but in some cases it is NP-hard. As an illustration, we apply our results in interval linear programming in the problem of testing boundedness of the optimal solution set.",
keywords = "Linear programming; Interval analysis; Interval system; Robustness; NP-
hardness",
}
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