Saeed Ketabchi, Hossein Moosaei, Mario Rosario Guarracino, and Milan Hladík. Separating two polyhedra utilizing alternative theorems and penalty function. In Dimitris E. Simos and others, editors, Learning and Intelligent Optimization, LNCS, pp. 27–39, Springer, Cham, 2022.
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The separation of two polyhedra by a family of parallel hyperplanes is a well-known problem with important applications in operations research,statistics and functional analysis. In this paper, we introduce a new algorithm for constructing a family of parallel hyperplanes that separates two disjoint polyhedra given by a system of linear inequalities. To do this, we consider the alternative system and introduce its dual problem using the alternative theorem. We can find its minimum-norm solution by combining the objective function and constraints into a penalty function. Since our objective function is only once differentiable, we propose an extension of Newton's method to solve the unconstrained objective optimization. The computational outcomes demonstrate the efficacy of the proposed method.
@inCollection{KetMoo2022a,
author = "Saeed Ketabchi and Hossein Moosaei and Mario Rosario Guarracino and Milan Hlad\'{\i}k",
title = "Separating two polyhedra utilizing alternative theorems and penalty function",
editor = "Dimitris E. Simos and others",
feditor = "Simos, Dimitris E. and Rasskazova, Varvara A. and Archetti, Francesco and Kotsireas, Ilias S. and Pardalos, Panos M.",
booktitle = "Learning and Intelligent Optimization",
fbooktitle = "Learning and Intelligent Optimization, 16th International Conference, LION 16
Milos Island, Greece, June 5-10, 2022, Revised Selected Papers",
publisher = "Springer",
address = "Cham",
series = "LNCS",
fseries = "Lecture Notes in Computer Science",
volume = "13621",
pages = "27-39",
year = "2022",
doi = "10.1007/978-3-031-24866-5_3",
isbn = "978-3-031-24866-5",
issn = "0302-9743",
url = "https://doi.org/10.1007/978-3-031-24866-5_3",
bib2html_dl_html = "https://link.springer.com/chapter/10.1007/978-3-031-24866-5_3",
bib2html_dl_pdf = "https://rdcu.be/c5uBU",
abstract = "The separation of two polyhedra by a family of parallel hyperplanes is a well-known problem with important applications in operations research,statistics and functional analysis. In this paper, we introduce a new algorithm for constructing a family of parallel hyperplanes that separates two disjoint polyhedra given by a system of linear inequalities. To do this, we consider the alternative system and introduce its dual problem using the alternative theorem. We can find its minimum-norm solution by combining the objective function and constraints into a penalty function. Since our objective function is only once differentiable, we propose an extension of Newton's method to solve the unconstrained objective optimization. The computational outcomes demonstrate the efficacy of the proposed method.",
keywords = "Polyhedra; Separating theorem; Theorems of alternative; Penalty function; Generalized",
}
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