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.
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Acknowledgments
H. Moosaei was supported by the Czech Science Foundation Grant 22-19353S. M. Hladík was supported by the Czech Science Foundation Grant P403-22-11117S.
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Ketabchi, S., Moosaei, H., Guarracino, M.R., Hladík, M. (2022). Separating Two Polyhedra Utilizing Alternative Theorems and Penalty Function. In: Simos, D.E., Rasskazova, V.A., Archetti, F., Kotsireas, I.S., Pardalos, P.M. (eds) Learning and Intelligent Optimization. LION 2022. Lecture Notes in Computer Science, vol 13621. Springer, Cham. https://doi.org/10.1007/978-3-031-24866-5_3
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