Open Access. © 2020 E. Kiseleva et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License Open Comput. Sci. 2020; 10:124–136 Research Article Elena Kiseleva, Olha Prytomanova, and Liudmyla Hart* Solving a Two-stage Continuous-discrete Problem of Optimal Partitioning-Allocation with Subsets Centers Placement https://doi.org/10.1515/comp-2020-0142 Received Sep 04, 2019; accepted Apr 06, 2020 Abstract: A two-stage continuous-discrete optimal partitioning-allocation problem is studied, and a method and an algorithm for its solving are proposed. This prob- lem is a generalization of a classical transportation prob- lem to the case when coordinates of the production points (collection, storage, processing) of homogeneous products are continuously allocated in the given domain and the production volumes at these points are unknown. These coordinates are found as a solution of the corresponding continuous optimal set-partitioning problem in a finite- dimensional Euclidean space with the placement (finding coordinates) of these subsets’ centers. Also, this problem generalizes discrete two-stage production-transportation problems to the case of continuously allocated consumers. The method and algorithm are illustrated by solving two model problems. Keywords: infinite-dimensional mathematical program- ming, the theory of optimal set-partitioning, transporta- tion problem, non-differentiable optimization, Shor’s r- algorithm. 2010 Mathematics Subject Classification: 49M29, 49J52, 49K35 *Corresponding Author: Liudmyla Hart: Department of Computa- tional Mathematics and Mathematical Cybernetics, Oles Honchar Dnipro National University (DNU), Dnipro, 49010, Ukraine; Email: ll_hart@ukr.net Elena Kiseleva: Faculty of Applied Mathematics, Oles Honchar Dnipro National University (DNU), Dnipro, 49010, Ukraine; Email: kiseleva47@ukr.net Olha Prytomanova: Department of Computational Mathematics and Mathematical Cybernetics, Oles Honchar Dnipro National University (DNU), Dnipro, 49010, Ukraine; Email: olgmp@ua.fm 1 Introduction The mathematical theory of optimal set-partitioning (OSP) in n-dimensional Euclidean space is a new non-classical area of infinite-dimensional mathematical programming operating with Boolean variables. Studies on the optimal set-partitioning topic were started in the 1970s by scientists H. Corley and S. Roberts [1, 2] and, independently, by mathematicians I. Beyko and E. Kiseleva [3, 4]. In their studies, H. Corley and S. Roberts were only able to get the necessary conditions for optimal- ity of partitions. At the same time, I. Beyko and E. Kise- leva managed to advance further and to develop numer- ical algorithms for finding optimal solutions based on similar necessary optimality conditions obtained by them. This was made possible by using effective methods of non-differentiable optimization (various variants of the Shor’s r-algorithm [5–8]) for the numerical solution of aux- iliary finite-dimensional optimization problems that arise during the development of methods for solving infinite- dimensional OSP problems. Numerous important theoretical and practical opti- mization problems, that are completely different, are re- ducible to continuous models of optimal set-partitioning. Here, let us indicate a few of them: the generalized Neumann-Pearson problems [9]; various global optimiza- tion problems [10]; a problem of forming optimal quadra- tures [9]; a problem of constructing the Dirichlet-Voronoi diagram and its generalizations [9, 11, 12]; optimal ball cov- ering continuous problems [13] and other optimal geomet- ric objects’ packing [14, 15], etc. Also, the OSP theory can be used for solving applied problems such as pattern recognition with a goal to min- imize the average function of wrong recognition; medical diagnostics aiming to minimize errors of diagnosis; service sector territorial planning; geological forecasting; environ- ment protection, e.g., a problem of providing environmen- tal safety of placing radioactive waste disposal facilities of nuclear power plants, taking into account an ecologi- cal structure of the placement domain; placement of am-