Abstract
The problem of approximating the set of all solutions to a system of nonlinear inequalities is studied. A method based on the concept of nonuniform coverings is proposed. It allows one to obtain an interior and exterior approximation of this set with a prescribed accuracy. The efficiency of the method is demonstrated by determining the workspace of a parallel robot.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. V. Lotov and A. I. Pospelov, “The modified method of refined bounds for polyhedral approximation of convex polytopes,” Comput. Math. Math. Phys. 48, 933–941 (2008).
G. K. Kamenev, “Method for polyhedral approximation of a ball with an optimal order of growth of the facet structure cardinality,” Comput. Math. Math. Phys. 54, 1201–1213 (2014).
G. K. Kamenev, “Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls,” Comput. Math. Math. Phys. 56, 744–755 (2016).
V. A. Bushenkov and A. V. Lotov, “Methods and algorithms for the analysis of linear systems based on constructing generalized sets of attainability,” Zh. Vychisl. Mat. Mat. Fiz. 20 (5), 1130–1141 (1980).
E. M. Voronov, A. P. Karpenko, O. G. Kozlova, and V. A. Fedin, “Numerical for constructing the attainability set of a dynamic system,” in Vestn. Bauman Mosk. Gos. Univ., Ser. Priborostroenie (Moscow, 2010) [in Russian].
Yu. G. Evtushenko and M. A. Posypkin, “Effective hull of a set and its approximation,” Dokl. Math. 90, 791–794 (2014).
V. A. Garanzha and L. N. Kudryavtseva, “Generation of three-dimensional Delaunay meshes from weakly structured and inconsistent data,” Comput. Math. Math. Phys. 52, 427–447 (2012).
Yu. G. Evtushenko, “A numerical method for finding the global extremum of a function (search on a nonuniform grid),” Zh. Vychisl. Mat. Mat. Fiz. 11, 1390–1403 (1971).
J. P. Evans and F. J. Gould, “Stability in nonlinear programming,” Oper. Res. 18 (1) 107–118 (1970).
L. A. Rybak, V. V. Erzhukov, and A. V. Chichvarin, Efficient Methods for solving Problems of the Kinematics and Dynamics for a Parallel Machine Tool Robot (Fizmatlit, Moscow, 2011) [in Russian].
L. A. Rybak and A. V. Sinev, “Synthesis of the multiconnected digital control of an active vibration isolation system for a two-dimensional planar problem,” Probl. Mashinostr. Nadezhnosti Mashin., No. 3, 10–17 (1997).
J. Huerta-Cepas, F. Serra, and P. Bork, “ETE 3: Reconstruction, analysis and visualization of phylogenomic data,” Mol. Biol. Evol. 2016. doi 10.1093/molbev/msw046
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.G. Evtushenko, M.A. Posypkin, L.A. Rybak, A.V. Turkin, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 8, pp. 1248–1254.
Rights and permissions
About this article
Cite this article
Evtushenko, Y.G., Posypkin, M.A., Rybak, L.A. et al. Finding sets of solutions to systems of nonlinear inequalities. Comput. Math. and Math. Phys. 57, 1241–1247 (2017). https://doi.org/10.1134/S0965542517080073
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517080073