Abstract
Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. I. Zukhovitskii and L. I. Avdeeva, Linear and Convex Programming (Saunders, Philadelphia, Pa., 1966; Nauka, Moscow, 1967).
V. P. Bulatov, Candidate’s Dissertation in Mathematics and Physics (Tomsk State Univ., Tomsk, 1967).
V. P. Bulatov, Embedding Methods in Optimization Problems (Nauka, Novosibirsk, 1977) [in Russian].
S. I. Zukhovitskii, R. A. Polyak, and M. E. Primak, “Two Methods for Finding Equilibrium Points in Concave n-Person Games,” Dokl. Akad. Nauk SSSR 185, 24–27 (1969).
E. I. Nenakhov and M. E. Primak, “On the Convergence of the Method of Chebyshev Centers and Some of Its Applications,” Kibernetika, No. 2, 60–65 (1986).
M. E. Primak, “On the Convergence of the Modified Method of Chebyshev Centers for Convex Programming Problems,” Kibernetika, No. 5, 100–102 (1977).
M. E. Primak, “On the Convergence of the Cutting Plane Method with Cleaning at Each Step,” Kibernetika, No. 1, 119–121 (1980).
J. Elzinda and T. G. Moore, “A Central Cutting Plane Algorithm for the Convex Programming Problems,” Math. Program. 8, 134–145 (1975).
V. Bulatov, T. Belykh, and A. Burdukovskaja, “Methods of the Chebyshev Points for Some Problems of Operations Research,” Proceedings of the 13th Baikal International School-Seminar on Optimization Methods and Their Applications (Irkutsk, 2005), Vol. 1, pp. 37–51.
F. A. Chernous’ko, “Optimal Guaranteed Estimates of Indeterminacies with the Aid of Ellipsoids II,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, 3–11 (1980).
A. I. Ovseevich and Yu. N. Reshetnyak, “Approximation of the Intersection of Ellipsoids in Problems of Guaranteed Estimation,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, 182–189 (1988).
E. G. Antsiferov, L. T. Ashchepkov, and V. P. Bulatov, Optimization Methods and Applications, Part 1: Mathematical Programming (Nauka, Novosibirsk, 1990) [in Russian].
A. Z. Gamm, “Optimization of Electric Power Interconnection Regimes under New Economic Conditions,” Elektrichestvo, No. 11, 1–8 (1993).
A. Z. Gamm, “Tradeoff Control of Economically Independent Electric Power Systems,” Izv. Ross. Akad. Nauk, Energ. 11, 46–57 (1993).
A. Z. Gamm, E. V. Tairova, and O. V. Khamisov, Pareto Optimal and Chebyshev Points in Tradeoff Control of Electric Power Systems (Inst. Sist. Energ., Sib. Otd. Ross. Akad. Nauk, Irkutsk, 1998) [in Russian].
E. G. Antsiferov and V. P. Bulatov, “Simplex Embedding Algorithm in Convex Programming,” Zh. Vychisl. Mat. Mat. Fiz. 27, 377–384 (1987).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © T.I. Belykh, V.P. Bulatov, E.N. Yas’kova, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 1, pp. 18–32.
Rights and permissions
About this article
Cite this article
Belykh, T.I., Bulatov, V.P. & Yas’kova, E.N. Methods of Chebyshev points of convex sets and their applications. Comput. Math. and Math. Phys. 48, 16–29 (2008). https://doi.org/10.1134/S0965542508010028
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542508010028