Abstract
The distance from a given point to the solution set of a system of strict and nonstrict inequalities described by convex functions is estimated. As consequences, estimates for the distance from a given point to the Lebesgue set of a convex function are obtained and sufficient conditions for convex-valued set-valued mappings to be covering are established.
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References
F. P. Vasil’ev and A. Yu. Ivanitskii, Linear Programming (Faktorial, Moscow, 1998) [in Russian].
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control (Nauka, Moscow, 1979) [in Russian].
A. F. Izmailov and M. V. Solodov, Numerical Optimization Methods (Fizmatlit, Moscow, 2005) [in Russian].
J.-Sh. Pang, “Error bounds in mathematical programming,” Math. Program. 79, 299–332 (1997).
F. Facchinei and J.-Sh. Pang, Finite-Dimensional Variational Inequalities and Complementary Problems (Springer, New York, 2003).
A. M. Ter-Krikorov, “Convex programming in a space adjoint to a Banach space and convex optimal control problems with phase constraints,” USSR Comput. Math. Math. Phys. 16 (2), 68–75 (1976).
A. V. Arutyunov, “Covering mappings in metric spaces and fixed points,” Dokl. Math. 76 (2), 665–668 (2007).
A. V. Arutyunov, “An iterative method for finding coincidence points of two mappings,” Comput. Math. Math. Phys. 52 (1), 1483–1486 (2012).
A. S. Lewis and J.-S. Pang, “Error bounds for convex inequality systems,” Proceedings of the Fifth International Symposium of Generalized Convexity (Luminy, 1996).
O. L. Mangasarian, “Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification,” Math. Program. Tech. Rep. 96–04 (Computer Sci. Dep., Univ. of Wisconsin, Madison, 1996).
F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].
A. D. Ioffe and V. M. Tikhomirov, Theory of Optimization Problems (Nauka, Moscow, 1974) [in Russian].
G. J. Zalmai, “A continuous-time generalization of Gordan’s transposition theorem,” J. Math. Anal. Appl. 110, 130–140 (1985).
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Original Russian Text © A.V. Arutyunov, S.E. Zhukovskiy, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 9, pp. 1486–1492.
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Arutyunov, A.V., Zhukovskiy, S.E. On estimates for solutions of systems of convex inequalities. Comput. Math. and Math. Phys. 55, 1444–1450 (2015). https://doi.org/10.1134/S0965542515070039
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DOI: https://doi.org/10.1134/S0965542515070039