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## Archive of Issues

Russia Nizhni Novgorod; Tambov
Year
2020
Volume
30
Issue
3
Pages
410-428
 Section Mathematics Title On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems Author(-s) Sumin M.I.ab Affiliations Nizhni Novgorod State Universitya, Tambov State Universityb Abstract We consider the regularization of the Lagrange principle (LP) in the convex constrained optimization problem with operator constraint-equality in a Hilbert space and with a finite number of functional inequality-constraints. The objective functional of the problem is not, generally speaking, strongly convex. The set of admissible elements of the problem is also embedded into a Hilbert space and is not assumed to be bounded. Obtaining a regularized LP is based on the dual regularization method and involves the use of two regularization parameters and two corresponding matching conditions at the same time. One of the regularization parameters is «responsible» for the regularization of the dual problem, while the other is contained in a strongly convex regularizing addition to the objective functional of the original problem. The main purpose of the regularized LP is the stable generation of generalized minimizing sequences that approximate the exact solution of the problem by function and by constraint, for the purpose of its practical stable solving. Keywords constrained optimization, instability, dual regularization, regularized Lagrange principle, generalized minimizing sequence UDC 519.853, 517.98 MSC 90C25, 90C46, 47A52, 65F22 DOI 10.35634/vm200305 Received 1 June 2020 Language Russian Citation Sumin M.I. On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 3, pp. 410-428. References Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal Control, New York: Plenum Press, 1987. https://doi.org/10.1007/978-1-4615-7551-1 Vasil'ev F.P. Metody optimizatsii (Optimization methods), vols. 1, 2, Moscow: Moscow Center for Continuous Mathematical Education, 2011. Sumin M.I. Regularized parametric Kuhn-Tucker theorem in a Hilbert space, Comput. Math. Math. Phys., 2011, vol. 51, no. 9, pp. 1489-1509. https://doi.org/10.1134/S0965542511090156 Sumin M.I. Stable sequential convex programming in a Hilbert space and its application for solving unstable problems, Comput. Math. Math. Phys., 2014, vol. 54, no. 1, pp. 22-44. https://doi.org/10.1134/S0965542514010138 Sumin M.I. Why regularization of Lagrange principle and Pontryagin maximum principle is needed and what it gives, Vestnik Tambovskogo Universiteta. Seriya Estestvennye i Tekhnicheskie Nauki, 2018, vol. 23, no. 124, pp. 757-775 (in Russian). https://doi.org/10.20310/1810-0198-2018-23-124-757-775 Sumin M.I. Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 279-296 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-279-296 Tikhonov A.N., Arsenin V.Ya. Solutions of Ill-Posed Problems, New York: Halsted Press, 1977. Krasnov M.L., Kiselev A.I., Makarenko G.I. Integral'nye uravneniya (Integral Equations), Moscow: Nauka, 1976. Sumin M.I. Duality-based regularization in a linear convex mathematical programming problem, Comput. Math. Math. Phys., 2007, vol. 47, no. 4, pp. 579-600. https://doi.org/10.1134/S0965542507040045 Gol'shtein E.G. Teoriya dvoistvennosti v matematicheskom programmirovanii i ee prilozheniya (Duality theory in mathematical programming and its applications), Moscow: Nauka, 1971. Warga J. Optimal control of differential and functional equations, New York: Academic Press, 1972. Translated under the title Optimal'noe upravlenie differentsial'nymi i funktsional'nymi uravneniyami, Moscow: Nauka, 1977. Sumin M.I. On the regularization of the classical optimality conditions in convex optimal control problems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 252-269 (in Russian). https://doi.org/10.21538/0134-4889-2020-26-2-252-269 Sumin M.I. On the stable sequential Lagrange principle in convex programming and its application for solving unstable problems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, vol. 19, no. 4, pp. 231-240 (in Russian). Kuterin F.A., Sumin M.I. On the regularized Lagrange principle in the iterative form and its application for solving unstable problems, Mathematical Models and Computer Simulations, 2017, vol. 9, no. 3, pp. 328-338. https://doi.org/10.1134/S2070048217030085 Kuterin F.A., Sumin M.I. The stable iterative Lagrange principle in convex programming as an instrument of solving unstable problems, Comput. Math. Math. Phys., 2017, vol. 57, no. 1, pp. 71-82. https://doi.org/10.1134/S0965542517010092 Full text