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A choice of the regularization parameter in solving convex extremal problems. (English. Russian original) Zbl 0947.90087

Comput. Math. Math. Phys. 37, No. 7, 869-870 (1997); translation from Zh. Vychisl. Mat. Mat. Fiz. 37, No. 7, 895-896 (1997).
The ill-posed problem of the minimization of a convex functional that is not uniformly convex is ill-posed is considered. The author uses a smoothing-functional method for solving this problem, which is based on the generalized residual principles. The aim this work is the construction of a generalized residual principle of a new form, which is based on another data. Such a problem arises from the fact that even for convex minimized functionals, one has to solve the problems of nonconvex minimization when using the above mentioned method. This sometimes presents unsurmountable difficulties in numerically implementing the method.

MSC:

90C25 Convex programming
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] A. S. Antipin, ”Continuous and Iterative Processes with Projection Operators and Projection Type,” in Problems in Cybernetics, Numerical Aspects of Analysis of Large-Scale Systems (Nauchn. sovet po kompleksnoi probleme ”Kibernetika” Akad. Nauk SSSR, Moscow, 1989), pp. 5–43 [in Russian].
[2] I. P. Ryazantseva, ”First-Order Continuous and Iterative Methods with a Generalized Projection Operator for Monotone Variational Inequalities in a Banach Space,” Zh. Vychisl. Mat. Mat. Fiz. 45, 400–410 (2005) [Comput. Math. Math. Phys. 45, 383–393 (2005)]. · Zbl 1086.47509
[3] Ya. I. Al’ber, Doctoral Dissertation in Mathematics and Physics (Novosibirsk. Gos. Univ., Novosibirsk, 1986).
[4] Ya. I. Al’ber, ”Generalized Projection Operators in Banach Spaces: Properties and Applications,” Funct. Differ. Equations 1(1), 1–21 (1994).
[5] A. Nedich, ”Regularized Continuous Method of Gradient Projection for Minimization Problems with Inexact Initial Data,” Vestn. Mosk. Gos. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 1, 3–10 (1994). · Zbl 0856.49025
[6] M. M. Vainberg, Variational Method and Monotone Operator Method in the Theory of Nonlinear Equations (Nauka, Moscow, 1972) [in Russian].
[7] J.-L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires (Dunod, Paris, 1969; Mir, Moscow, 1972).
[8] V. V. Yurgelas, Candidate’s Dissertation in Mathematics and Physics (Voronezh Gos. Univ., Voronezh, 1983).
[9] H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974; Mir, Moscow, 1978). · Zbl 0289.47029
[10] T. Figiel, ”On the Moduli of Convexity and Smoothness,” Studia Math. 56(2), 121–155 (1976). · Zbl 0344.46052
[11] J. Diestel, Geometry of Banach Spaces (Springer-Verlag, Berlin, 1975; Vishcha Shkola, Kiev, 1980).
[12] A. I. Notik, Candidate’s Dissertation in Mathematics and Physics (Voronezh, 1986).
[13] V. V. Vladimirov, Yu. E. Nesterov, and Yu. N. Chekanov, ”On Uniformly Convex Functional,” Vestn. Mosk. Gos. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 3, 12–23 (1972).
[14] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Graylock, Albany, N.Y., 1961; Nauka, Moscow, 1976). · Zbl 0103.08801
[15] D. Pascali and S. Sburlan, Nonlinear Operators of Monotone Type (R.S.R., Bucharest, 1978). · Zbl 0423.47021
[16] F. P. Vasil’ev, Numerical Solution Methods for Extremal Problems (Nauka, Moscow, 1988) [in Russian].
[17] I. P. Ryazantseva, Doctoral Dissertation in Mathematics and Physics (Novosibirsk Gos. Univ., Novosibirsk, 1996).
[18] F. P. Vasil’ev, Solution Methods for Extremal Problems (Nauka, Moscow, 1981) [in Russian].
[19] I. P. Ryazantseva, ”A Continuous First-Order Regularization Method for Monotone Variational Inequalities in a Banach Space,” Differ. Uravn. 39, 113–117 (2003) [Differ. Equations 39, 121–126 (2003)]. · Zbl 1175.49011
[20] I. P. Ryazantseva and O. Yu. Bubnova, ”Second-Order Continuous Method for Nonlinear Accretive Equations in a Banach Space,” Tr. Srednevolzhskogo Mat. O-va 3/4(1), 327–334 (2002). · Zbl 1091.47512
[21] Ya. Alber, D. Butnariu, and I. Ryazantseva, ”Regularization Methods for Ill-Posed Inclusions and Variational Inequalities with Domain Perturbations,” J. Nonlinear Convex Anal. 2(1), 53–79 (2001). · Zbl 1003.47047
[22] I. P. Ryazantseva, ”How to Approximate Solutions of Variational Inequalities with Monotone Mappings in a Banach Space if the Data Are Known Approximately,” Differ. Uravn. 40, 1108–1117 (2004) [Differ. Equations 40, 1174–1183 (2004)]. · Zbl 1078.47039
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