Abstract
An operator regularization method is considered for ill-posed vector optimization of weakly lower semicontinuous essentially convex functionals on reflexive Banach spaces. The regularization parameter is chosen by a modified generalized discrepancy principle. A condition for the estimation of the convergence rate of regularized solutions is derived.
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References
R.E. Steuer, Multiple-Criteria Optimization: Theory, Computation, and Applications (Wiley, New York, 1986).
K. Miettinen, Nonlinear Multiobjective Optimization (Kluwer Academic, Boston, 1999).
S. Schaffler, R. Schultz, and K. Weinzierl, “Stochastic Method for the Solution of Unconstrained Vector Optimization Problems,” J. Optim. Theory Appl. 114, 209–222 (2002).
M. M. Vainberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations (Nauka, Moscow, 1972) [in Russian].
Ya. I. Alber, “On Solving Nonlinear Equations Involving Monotone Operators in Banach Spaces,” Sib. Mat. Zh. 26, 3–11 (1975).
Ya. I. Alber and I. P. Ryazantseva, “On Solutions of Nonlinear Problems Involving Monotone Discontinuous Operators,” Differ. Uravn. 25, 331–342 (1979).
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Noordhoff Int. Publ. Leyden Ed. Acad. Bucuresti, Romania, Netherlands, 1976).
E. F. Browder, “Existence and Approximation of Solutions of Nonlinear Variational Inequalities,” Proc. Natl. Acad. Sci. USA 56, 1080–1086 (1966).
I. P. Ryazantseva, “On Solving Variational Inequalities with Monotone Operators by Regularization Method,” Zh. Vychisl. Mat. Fiz. 23, 479–483 (1983).
I. P. Ryazantseva, “Operator Method of Regularization for Problems of Optimal Programming with Monotone Maps,” Sib. Mat. Zh. 24, 214 (1983).
H. W. Engl, “Discrepancy Principle for Tikhonov Regularization of Ill-Posed Problems Leading to Optimal Convergence Rates,” J. Optim. Theory Appl. 52, 209–215 (1987).
Nguyen Buong, “Generalized Discrepancy Principle and Ill-Posed Equations Involving Accretive Operators,” J. Nonlinear Funct. Anal. Appl., Korea 9, 73–78 (2004).
Nguyen Buong and Pham Van Loi, “On Parameter Choice and Convergence Rates in Regularization for a Class of Ill-Posed Variational Inequalities,” Zh. Vychisl. Mat. Mat. Fiz. 44, 1735–1744 (2004) [Comput. Math. Math. Phys. 44, 1649–1658 (2004)].
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Buong, N. Regularization for unconstrained vector optimization of convex functionals in Banach spaces. Comput. Math. and Math. Phys. 46, 354–360 (2006). https://doi.org/10.1134/S096554250603002X
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DOI: https://doi.org/10.1134/S096554250603002X