On ill-posed problems, extremals of the Tikhonov functional and the regularized Lagrange principles

M. I. Sumin

Vestnik rossijskih universitetov. Matematika, Tome 27 (2022) no. 137, pp. 58-79

Résumé

The problem of finding a normal solution to an operator equation of the first kind on a pair of Hilbert spaces is classical in the theory of ill-posed problems. In accordance with the theory of regularization, its solutions are approximated by the extremals of the Tikhonov functional. From the point of view of the theory of problems for constrained extremum, the problem of minimizing a functional, equal to the square of the norm of an element, with an operator equality constraint (that is, given by an operator with an infinite-dimensional image) is equivalent to the classical ill-posed problem. The paper discusses the possibility of regularizing the Lagrange principle (LP) in the specified constrained extremum problem. This regularization is a transformation of the LP that turns it into a universal tool of stable solving ill-posed problems in terms of generalized minimizing sequences (GMS) and preserves its “general structural arrangement” based on the constructions of the classical Lagrange function. The transformed LP “contains” the classical analogue as its limiting variant when the numbers of the GMS elements tend to infinity. Both non-iterative and iterative variants of the regularization of the LP are discussed. Each of them leads to stable generation of the GMS in the original constrained extremum problem from the extremals of the regular Lagrange functional taken at the values of the dual variable generated by the corresponding procedure for the regularization of the dual problem. In conclusion, the article discusses the relationship between the extremals of the Tikhonov and Lagrange functionals in the considered classical ill-posed problem.

Détail

Keywords: ill-posed problem, linear operator equation, Tikhonov's regularization method, constrained minimization, operator equality constraint, generalized minimizing sequence, regularizing algorithm, iterative regularization, dual regularization, regularized Lagrange principle.
Mots-clés : Lagrange multiplier rule

Bibliographie

[1] A. N. Tikhonov, “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Doklady AN SSSR, 151:3 (1963), 501–504 | Zbl

[2] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1974; A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems, Winston; Halsted Press, Washington; New York, 1977 | MR | Zbl

[3] A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, A. G. Yagola, Regularizing Algorithms and A Priori Information, Nauka Publ., Moscow, 1983 (In Russian) | MR

[4] A. B. Bakushinskii, A. V. Goncharskii, Incorrect Problems. Numerical Methods and Applications, Moscow University Publishing House, Moscow, 1989 (In Russian)

[5] E. G. Golshtein, Duality Theory in Mathematical Programming and its Applications, Nauka Publ., Moscow, 1971 (In Russian)

[6] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972 | MR | Zbl

[7] F. P. Vasil’ev, Optimization methods: in 2 books, MCCME, Moscow, 2011 (In Russian)

[8] M. I. Sumin, “Regulyarizovannaya parametricheskaya teorema Kuna–Takkera v gilbertovom prostranstve”, Zhurn. vychisl. matem. i matem. fiz., 51:9 (2011), 1594–1615 | MR | Zbl

[9] M. I. Sumin, “Regularized Lagrange principle and Pontryagin maximum principle in optimal control and in inverse problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 1, 2019, 279–296

[10] V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimalnoe upravlenie, Nauka, M., 1979 ; V. M. Alekseev, V. M. Tikhomirov, S. V. Fomin, Optimal control, Plenum Press, New York, 1987 | MR | MR

[11] E. R. Avakov, G. G. Magaril-Ilyaev, V. M. Tikhomirov, “O printsipe Lagranzha v zadachakh na ekstremum pri nalichii ogranichenii”, Uspekhi matem. nauk, 68:3(411) (2013), 5–38 | MR | Zbl

[12] M. I. Sumin, “Nondifferential Kuhn–Tucker theorems in constrained extremum problems via subdifferentials of nonsmooth analysis”, Russian Universities Reports. Mathematics, 25:131 (2020), 307–330 (In Russian) | Zbl

[13] V. A. Trenogin, Functional Analysis, Nauka Publ., Moscow, 1980 (In Russian) | MR

[14] J. -P. Aubin, L’analyse Non Lineaire Et Ses Motivations Economiques, Masson, Paris–New York, 1984 | MR

[15] P. D. Loewen, Optimal Control via Nonsmooth Analysis, v. 2, CRM Proceedings Lecture Notes, Amer. Math. Soc., Providence, RI, 1993 | DOI | MR | Zbl

[16] M. I. Sumin, “On the regularization of the classical optimality conditions in convex optimal control problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 2, 2020, 252–269 (In Russian)

[17] M. I. Sumin, “Regulyarizatsiya v lineino-vypukloi zadache matematicheskogo programmirovaniya na osnove teorii dvoistvennosti”, Zhurn. vychisl. matem. i matem. fiz., 47:4 (2007), 602–625 | MR | Zbl

[18] M. I. Sumin, “Regulyarizovannyi gradientnyi dvoistvennyi metod resheniya obratnoi zadachi finalnogo nablyudeniya dlya parabolicheskogo uravneniya”, Zhurn. vychisl. matem. i matem. fiz., 44:11 (2004), 2001–2019 | MR | Zbl