Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

F. A. Kuterin   ; M. I. Sumin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 1, pp. 55-68

Résumé

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.

Bibliographie

[1] Vasilev F. P., Metody optimizatsii, v 2-kh kn., MTsNMO, M., 2011

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

[3] Sumin M. I., Nekorrektnye zadachi i metody ikh resheniya. Materialy k lektsiyam dlya studentov starshikh kursov, Uchebnoe posobie, Izdatelstvo Nizhegorodskogo gosuniversiteta, Nizhnii Novgorod, 2009

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

[5] Sumin M. I., “On the stable sequential Kuhn–Tucker theorem and its applications”, Appl. Math., 3:10A, special iss. “Optimization” (2012), 1334–1350 | DOI

[6] Sumin M. I., “Ustoichivoe sekventsialnoe vypukloe programmirovanie v gilbertovom prostranstve i ego prilozhenie k resheniyu neustoichivykh zadach”, Zh. vychisl. matem. i matem. fiz., 54:1 (2014), 25–49 | DOI

[7] Errou K. Dzh., Gurvits L., Udzava Kh., Issledovaniya po lineinomu i nelineinomu programmirovaniyu, IL, M., 1962

[8] Minu M., Matematicheskoe programmirovanie. Teoriya i algoritmy, Nauka, M., 1990

[9] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977

[10] Sumin M. I., “Suboptimalnoe upravlenie sistemami s raspredelennymi parametrami: svoistva normalnosti, subgradientnyi dvoistvennyi metod”, Zh. vychisl. matem. i matem. fiz., 37:2 (1997), 162–178 | Zbl

[11] Sumin M. I., “Suboptimalnoe upravlenie sistemami s raspredelennymi parametrami: minimiziruyuschie posledovatelnosti, funktsiya znachenii”, Zh. vychisl. matem. i matem. fiz., 37:1 (1997), 23–41

[12] Loewen P. D., Optimal control via nonsmooth analysis, CRM Proc. and Lecture Notes, 2, Amer. Math. Soc., Providence, RI, 1993 | MR | Zbl

[13] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1979

[14] Oben Zh.-P., Nelineinyi analiz i ego ekonomicheskie prilozheniya, Mir, M., 1988

[15] Bakushinskii A. B., “Metody resheniya monotonnykh variatsionnykh neravenstv, osnovannye na printsipe iterativnoi regulyarizatsii”, Zh. vychisl. matem. i matem. fiz., 17:6 (1977), 1350–1362 | Zbl

[16] Bakushinskii A. B., Goncharskii A. V., Iteratsionnye metody resheniya nekorrektnykh zadach, Nauka, M., 1989 | MR

[17] Sumin M. I., “Iterativnaya regulyarizatsiya gradientnogo dvoistvennogo metoda dlya resheniya uravneniya Fredgolma pervogo roda”, Vestnik Nizhegorodskogo universiteta. Ser. Matematika, 2004, no. 2, 193–209

[18] Krasnov M. L., Kiselev A. I., Makarenko G. I., Integralnye uravneniya, Nauka, M., 1976