Regularized first-order methods for monotone variational inequalities with generalized projection operators

I. P. Ryazantseva

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 11, pp. 1954-1962

Résumé

For a certain class of Banach spaces, variational inequalities with monotone operators are examined under the assumption that the data are given approximately. Regularized first-order methods (a continuous and an iterative one) are constructed in the form of equations containing generalized projection operators. Sufficient conditions are obtained for the strong convergence of these methods to the normal solution of the original problem.

Bibliographie

[1] Alber Ya. I., Metody resheniya nelineinykh operatornykh uravnenii i variatsionnykh neravenstv v banakhovykh prostranstvakh, Dis. ...dokt. fiz.-matem. nauk, NGU, Novosibirsk, 1986

[2] Bakushinskii A. B., Goncharskii A. V., Nekorrektnye zadachi. Chislennye metody i prilozheniya, Izd-vo MGU, M., 1989

[3] Alber Ya., Butnariu D., Ryazantseva I., “Regularization methods for ill-posed inclusions and variational inequalities with domain perturbations”, J. Nonlinear and Convex Analys., 2:1 (2001), 53–79 | MR | Zbl

[4] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[5] Antipin A. S., “Nepreryvnye i iterativnye protsessy s operatorami proektirovaniya i tipa proektirovaniya”, Vopr. kibernetiki. Vychisl. vopr. analiza bolshikh sistem, Nauchn. sovet po kompleksnoi probleme “Kibernetika” AN SSSR, M., 1989, 5–43 | MR

[6] Nedich A., “Regulyarizovannyi nepreryvnyi metod proektsii gradienta dlya zadach minimizatsii s netochnymi iskhodnymi dannymi”, Vestn. MGU. Ser. 15, 1994, no. 1, 3–10 | MR | Zbl

[7] Vasilev F. P., Nedich A., “Ob odnom variante regulyarizovannogo metoda proektsii gradienta”, Zh. vychisl. matem. i matem. fiz., 34:4 (1994), 511–519 | MR

[8] Alber Ya. I., “Generalized projection operators in Banach spaces: Properties and applications”, Funct. Different. Equations, 1:1 (1994), 1–21 | MR

[9] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii, Nauka, M., 1972 | MR

[10] Serb I., “Estimates of the modulus of convexity and smoothness of a Banach spaces”, Mathematica, 34:1 (1992), 61–70 | MR | Zbl

[11] Figiel T., “On the moduli of convexity and smoothness”, Studia Math., 56:2 (1976), 121–155 | MR | Zbl

[12] Distel D., Geometriya banakhovykh prostranstv, Vischa shkola, Kiev, 1980 | MR

[13] Ryazantseva I. P., “Nepreryvnye i iterativnye metody pervogo poryadka s operatorom obobschennogo proektirovaniya dlya monotonnykh variatsionnykh neravenstv v banakhovom prostranstve”, Zh. vychisl. matem. i matem. fiz., 45:3 (2005), 400–410 | MR | Zbl

[14] Ryazantseva I. P., “Ob odnom sposobe approksimatsii reshenii variatsionnykh neravenstv s monotonnymi operatorami v banakhovom prostranstve pri priblizhennom zadanii dannykh”, Differents. ur-niya, 40:8 (2004), 1108–1117 | MR | Zbl

[15] Alber Ya. I., “A new approach to investigation of evolution differential equations in Banach space”, Nonlinear. Analys., 23:9 (1994), 1115–1134 | DOI | MR | Zbl

[16] Ryazantseva I. P., Bubnova O. Yu., “Nepreryvnyi metod vtorogo poryadka dlya nelineinykh akkretivnykh uravnenii v banakhovom prostranstve”, Tr. Srednevolzhskogo matem. ob-va, 3–4:1 (2002), 327–334

[17] Alber Ya. I., Ryazantseva I. P., “Minimizatsiya vypuklykh funktsionalov”, Tezisy dokl. Vses. konf. po ekstremalnym zadacham i ikh prilozh., Tallin, 1973, 18–19

[18] Ryazantseva I. P., “Nepreryvnyi metod regulyarizatsii pervogo poryadka dlya monotonnykh variatsionnykh neravenstv v banakhovom prostranstve”, Differents. ur-niya, 39:1 (2003), 113–117 | MR | Zbl

[19] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[20] Apartsin A. S., “K postroeniyu iteratsionnykh protsessov v gilbertovom prostranstve”, Tr. po prikl. matem. i kibernetike, Irkutsk, 1972, 7–14

[21] Lindenstrauss J., Tzafriri L., Classical Banach spaces, v. II, Springer, Berlin etc., 1979 | MR