Geodesic
New numerical methods and some applied aspects of the $p$-regularity theory
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 11, pp. 1987-2000 
O. A. Brezhneva; Yu. G. Evtushenko; A. A. Tret'yakov. New numerical methods and some applied aspects of the $p$-regularity theory. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 11, pp. 1987-2000. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a5/
@article{ZVMMF_2006_46_11_a5,
     author = {O. A. Brezhneva and Yu. G. Evtushenko and A. A. Tret'yakov},
     title = {New numerical methods and some applied aspects of the $p$-regularity theory},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1987--2000},
     year = {2006},
     volume = {46},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a5/}
}
TY  - JOUR
AU  - O. A. Brezhneva
AU  - Yu. G. Evtushenko
AU  - A. A. Tret'yakov
TI  - New numerical methods and some applied aspects of the $p$-regularity theory
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 1987
EP  - 2000
VL  - 46
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a5/
LA  - ru
ID  - ZVMMF_2006_46_11_a5
ER  - 
%0 Journal Article
%A O. A. Brezhneva
%A Yu. G. Evtushenko
%A A. A. Tret'yakov
%T New numerical methods and some applied aspects of the $p$-regularity theory
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 1987-2000
%V 46
%N 11
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_11_a5/
%G ru
%F ZVMMF_2006_46_11_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

The theory of $p$-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu. G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the $p$-factor method for solving ODEs with a small parameter.

[1] Brezhneva O. A., Tretyakov A. A., Metody resheniya suschestvenno nelineinykh zadach, VTs RAN, M., 2000

[2] Tret'yakov A. A., Marsden J. E., “Factor-analysis of nonlinear mapping: $p$-reularity theory”, Communs Pure and Appf Analys., 2 (2003), 425–445 | DOI | MR

[3] Izmailov A. F., Tretyakov A. A., 2-regulyarnye resheniya nelineinykh zadach, Fizmatgiz, M., 1999 | MR

[4] Evtushenko Yu., “Generalized Lagrange multiplier technique for nonlinear programming”, J. Optimizat. Theory and Applic., 21:2 (1977), 121–135 | DOI | MR | Zbl

[5] Tretyakov A. A., “Teorema o neyavnoi funktsii v vyrozhdennykh zadachakh”, Uspekhi matem. nauk, 42:5 (1987), 215–216 | MR

[6] Belash K. N., Tretyakov A. A., “Metody resheniya vyrozhdennykh zadach”, Zh. vychisl. matem. i matem. fiz., 28:7 (1988), 1097–1102 | MR | Zbl

[7] Izmailov A. F., Tretyakov A. A., Faktor-analiz nelineinykh otobrazhenii, Nauka, M., 1994 | Zbl

[8] Tretyakov A. A., Struktury nelineinykh vyrozhdennykh otobrazhenii i ikh primenenie k postroeniyu chislennykh metodov, Diss. $\dots$ dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1987, 167 pp.

[9] Mangasarian O. L., “A Newton method for linear programming”, J. Optimizat. Theory and Appl., 121 (2004), 1–18 | DOI | MR | Zbl

[10] Golikov A. I., Evtushenko Yu. G., Mollaverdi H., “Primenenie metoda Nyutona k resheniyu zadach lineinogo programmirovaniya bolshoi razmernosti”, Zh. vychisl. matem. i matem. fiz., 44:9 (2004), 1564–1573 | MR | Zbl

[11] Facchinei F., Fisher A., Kanzow C., “On the accurate identification of active constraints”, SIAM J. Optimizat, 9 (1998), 14–32 | DOI | MR | Zbl

[12] Elsgolts L. E., Obyknovennye differentsialnye uravneniya, Gostekhteorizdat, M., L., 1950

[13] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971 | MR