Geodesic
On the attraction of Newton’s method to critical Lagrange multipliers
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1272-1286 
E. I. Uskov. On the attraction of Newton’s method to critical Lagrange multipliers. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1272-1286. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a4/
@article{ZVMMF_2013_53_8_a4,
     author = {E. I. Uskov},
     title = {On the attraction of {Newton{\textquoteright}s} method to critical {Lagrange} multipliers},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1272--1286},
     year = {2013},
     volume = {53},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a4/}
}
TY  - JOUR
AU  - E. I. Uskov
TI  - On the attraction of Newton’s method to critical Lagrange multipliers
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2013
SP  - 1272
EP  - 1286
VL  - 53
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a4/
LA  - ru
ID  - ZVMMF_2013_53_8_a4
ER  - 
%0 Journal Article
%A E. I. Uskov
%T On the attraction of Newton’s method to critical Lagrange multipliers
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2013
%P 1272-1286
%V 53
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a4/
%G ru
%F ZVMMF_2013_53_8_a4

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

The attraction of dual trajectories of Newton’s method for the Lagrange system to critical Lagrange multipliers is analyzed. This stable effect, which has been confirmed by numerical practice, leads to the Newton–Lagrange method losing its superlinear convergence when applied to problems with irregular constraints. At the same time, available theoretical results are of “negative” character; i.e., they show that convergence to a noncritical multiplier is not possible or unlikely. In the case of a purely quadratic problem with a single constraint, a “positive” result is proved for the first time demonstrating that the critical multipliers are attractors for the dual trajectories. Additionally, the influence exerted by the attraction to critical multipliers on the convergence rate of direct and dual trajectories is characterized.

[1] Izmailov A. F., “Ob analiticheskoi i vychislitelnoi ustoichivosti kriticheskikh mnozhitelei Lagranzha”, Zh. vychisl. matem. i matem. fiz., 45:6 (2005), 966–982 | MR | Zbl

[2] Izmailov A. F., Solodov M. V., “On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions”, Math. Program., 117 (2009), 271–304 | DOI | MR | Zbl

[3] Izmailov A. F., Solodov M. V., “Examples of dual behaviour of Newton-type methods on optimization problems with generate constraints”, Comput. Optim. Appl., 42:2 (2009), 231–264 | DOI | MR | Zbl

[4] Izmailov A. F., Solodov M. V., “On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers”, Math. Program., 126 (2001), 231–257 | DOI | MR

[5] Izmailov A. F., Uskov E. I., “Effekt prityazheniya metoda Nyutona–Lagranzha k kriticheskim mnozhitelyam Lagranzha: polnyi analiz v odnomernom sluchae”, Teoreticheskie i prikladnye zadachi nelineinogo analiza, VTs RAN, M., 2012, 53–71

[6] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[7] Voevodin V. V., Kuznetsov Yu. A., Matritsy i vychisleniya, Nauka, M., 1984 | MR | Zbl