Geodesic
Convergence of the gradient projection method and Newton's method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1733-1749 
Yu. A. Chernyaev. Convergence of the gradient projection method and Newton's method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1733-1749. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a4/
@article{ZVMMF_2016_56_10_a4,
     author = {Yu. A. Chernyaev},
     title = {Convergence of the gradient projection method and {Newton's} method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1733--1749},
     year = {2016},
     volume = {56},
     number = {10},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a4/}
}
TY  - JOUR
AU  - Yu. A. Chernyaev
TI  - Convergence of the gradient projection method and Newton's method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2016
SP  - 1733
EP  - 1749
VL  - 56
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a4/
LA  - ru
ID  - ZVMMF_2016_56_10_a4
ER  - 
%0 Journal Article
%A Yu. A. Chernyaev
%T Convergence of the gradient projection method and Newton's method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2016
%P 1733-1749
%V 56
%N 10
%U http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a4/
%G ru
%F ZVMMF_2016_56_10_a4

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

The gradient projection method and Newton's method are generalized to the case of nonconvex constraint sets representing the set-theoretic intersection of a spherical surface with a convex closed set. Necessary extremum conditions are examined, and the convergence of the methods is analyzed.

[1] Vasilev F. P., Chislennye metody resheniya ekstremalnykh zadach, Nauka, M., 1980

[2] Karmanov V. G., Matematicheskoe programmirovanie, Nauka, M., 1975 | MR

[3] Nedich A., “Trekhshagovyi metod proektsii gradienta dlya zadach minimizatsii”, Izvestiya vuzov. Matematika, 1993, no. 10, 32–37 | MR

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

[5] Antipin A. S., “Ob otsenkakh skorosti skhodimosti metoda proektsii gradienta”, Izvestiya vuzov. Matematika, 1995, no. 6, 16–24 | MR

[6] Pshenichnyi B. N., Danilin Yu. M., Chislennye metody v ekstremalnykh zadachakh, Nauka, M., 1975 | MR

[7] Vasilev F. P., Yachimovich M., “Ob iterativnoi regulyarizatsii metoda Nyutona”, Zh. vychisl. matem. i matem. fiz., 21:3 (1981), 775–778 | MR

[8] Vasilev F. P., “O regulyarizatsii metoda Nyutona pri netochnom zadanii iskhodnykh dannykh”, Trudy Matematicheskogo instituta AN SSSR, 167, 1985, 53–59 | MR | Zbl

[9] Bogomolov N. A., Karmanov V. G., “O metode vychisleniya statsionarnykh tochek obschei zadachi nelineinogo programmirovaniya”, Zh. vychisl. matem. i matem. fiz., 17:1 (1977), 72–78 | MR | Zbl

[10] Zabotin V. I., Minnibaev T. F., “Metod vozmozhnykh napravlenii dlya zadach matematicheskogo programmirovaniya s predvypuklymi ogranicheniyami”, Zh. vychisl. matem. i matem. fiz., 48:2 (2008), 255–263 | MR | Zbl

[11] Bertsekas D., Uslovnaya optimizatsiya i metody mnozhitelei Lagranzha, Radio i svyaz, M., 1987 | MR

[12] Golshtein E. G., Tretyakov N. V., Modifitsirovannye funktsii Lagranzha: Teoriya i metody optimizatsii, Nauka, M., 1989 | MR

[13] Golikov A. I., Zhadan V. G., “Iterativnye metody resheniya zadach nelineinogo programmirovaniya s ispolzovaniem modifitsirovannykh funktsii Lagranzha”, Zh. vychisl. matem. i matem. fiz., 20:4 (1980), 874–888 | MR

[14] Zhadan V. G., “Modifitsirovannye funktsii Lagranzha v nelineinom programmirovanii”, Zh. vychisl. matem. i matem. fiz., 22:2 (1982), 296–308 | MR | Zbl

[15] Pshenichnyi B. N., Sobolenko L. A., “Uskorenie skhodimosti metoda linearizatsii dlya zadachi uslovnoi minimizatsii”, Zh. vychisl. matem. i matem. fiz., 20:3 (1980), 605–614 | MR | Zbl

[16] Golikov A. I., Zhadan V. G., “Dve modifikatsii metoda linearizatsii v nelineinom programmirovanii”, Zh. vychisl. matem. i matem. fiz., 23:2 (1983), 314–325 | MR | Zbl

[17] Antipin A. S., Nedich A., Yachimovich M., “Trekhshagovyi metod linearizatsii dlya zadach minimizatsii”, Izvestiya vuzov. Matematika, 1994, no. 12, 3–7

[18] Antipin A. S., Nedich A., Yachimovich M., “Dvukhshagovyi metod linearizatsii dlya zadach minimizatsii”, Zh. vychisl. matem. i matem. fiz., 36:4 (1996), 18–25 | MR | Zbl

[19] Chernyaev Yu. A., “Skhodimost metoda proektsii gradienta dlya odnogo klassa nevypuklykh zadach matematicheskogo programmirovaniya”, Izvestiya vuzov. Matematika, 2005, no. 12(523), 76–79

[20] Chernyaev Yu. A., “Obobschenie metoda Nyutona na klass nevypuklykh zadach matematicheskogo programmirovaniya”, Izvestiya vuzov. Matematika, 2008, no. 1(548), 78–82

[21] Chernyaev Yu. A., “Obobschenie metoda proektsii gradienta i metoda Nyutona na ekstremalnye zadachi s ogranicheniem v vide gladkoi poverkhnosti”, Zh. vychisl. matem. i matem. fiz., 55:9 (2015), 1493–1502 | DOI | Zbl

[22] Dulliev A. M., “Relaksatsionnyi metod minimizatsii gladkoi funktsii na obobschennom segmente sfery”, Zh. vychisl. matem. i matem. fiz., 54:2 (2014), 208–223 | DOI | Zbl

[23] Dulliev A. M., Zabotin V. I., “Iteratsionnyi algoritm proektirovaniya tochki na nevypukloe mnogoobrazie v lineinom normirovannom prostranstve”, Zh. vychisl. matem. i matem. fiz., 44:5 (2004), 827–830 | MR | Zbl

[24] Loran P. Zh., Approksimatsiya i optimizatsiya, Mir, M., 1975

[25] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980 | MR