Geodesic
On the construction of regularizing algorithms for the correction of improper convex programming problems
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 234-243 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider convex programming methods with a possibly inconsistent constraint system. Such problems constitute an important class of improper models of convex optimization and often arise in the mathematical modeling of real-life operations research statements. Since improper problems arise rather frequently, the theory and methods of their numerical approximation (correction) should be developed, which would allow to design objective procedures that resolve inconsistent constraints, turn an improper model into a family of feasible problems, and choose an optimal correction among them. In the present paper, an approximating problem is constructed by the variation of the right-hand sides of the constraints with respect to some vector norm. The type of the norm defines the form of a penalty function, and the minimization of the penalty function together with a stabilizing term is the core of each specific method of optimal correction of improper problems. The Euclidean norm implies the application of a quadratic penalty, whereas a piecewise linear (Chebyshev of octahedral) norm is concerned with the use of an exact penalty function. The proposed algorithms may also be interpreted as (Tikhonov) regularization methods for convex programming problems with inaccurate input information. Convergence conditions are formulated for the methods under consideration and convergence bounds are established.
Keywords: convex programming, improper problem, Tikhonov regularization method, penalty function methods.
Mots-clés : optimal correction 
@article{TIMM_2017_23_3_a20,
     author = {V. D. Skarin},
     title = {On the construction of regularizing algorithms for the correction of improper~convex programming problems},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {234--243},
     year = {2017},
     volume = {23},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_3_a20/}
}
TY  - JOUR
AU  - V. D. Skarin
TI  - On the construction of regularizing algorithms for the correction of improper convex programming problems
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2017
SP  - 234
EP  - 243
VL  - 23
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2017_23_3_a20/
LA  - ru
ID  - TIMM_2017_23_3_a20
ER  - 
%0 Journal Article
%A V. D. Skarin
%T On the construction of regularizing algorithms for the correction of improper convex programming problems
%J Trudy Instituta matematiki i mehaniki
%D 2017
%P 234-243
%V 23
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2017_23_3_a20/
%G ru
%F TIMM_2017_23_3_a20
V. D. Skarin. On the construction of regularizing algorithms for the correction of improper convex programming problems. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 234-243. http://geodesic.mathdoc.fr/item/TIMM_2017_23_3_a20/

[1] Eremin I. I., Mazurov V. D., Astafev N. N., Nesobstvennye zadachi lineinogo i vypuklogo programmirovaniya, Nauka, M., 1983, 336 pp. | MR

[2] Vasilev F. P., Metody optimizatsii, Faktorial, M., 2002, 824 pp.

[3] Skarin B. D., “O primenenii odnogo metoda regulyarizatsii dlya korrektsii nesobstvennykh zadach vypuklogo programmirovaniya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:3 (2012), 230–241 | MR

[4] Antipin A. S., “Metod regulyarizatsii v zadachakh vypuklogo programmirovaniya”, Ekonomika i mat. metody, 11:2 (1975), 336–342

[5] Eremin I. I., Astafev N. N., Vvedenie v teoriyu lineinogo i vypuklogo programmirovaniya, Nauka, M., 1976, 192 pp. | MR

[6] Skarin B. D., “O metode regulyarizatsii dlya protivorechivykh zadach vypuklogo programmirovaniya”, Izv. vuzov. Matematika, 1995, no. 12, 81–88 | MR | Zbl

[7] Eremin I. I., “K metodu shtrafov v matematicheskom programmirovanii”, Dokl. RAN, 346:4 (1996), 459–461 | MR | Zbl

[8] Zukhovitskii S. I., Avdeeva L. I., Lineinoe i vypukloe programmirovanie, Nauka, M., 1967, 460 pp. | MR

[9] Skarin B. D., “O metode barernykh funktsii i algoritmakh korrektsii nesobstvennykh zadach vypuklogo programmirovaniya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:2 (2008), 115–128 | MR | Zbl