Geodesic
Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 10-19
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The notions of upper and lower exhausters and coexhausters are discussed and necessary conditions for an unconstrained extremum of a nonsmooth function are derived. The necessary minimum conditions are formulated in terms of an upper exhauster (coexhauster) and the necessary maximum conditions are formulated in terms of a lower exhauster (coexhauster). This involves the problem of transforming an upper exhauster (coexhauster) into a lower exhauster (coexhauster) and vice versa. The transformation is carried out by means of a conversion operation (converter). Second-order approximations obtained with the help of second-order (upper and lower) coexhausters are considered. It is shown how a second-order upper coexhauster can be converted to a lower coexhauster and vice versa. This problem is reduced to using a first-order conversion operator but in a space of a higher dimension. The obtained result allows one to construct second-order methods for the optimization of nonsmooth functions (Newton-type methods).
Keywords: nonsmooth analysis, nondifferentiable optimization, exhauster, coexhauster
Mots-clés : converter. 
@article{TIMM_2009_15_4_a1,
     author = {M. E. Abbasov and V. F. Demyanov},
     title = {Extremum conditions for a~nonsmooth function in terms of exhausters and coexhausters},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {10--19},
     year = {2009},
     volume = {15},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a1/}
}
TY  - JOUR
AU  - M. E. Abbasov
AU  - V. F. Demyanov
TI  - Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2009
SP  - 10
EP  - 19
VL  - 15
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a1/
LA  - ru
ID  - TIMM_2009_15_4_a1
ER  - 
%0 Journal Article
%A M. E. Abbasov
%A V. F. Demyanov
%T Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters
%J Trudy Instituta matematiki i mehaniki
%D 2009
%P 10-19
%V 15
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a1/
%G ru
%F TIMM_2009_15_4_a1
M. E. Abbasov; V. F. Demyanov. Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 10-19. http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a1/

[1] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 472 pp.

[2] Demyanov V. F., Malozemov V. N., Vvedenie v minimaks, Nauka, M., 1972, 368 pp. | MR

[3] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 288 pp. | MR | Zbl

[4] Shor N. Z., “O klasse pochti-differentsiruemykh funktsii i odnom metode minimizatsii funktsii etogo klassa”, Kibernetika, 1972, no. 4, 65–70 | Zbl

[5] Kusraev A. G., Kututeladze S. S., Subdifferentsialy. Teoriya i prilozheniya, Nauka, Novosibirsk, 1992, 270 pp. | MR | Zbl

[6] Pshenichnyi B. N., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980, 320 pp. | MR | Zbl

[7] Demyanov V. F., Rubinov A. M., “Elementy kvazidifferentsialnogo ischisleniya”, Negladkie zadachi teorii optimizatsii i upravleniya, ed. V. F. Demyanov, Izd-vo Leningr. un-ta, L., 1982, 5–127 | MR

[8] Demyanov V. F., “Exhausters of a positively homogeneous function”, Optimization, 45 (1999), 13–29 | DOI | MR | Zbl

[9] Demyanov V. F., Rubinov A. M., “Exhaustive families of approximations revisited”, From Convexity to Nonconvexity, Nonconvex Optim. Appl., 55, eds. R. P. Gilbert, P. D. Panagiotopoulos, P. M. Pardalos, Kluwer Scient. Publ., Dordrecht, 2001, 43–50 | MR | Zbl

[10] Demyanov V. F., “Exhausters and convexificators – new tools in nonsmooth Analysis”, Quasidifferentiability and Related Topics, eds. V. Demyanov, A. Rubinov, Kluwer Acad. Publ., Dordrecht, 2000, 85–137 | MR | Zbl

[11] Demyanov V. F., Roschina V. A., “Optimality conditions in terms of upper and lower exhausters”, Optimization, 55:5/6 (2006), 525–540 | DOI | MR | Zbl

[12] Demyanov V. F., Roschina V. A., “Obobschennye subdifferentsialy i ekzostery v negladkom analize”, Dokl. RAN, 416:1 (2007), 18–21 | MR

[13] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 432 pp. | MR

[14] Castellani M., “A dual representation for proper positively homogeneous functions”, J. Global Optim., 16:4 (2000), 393–400 | DOI | MR | Zbl