Geodesic
The subdifferential and the directional derivatives of the maximum of a~family of convex functions.~II
Izvestiya. Mathematics , Tome 65 (2001) no. 1, pp. 99-121.

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The paper deals with calculating the directional derivatives and the subdifferential of the maximum of convex functions with no compactness conditions on the indexing set. We apply our results to the problems of minimax theory in which the Lagrange function is not assumed to be concave. We also apply these results to the duality theory of non-convex extremum problems, and strengthen earlier results of Yakubovich, Matveev and the author. We illustrate our results by investigating a problem of optimal design of experiments. 
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V. N. Solov'ev. The subdifferential and the directional derivatives of the maximum of a~family of convex functions.~II. Izvestiya. Mathematics , Tome 65 (2001) no. 1, pp. 99-121. http://geodesic.mathdoc.fr/item/IM2_2001_65_1_a6/

[1] Blagodatskikh V. I., Teoriya differentsialnykh vklyuchenii, Ch. 1, Izd-vo MGU, M., 1979

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

[3] Dubovitskii A. Ya., Milyutin A. A., “Zadachi na ekstremum pri nalichii ogranichenii”, ZhVM i MF, 5:3 (1965), 395–453 | MR | Zbl

[4] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[5] Ioffe A. D., Levin V. L., “Subdifferentsialy vypuklykh funktsii”, Tr. Mosk. matem. obsch-va, 26 (1972), 3–72 | MR

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

[7] Kusraev A. G., Kutateladze S. S., Subdifferentsialy. Teoriya i prilozheniya, Nauka, Novosibirsk, 1992 | MR | Zbl

[8] Levin V. L., “Ob odnoi zadache vypuklogo analiza, voznikayuschei v teorii optimalnogo upravleniya”, Matem. zametki, 47 (1990), 45–51 | MR | Zbl

[9] Levin V. L., “Primenenie teoremy E. Khelli v vypuklom programmirovanii, zadachakh nailuchshego priblizheniya i smezhnykh voprosakh”, Matem. sb., 79:2 (1969), 250–263 | MR | Zbl

[10] Levin V. L., Vypuklyi analiza v prostranstvakh izmerimykh funktsii, Nauka, M., 1985 | MR

[11] Levin V. L., “O nekotorykh svoistvakh opornykh funktsionalov”, Matem. zametki, 4:6 (1968), 685–696 | MR | Zbl

[12] Minchenko L. I., Borisenko O. F., Differentsialnye svoistva marginalnykh funktsii i ikh prilozheniya k zadacham optimizatsii, Nauka i tekhnika, Minsk, 1992 | MR | Zbl

[13] Oben Zh. P., Ekland I., Prikladnoi nelineinyi analiz, Mir, M., 1988 | MR

[14] Pshenichnyi B. N., Neobkhodimye usloviya ekstremuma, 2-e izd., Nauka, M., 1982 | MR

[15] Rokafellar R. T., Vypuklyi analiz, Nauka, M., 1973

[16] Silin D. B., “Subdifferentsialy vypuklykh funktsii i integraly ot mnogoznachnykh otobrazhenii”, Vestn. Mosk. un-ta. Ser. vychis. matematika i kibernetika, 1984, no. 1, 55–59 | MR | Zbl

[17] Solovev V. N., “Dvoistvennost nevypuklykh ekstremalnykh zadach”, DAN SSSR, 314:1 (1990), 135–138 | Zbl

[18] Solovev V. N., “O subdifferentsiale maksimuma semeistva vypuklykh funktsii”, UMN, 51:2 (1996), 175–176 | MR | Zbl

[19] Solovev V. N., “O subdifferentsiale maksimuma vypuklykh funktsii”, Dokl. RAN, 357:1 (1997), 24–25 | MR | Zbl

[20] Solovev V. N., “Dvoistvennye ekstremalnye zadachi i ikh primeneniya k zadacham minimaksnogo otsenivaniya”, UMN, 52:4 (1997), 49–86 | MR | Zbl

[21] Solovev V. N., “O subdifferentsiale i proizvodnykh po napravleniyam maksimuma semeistva vypuklykh funktsii”, Izv. RAN. Ser. matem., 62:4 (1998), 173–200 | MR | Zbl

[22] Bakhshiyan V. Ts., Solovev V. N., “Teoriya i algoritmy resheniya zadach $L$- i $\operatorname {MV}$-optimalnogo planirovaniya eksperimenta”, Avtomatika i telemekhanika, 1998, no. 8, 80–96 | MR | Zbl

[23] Sukharev A. G., Timokhov A. V., Fedorov V. V., Kurs metodov optimizatsii, Nauka, M., 1986 | MR | Zbl

[24] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979 | MR

[25] Clarke F. H., Ledyaev Yu. S., Stern R. J., Wolenski P. R., Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178, Springer-Verlag, N. Y., 1998 | MR | Zbl

[26] Lindberg P. O., Optimeringslära, en Introduktion, Kungl Tekniska Högskolan, Institutionen för Matematik, Stockholm, 1988 | MR

[27] Mangasarian O. L., “Pseudo-Convex Functions”, SIAM J. Control. Ser. A, 3 (1965), 281–290 | DOI | MR | Zbl

[28] Matveev A. S., Yakubovich V. A., “Nonconvex Problems of Global Optimization: Linear-Quadratic Control Problems with Quadratic Constraints”, Dynamics and Control, 7 (1997), 99–134 | DOI | MR | Zbl

[29] Rissanen J., “On Duality without Convexity”, J. Math. Anal. and Appl., 18 (1967), 269–275 | DOI | MR | Zbl

[30] Soloviov V., “Lagrangian's Duality Scheme in the Absence of Convexity”, Optimal Control and Appendices, International Conference Dedicated to the 90th Anniversary of L. S. Pontryagin. Abstracts (Moscow State University, 1998, August 31 – September 6), 180–181

[31] Valadier M., “Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes”, C. R. Acad. Sci. Paris. Série A, 268 (1969), 39–42 | MR | Zbl

[32] Volle M., “Sous-différentiel d'une enveloppe supérieure de fonctions convexes”, C. R. Acad. Sci. Paris. Série I, 317 (1993), 845–849 | MR | Zbl

[33] Borwein J. M., Zhu Q., A survey of smooth subdifferential calculus with applications, CECM Research Report 98:105