Stability theorem and extremum conditions for abnormal problems

E. R. Avakov

Informatics and Automation, Optimal control, Tome 291 (2015), pp. 10-29.

Résumé

We prove a generalized inverse function theorem in a neighborhood of a singular point of a mapping. As corollaries to this theorem, we obtain an inverse function theorem, an error bound theorem, and a tangent cone theorem that extend and strengthen the corresponding classical results in the irregular case. Using these corollaries, we establish necessary extremum conditions that are meaningful for abnormal problems.

Bibliographie

[1] Dmitruk A.V., Milyutin A.A., Osmolovskii N.P., “Teorema Lyusternika i teoriya ekstremuma”, UMN, 35:6 (1980), 11–46 | MR | Zbl

[2] Tikhomirov V.M., “Teorema Lyusternika o kasatelnom prostranstve i nekotorye ee modifikatsii”, Optimalnoe upravlenie, Matematicheskie voprosy upravleniya proizvodstvom, 7, Izd-vo MGU, M., 1977, 22–30

[3] Robinson S.M., “Regularity and stability for convex multivalued functions”, Math. Oper. Res., 1 (1976), 130–143 | DOI | MR | Zbl

[4] Ioffe A.D., “Metricheskaya regulyarnost i subdifferentsialnoe ischislenie”, UMN, 55:3 (2000), 103–162 | DOI | MR | Zbl

[5] Bonnans J.F., Shapiro A., Perturbation analysis of optimization problems, Springer, New York, 2000 | MR | Zbl

[6] Mordukhovich B.S., Variational analysis and generalized differentiation, V. 1, Springer, Berlin, 2006 | MR

[7] Avakov E.R., “Usloviya ekstremuma dlya gladkikh zadach s ogranicheniyami tipa ravenstv”, ZhVMiMF, 25:5 (1985), 680–693 | MR | Zbl

[8] Avakov E.R., “Teoremy ob otsenkakh v okrestnosti osoboi tochki otobrazheniya”, Mat. zametki, 47:5 (1990), 3–13 | MR | Zbl

[9] Arutyunov A.V., “Teorema o neyavnoi funktsii bez apriornykh predpolozhenii normalnosti”, ZhVMiMF, 46:2 (2006), 205–215 | MR | Zbl

[10] Izmailov A.F., “O nekotorykh obobscheniyakh lemmy Morsa”, Tr. MIAN, 220 (1998), 142–156 | MR | Zbl

[11] Izmailov A.F., “Teoremy o predstavlenii semeistv nelineinykh otobrazhenii i teoremy o neyavnoi funktsii”, Mat. zametki, 67:1 (2000), 57–68 | DOI | MR | Zbl

[12] Arutyunov A.V., “Teorema o neyavnoi funktsii na konuse v okrestnosti anormalnoi tochki”, Mat. zametki, 78:4 (2005), 619–621 | DOI | MR | Zbl

[13] Sussmann H.J., “High-order open mapping theorems”, Directions in mathematical systems theory and optimization, Springer, Berlin, 2003, 293–316 | DOI | MR | Zbl

[14] Avakov E.R., Arutyunov A.V., “Teorema ob obratnoi funktsii i usloviya ekstremuma dlya anormalnykh zadach s nezamknutym obrazom”, Mat. sb., 196:9 (2005), 3–22 | DOI | MR | Zbl

[15] Avakov E.R., “Neobkhodimye usloviya ekstremuma dlya gladkikh anormalnykh zadach s ogranicheniyami tipa ravenstv i neravenstv”, Mat. zametki, 45:6 (1989), 3–11 | MR | Zbl

[16] Agrachev A.A., Gamkrelidze R.V., “Vychislenie eilerovoi kharakteristiki peresecheniya veschestvennykh kvadrik”, DAN SSSR, 299:1 (1988), 11–14 | MR | Zbl

[17] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[18] Bliss G.A., Lektsii po variatsionnomu ischisleniyu, Izd-vo inostr. lit., M., 1950

[19] Avakov E.R., “Neobkhodimye usloviya pervogo poryadka dlya anormalnykh zadach variatsionnogo ischisleniya”, Dif. uravneniya, 27:5 (1991), 739–745 | MR | Zbl

[20] Arutyunov A.V., Avakov E.R., Izmailov A.F., “Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications”, Math. Program. A, 114:1 (2008), 37–68 | DOI | MR | Zbl