Optimality conditions in vector optimization problems with a nonsolid cone of positive elements

V. V. Gorokhovik

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 7, pp. 1192-1214

Bibliographie

[1] Jameson G., Ordered linear spaces, Springer, Berlin, 1970 | MR | Zbl

[2] Peressini A.L., Ordered topological vector spaces, Harper and Row, New York, 1967 | MR | Zbl

[3] Gorokhovik V.V., Vypuklye i negladkie zadachi vektornoi optimizatsii, Nauka i tekhnika, Minsk, 1990

[4] Levitin E.S., Milyutin A.A., Osmolovskii N.P., “O neobkhodimykh usloviyakh lokalnogo minimuma v zadache s ogranicheniyami”, Dokl. AN SSSR, 210:5 (1973), 1022–1025 | MR | Zbl

[5] Levitin E.S., Milyutin A.A., Osmolovskii N.P., “Ob usloviyakh lokalnogo minimuma v zadache s ogranicheniyami”, Matem. ekonomika i funkts. analiz, 1974, 139–202, Nauka, M.

[6] Levitin E.S., Milyutin A.A., Osmolovskii N.P., “Usloviya vysshikh poryadkov v zadachakh s ogranicheniyami”, Uspekhi matem. nauk, 33:6 (1978), 85–148 | MR

[7] Levitin E.S., Milyutin A.A., Osmolovskii N.P., “Teoriya uslovii vysshikh poryadkov v gladkikh zadachakh na ekstremum s ogranicheniyami”, Teoreticheskie i prikladnye voprosy optimalnogo upravleniya, Nauka, Sib. otd-nie, Novosibirsk, 1985, 4–40

[8] Gorokhovik V.V., Rachkovskii N.N., “Sobstvennaya minimalnost v preduporyadochennykh vektornykh prostranstvakh”, Vestsi AN Belarusi. Ser. fiz.-mat. navuk, 1993, no. 4, 10–15

[9] Vao T.Q., Mordukhovich B.S., “Relative Pareto minimizers for multiobjective problems: existence and optimality conditions”, Math. Program. Ser. A, 122:2 (2010), 301–347 | DOI | MR | Zbl

[10] Polyak B.E., Vvedenie v optimizatsiyu, Nauka, M., 1983

[11] Vednarczuk E., “Weak sharp efficiency and growth condition for vector-valued functions with applications”, Optimization, 53:5–6 (2004), 455–474 | DOI | MR

[12] Burke J.V., Ferris M.S., “Weak sharp minima in mathematical programming”, SIAM J. Control Optim., 31:5 (1993), 1340–1359 | DOI | MR | Zbl

[13] Burke J.V., Deng S., “Weak sharp minima revisited. Part I: basic theory”, Control Cybernet., 31:3 (2002), 439–469 | MR | Zbl

[14] Burke J.V., Deng S., “Weak sharp minima revisited, part II: application to linear regularity and error bounds”, Math. Program., 104:2–3 (2005), 235–261 | DOI | MR | Zbl

[15] Jiménez B., “Strict efficiency in vector optimization”, J. Math. Anal. Appl., 265:2 (2002), 264–284 | DOI | MR | Zbl

[16] Studniarski M., Ward D.E., “Weak sharp minima: characterizations and sufficient conditions”, SIAM J. Control Optimizat., 38:1 (1999), 219–236 | DOI | MR | Zbl

[17] Demyanov V.F., Rubinov A.M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1974

[18] Mordukhovich B.S., Variational analysis and generalized differentiation, v. I, Basic theory, Springer, Berlin et al., 2005 | MR

[19] Rockafellar R.V., Wets R.J.-B., Variational analysis, Springer-Verlag, Berlin, 1998 | MR | Zbl

[20] Arutyunov A.V., Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi, Faktorial, M., 1997

[21] Vasilev F.P., Metody optimizatsii, Faktorial, M., 2002

[22] Gorokhovik V.V., “K usloviyam lokalnogo minimuma vtorogo poryadka v gladkikh zadachakh optimizatsii s abstraktnymi ogranicheniyami”, Izv. NAN Belarusi. Ser. fiz.-mat. nauk, 2005, no. 3, 5–12

[23] Gorokhovik V.V., “Asimptoticheski kasatelnyi konus vtorogo poryadka k mnozhestvam i usloviya lokalnogo minimuma v zadache optimizatsii s ogranicheniyami”, Vestn. Sankt-Peterburgskogo un-ta. Ser. informatika i teor. upravleniya, 2006, no. 1, 34–41 | MR

[24] Izmailov A.F., Tretyakov A.A., Faktor-analiz nelineinykh otobrazhenii, Nauka, M., 1994

[25] Izmailov A.F., Tretyakov A.A., 2-regulyarnye resheniya nelineinykh zadach, Fizmatlit, M., 1999

[26] Arutyunov A.V., “Second-order conditions in extremal problems. The abnormal points”, Trans. of American Math. Soc., 350:11 (1998), 4341–4365 | DOI | MR | Zbl

[27] Ben-Tal A., “Second order and related extremality conditions in nonlinear programming”, J. Optimizat. Theory and Appl., 31:2 (1980), 143–165 | DOI | MR | Zbl

[28] Ben-Tal A., Zowe J., “A unified theory of first and second order conditions for extremum problems in topological vector spaces”, Math. Programming Study, 19 (1982), 39–76 | DOI | MR

[29] Bonnans J.F., Shapiro A., Perturbation analysis of optimization problems, Springer, Berlin, 2000 | MR

[30] Cambini R., Martein L., Vlach M., “Second order tangent sets and optimality conditions”, Math. Japonica, 49:3 (1999), 451–461 | MR | Zbl

[31] Gutiérrez C., Jiménez V., Novo V., “New second-order directional derivative and optimality conditions in scalar and vector optimization”, J. Optimizat. Theory Appl., 142:1 (2009), 85–106 | DOI | MR | Zbl

[32] Ioffe A.D., “Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first order conditions”, SIAM J. Control and Optimizat., 17:2 (1979), 245–250 | DOI | MR | Zbl

[33] Ioffe A.D., “Necessary and sufficient conditions for a local minimum. 2: Conditions of Levitin–Miljutin–Osmolovskii type”, SIAM J. Control and Optimizat., 17:2 (1979), 251–265 | DOI | MR | Zbl

[34] Joffe A.D., “Necessary and sufficient conditions for a local minimum. 3: Second order conditions and augmented duality”, SIAM J. Control and Optimizat., 17:2 (1979), 266-288 | DOI | MR

[35] Penot J.P., “Second order conditions for optimization problems with constraints”, SIAM J. Control and Optimizat., 37:1 (1998), 303–318 | DOI | MR

[36] Shapiro A., “Semi-infinite programming, duality, discretization and optimality conditions”, Optimization, 58:2 (2009), 133–161 | DOI | MR | Zbl

[37] Bakhtin V.I., Gorokhovik V.V., “Usloviya optimalnosti vtorogo poryadka v zadachakh vektornoi optimizatsii s geometricheskimi ogranicheniyami”, Dokl. NAN Belarusi, 53:1 (2009), 33–38 | MR

[38] Bakhtin V.I., Gorokhovik V.V., “Usloviya optimalnosti pervogo i vtorogo poryadkov v zadachakh vektornoi optimizatsii na metricheskikh prostranstvakh”, Tr. In-ta matem. i mekh. Uralskogo otd. RAN, 15:4 (2009), 32–43 | Zbl

[39] Gorokhovik B.V., Gorokhovik S.Ya., Marinkovich B., “Neobkhodimye usloviya optimalnosti v gladkoi zadache upravleniya diskretnoi sistemoi s vektornym pokazatelem kachestva”, Tr. In-ta matem. NAN Belarusi, 17:1 (2009), 27–40 | Zbl

[40] Gfrerer H., “Second-order optimality conditions for scalar and vector optimization problems in Banach spaces”, SIAM J. Control Optimizat., 45:3 (2006), 972–997 | DOI | MR | Zbl

[41] Gfrerer H., “Second-order necessary conditions for nonlinear optimization problems with abstract constraints: the degenerate case”, SIAM J. Optimizat., 18:2 (2007), 589–612 | DOI | MR | Zbl

[42] Gutiérrez C., Jiménez V., Novo V., “On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming”, Math. Program. Ser. B, 123:1 (2010), 199–223 | DOI | MR | Zbl

[43] Jiménez V., Novo V., “Second-order necessary conditions in set constrained differentiable vector optimization”, Math. Methods Oper. Res., 58:2 (2003), 299–317 | DOI | MR | Zbl

[44] Jiménez B., Novo V., “Optimality conditions in differentiable vector optimization via second-order tangent sets”, Appl. Math. Optimizat., 49:2 (2004), 123–144 | MR | Zbl

[45] Hiriart-Urruty J.B., “Tangent cones, generalized gradients, and mathematical programming in Banach spaces”, Math. Oper. Research, 4:1 (1979), 79–97 | DOI | MR | Zbl

[46] Hiriart-Urruty J.B., “New concepts in nondifferentiable programming”, Bull. Soc. Math. France Mém., 60 (1979), 57–85 | MR | Zbl

[47] Krasnoselskii M.A., Burd V.Sh., Kolesov Yu.S., Nelineinye pochti periodicheskie kolebaniya, Nauka, M., 1970

[48] Göpfert A., Riahi H., Tammer Ghr., Zǎlinescu C., Variational methods in partially ordered spaces, Springer, New York, 2003 | MR

[49] Bednarczuk E., “Hölder-like properties of minimal points in vector optimization”, Control Cybernet., 31:3 (2002), 423–438 | MR | Zbl

[50] Durea M., Dutta J., Tammer G., “Lagrange multipliers for $\varepsilon$-Pareto solutions in vector optimization with nonsolid cones in Banach spaces”, J. Optimizat. Theory Appl., 145:1 (2010), 196–211 | DOI | MR | Zbl

[51] Ha T.X.D., “Lagrange multipliers for set-valued optimization problems associated with coderivatives”, J. Math. Anal. Appl., 311:2 (2005), 647–663 | DOI | MR | Zbl

[52] Ha T.X.D., “Optimality conditions for several types of efficient solutions of set-valued optimization problems”, Nonlinear Analysis and Variat. Problems, Optimization and its applications, 35, Part 2, Springer, Berlin, 2010, 305–324 | DOI | MR | Zbl

[53] Zafforoni A., “Degrees of efficiency and degrees of minimality”, SIAM J. Control and Optimizat., 42:3 (2003), 1071–1086 | DOI | MR

[54] Tammer Chr., Zǎlinescu C., “Lipschitz properties of the scalarization function and applications”, Optimization, 59:2 (2010), 305–319 | DOI | MR | Zbl

[55] Gorokhovik V.V., Rachkovskii N.N., Usloviya pervogo i vtorogo poryadka lokalnoi sobstvennoi minimalnosti v zadachakh vektornoi optimizatsii, Preprint No50(450), In-t matem. AN BSSR, Minsk, 1990

[56] Dubovitskii A.Ya., Milyutin A.A., “Zadachi na ekstremum pri nalichii ogranichenii”, Zh. vychisl. matem. i matem. fiz., 5:3 (1965), 395–453 | MR

[57] Demyanov V.F., Rubinov A.M., Priblizhennye metody resheniya ekstremalnykh zadach, Izd-vo Leningradskogo un-ta, L., 1968

[58] Aubin J.-P., Frankowska H., Set-valued analysis, Birkhauser, Boston, 1990 | MR | Zbl

[59] Alekseev V.M., Tikhomirov V.M., Fomin S.V., Optimalnoe upravlenie, Nauka, M., 1979

[60] Ioffe A.D., Tikhomirov V.M., Teoriya ekstremalnykh zadach, Nauka, M., 1990

[61] Danford N., Shvarts Dzh.E., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[62] Fabian M., Habala P., Haék P., Zizler V., Functional analysis and infinite-dimensional geometry, Springer, New York, 2001 | MR

[63] Gorokhovik V.V., “Kasatelnye vektory vtorogo poryadka k mnozhestvam i usloviya minimalnosti dlya tochek podmnozhestv uporyadochennykh normirovannykh prostranstv”, Tr. In-ta matem. HAH Belarusi, 14:2 (2006), 35–47 | MR

[64] Bonnans J.F., Cominetti R., Shapiro A., “Second order optimality conditions based on parabolic second order tangent sets”, SIAM J. Optimizat., 9:2 (1999), 466–492 | DOI | MR | Zbl