Borwein–Preiss vector variational principle
Tóm tắt
This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:
10.1006/jmaa.2000.6813
). We introduce and characterize two seemingly new natural concepts of
$$\varepsilon $$
-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function.
Tài liệu tham khảo
Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybernet. 36(3), 531–562 (2007)
Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal. 75(3), 1124–1140 (2012). doi:10.1016/j.na.2011.05.098
Bednarczuk, E.M., Kruger, A.Y.: Error bounds for vector-valued functions on metric spaces. Vietnam J. Math. 40(2–3), 165–180 (2012)
Bednarczuk, E.M., Przybyła, M.J.: The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors. SIAM J. Optim. 18(3), 907–913 (2007). doi:10.1137/060658989
Bednarczuk, E.M., Zagrodny, D.: Vector variational principle. Arch. Math. (Basel) 93(6), 577–586 (2009). doi:10.1007/s00013-009-0072-x
Bednarczuk, E.M., Zagrodny, D.: A smooth vector variational principle. SIAM J. Control Optim. 48(6), 3735–3745 (2010). doi:10.1137/090758271
Bonnel, H.: Remarks about approximate solutions in vector optimization. Pac. J. Optim. 5(1), 53–73 (2009)
Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Am. Math. Soc. 303(2), 517–527 (1987). doi:10.2307/2000681
Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)
Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338(1), 105–122 (1993). doi:10.2307/2154446
Chen, G.Y., Huang, X.X., Yang, X.: Vector Optimization. Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)
Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64. Longman Scientific & Technical, Harlow (1993)
Deville, R., Ivanov, M.: Smooth variational principles with constraints. Arch. Math. (Basel) 69(5), 418–426 (1997). doi:10.1007/s000130050140
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A view from Variational Analysis. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2014)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Fabian, M., Hájek, P., Vanderwerff, J.: On smooth variational principles in Banach spaces. J. Math. Anal. Appl. 197(1), 153–172 (1996). doi:10.1006/jmaa.1996.0013
Fabian, M., Mordukhovich, B.S.: Nonsmooth characterizations of Asplund spaces and smooth variational principles. Set Valued Anal. 6(4), 381–406 (1998). doi:10.1023/A:1008799412427
Finet, C., Quarta, L., Troestler, C.: Vector-valued variational principles. Nonlinear Anal. 52(1), 197–218 (2003). doi:10.1016/S0362-546X(02)00103-7
Georgiev, P.G.: Parametric Borwein-Preiss variational principle and applications. Proc. Am. Math. Soc. 133(11), 3211–3225 (2005). doi:10.1090/S0002-9939-05-07853-6
Göpfert, A., Henkel, E.C., Tammer, C.: A smooth variational principle for vector optimization problems. In: Recent Developments in Optimization (Dijon, 1994), Lecture Notes in Econom. and Math. Systems, vol. 429, pp. 142–154. Springer, Berlin (1995). doi:10.1007/978-3-642-46823-0_12
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer, New York (2003)
Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions forTanaka’s approximate solutions in vector optimization. In:Generalized Convexity and Related Topics, Lecture Notes inEconomics and Mathematical Systems, vol. 583, pp. 279–295.Springer, Berlin (2006)
Gutiérrez, C., Jiménez, B., Novo, V.: A set-valued Ekeland’s variational principle in vector optimization. SIAM J. Control Optim. 47(2), 883–903 (2008). doi:10.1137/060672868
Gutiérrez, C., Jiménez, B., Novo, V.: Equivalent \(\varepsilon \)-efficiency notions in vector optimization. TOP 20(2), 437–455 (2012). doi:10.1007/s11750-011-0223-7
Ha, T.X.D.: Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124(1), 187–206 (2005). doi:10.1007/s10957-004-6472-y
Ioffe, A.D., Tikhomirov, V.M.: Some remarks on variational principles. Math. Notes 61(2), 248–253 (1997). doi:10.1007/BF02355736
Isac, G.: Ekeland’s principle and nuclear cones: a geometrical aspect. Math. Comput. Model. 26(11), 111–116 (1997). doi:10.1016/S0895-7177(97)00223-9
Khan, A.A., Tammer, C.: Set-Valued Optimization: An Introduction with Applications. Vector Optimization. Springer, Heidelberg (2015). doi:10.1007/978-3-642-54265-7
Khanh, P.Q.: On Caristi–Kirk’s theorem and Ekeland’s variational principle for Pareto extrema. Bull. Polish Acad. Sci. Math. 37(1–6), 33–39 (1989)
Khanh, P.Q., Quy, D.N.: A generalized distance and enhanced Ekeland’s variational principle for vector functions. Nonlinear Anal. 73(7), 2245–2259 (2010). doi:10.1016/j.na.2010.06.005
Khanh, P.Q., Quy, D.N.: Versions of Ekeland’s variational principle involving set perturbations. J. Global Optim. 57(3), 951–968 (2013). doi:10.1007/s10898-012-9983-3
Krasnoselski, M.A., Lifshits, E.A., Sobolev, A.V.: Positive Linear Systems. The Method of Positive Operators. Nauka, Moscow (1985)
Kruger, A.Y., Plubtieng, S., Seangwattana, T.: Borwein–Preiss variational principle revisited. J. Math. Anal. Appl. 435(2), 1183–1193 (2016)
Kutateladze, S.S.: Convex \(\varepsilon \)-programming. Sov. Math. Dokl. 20, 391–393 (1979)
Li, Y., Shi, S.: A generalization of Ekeland’s \(\epsilon \)-variational principle and its Borwein–Preiss smooth variant. J. Math. Anal. Appl. 246(1), 308–319 (2000). doi:10.1006/jmaa.2000.6813
Liu, C.G., Ng, K.F.: Ekeland’s variational principle for set-valued functions. SIAM J. Optim. 21(1), 41–56 (2011). doi:10.1137/090760660
Loewen, P.D., Wang, X.: A generalized variational principle. Canad. J. Math. 53(6), 1174–1193 (2001). doi:10.4153/CJM-2001-044-8
Loridan, P.: \(\epsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43(2), 265–276 (1984). doi:10.1007/BF00936165
Németh, A.B.: A nonconvex vector minimization problem. Nonlinear Anal. 10(7), 669–678 (1986). doi:10.1016/0362-546X(86)90126-4
Penot, J.P.: Genericity of well-posedness, perturbations and smooth variational principles. Set Valued Anal. 9(1–2), 131–157 (2001). doi:10.1023/A:1011223206312
Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2 edn. Springer, Berlin (1993)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)
Sitthithakerngkiet, K., Plubtieng, S.: Vectorial form of Ekeland-type variational principle. Fixed Point Theory Appl. 2012(127), 1–11 (2012). doi:10.1186/1687-1812-2012-127
Tammer, C.: A generalization of Ekeland’s variational principle. Optimization 25(2–3), 129–141 (1992). doi:10.1080/02331939208843815
Tammer, C., Zălinescu, C.: Vector variational principles for set-valued functions. Optimization 60(7), 839–857 (2011). doi:10.1080/02331934.2010.522712