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2012 | OriginalPaper | Buchkapitel

12. The Fermat Rule and Lagrange Multiplier Rule for Various Efficient Solutions of Set-Valued Optimization Problems Expressed in Terms of Coderivatives

verfasst von : Truong Xuan Duc Ha

Erschienen in: Recent Developments in Vector Optimization

Verlag: Springer Berlin Heidelberg

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Abstract

Recently, the interest toward set-valued optimization problems, i.e., optimization problems with set-valued objective and/or constrain maps, has grown up. Exploiting advanced tools of variational analysis and set-valued analysis, the authors obtained a number of (necessary and/or sufficient) optimality conditions expressed in terms of generalized derivatives such as contingent derivative [ 2, 13, 62, 66], first-order derivative [14, 15], second-order derivative [ 28, 40], epiderivative and generalized epiderivative [10, 22, 24, 39, 43, 44], subgradient/subdifferential [4, 5, 39, 67] and coderivatives [ 2, 4, 5, 18, 27, 28, 29, 33, 39, 72, 73].

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Vorheriges Kapitel Vector Variational Principles for Set-Valued Functions

Nächstes Kapitel Extended Pareto Optimality in Multiobjective Problems

Aubin, J.P.: Lipschitz behavior of solutions to nonconvex problems. Math. Oper. Res. 9, 87–111 (1984)CrossRef

Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control and Cybernetics 36, 531–562 (2007)

Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Global Optim. 43, 533–552 (2009)CrossRef

Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Prog. 122Ser. A, 301–347 (2010)

Benson, H.P.: An improved definition of proper efficiency for vector minimization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1978)CrossRef

Borwein, J.M.: Epi-Lipschitz-like sets in Banach spaces: theorems and examples. Nonlinear Anal. 11, 1207–1217 (1987)CrossRef

Borwein, J.M., Zhu, Q.J.: A survey of subdifferential calculus with applications, Nonlinear Anal. 38, 687–773 (1999)CrossRef

Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Amer. Math. Soc. 338, 105–122 (1993)CrossRef

10.

Chen, G.Y., Jahn, J.: Optimality conditions for set-valued optimization problems. Math. Meth. Oper. Res. 48, 1187–1200 (1998)

11.

Ciligot-Travain, M.: On Lagrange-Kuhn-Tucker multiplier for Pareto optimization problems. Numer. Func. Anal. Optim. 15, 689–693 (1994)CrossRef

12.

Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

13.

Corley, H.W.: Optimality conditions for maximization of set-valued functions. J. Optim. Theory Appl. 58, 1–10 (1988)CrossRef

14.

Crepsi, G.P., Ginchev, I., Rocca, M.: First-order optimality conditions in set-valued optimization. Math. Meth. Oper. Res. 63, 87–106 (2006)CrossRef

15.

Durea, M.: First and second-order Lagrange claims for set-valued maps. J. Optim. Theory Appl. 133, 111–116 (2007)CrossRef

16.

Dutta, J., Tammer, C.: Lagrangian conditions for vector optimization in Banach spaces. Math. Meth. Oper. Res. 64, 521–540 (2006)CrossRef

17.

Edgeworth, F.Y.: Mathematical Psychics. Kegan Paul, London (1881)

18.

El Abdouni, B., Thibault, L.: Optimality conditions for problems with set-valued objectives.J. Appl. Anal. 2, 183–201 (1996)

19.

Fabian, M.: Subdifferentiability and trustworthiness in line of a new variational principle of Bowein and Preiss. Acta Univ. Carolin. Math. Phys. 30, 51–56 (1989)

20.

Gerth (Tammer), C., Weidner, P.W. : Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297–320 (1990)

21.

Gong, X.H.: Optimality conditions for Henig and globally properly efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–37 (2005)CrossRef

22.

Gong, X.H., Dong, H.B., Wang, S.Y.: Optimality conditions for properly efficient solutions of vector set-valued optimization. J. Math. Anal. Appl. 284, 332–350 (2003)CrossRef

23.

Gorokhovich, V.V.: Convex and Nonsmooth Problems of Vector Optimization. Nauka i Tekhnika, Minsk (1990)

24.

Gotz, A., Jahn, J.: The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim. 10, 331–344 (1999)CrossRef

25.

Guerraggio. A., Molno, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)

26.

Ha, T.X.D.: The Ekeland variational principle for set-valued maps involving coderivatives. J. Math. Anal.Appl. 286, 509–523 (2003)CrossRef

27.

Ha, T.X.D.: Lagrange multipliers for set-valued optimization problems associated with coderivatives. J. Math. Anal. Appl. 311, 647–663 (2005)CrossRef

28.

Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassias, Th.M., Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems. pp. 305–324. Springer, Heidelberg (2010)CrossRef

29.

Ha, T.X.D.: Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems. Preprint 10/04, Institute of Mathematics, Hanoi (2010)

30.

Henig, M.I.: Proper efficiency with respect to the cones. J. Optim. Theory Appl. 36, 387–407 (1982)CrossRef

31.

Hiriart-Urruty, J.B.: New concepts in nondifferentiable programming. Bull. Soc. Math. France 60, 57–85 (1979)

32.

Holmes, R. B. : Geometric Functional Analysis and its Applications. Graduated Texts in Mathematics 24, Springer, New York (1975)

33.

Huang, H.: The Lagrange multiplier rule for super efficiency in vector optimization. J. Math. Anal. Appl. 342, 503–513 (2008)CrossRef

34.

Hurwicz, L.: Programming in linear spaces. In: Arrow, K.J., Hurwicz, L., Uzawa, H. (eds.) Standford University Press, Standford, California (1958)

35.

Ioffe, A.D.: Approximate subdifferentials and applications 2: Functions on locally convex spaces. Mathematica 33, 111–128 (1986)

36.

Ioffe, A.D.: Approximate subdifferentials and applications 3: Metric theory, Mathematica 36, 1–38 (1989)

37.

Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surveys 55, 501–558 (2000)CrossRef

38.

Ioffe, A.D., Penot, J.-P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Well-posedness and stability of variational problems. Serdica Math. J. 223, 257–282 (1996)

39.

Isac, G., Khan, A.A.: Dubovitskii-Milyutin approach in set-valued optimization. SIAM J. Control Optim. 47, 144–162 (2008)CrossRef

40.

Isac, G., Khan, A.A.: Second-order optimality conditions in set-valued optimization by a new tangential derivative. Acta Math. Vietnam. 34, 8190 (2009)

41.

Jahn, J.: Mathematical Vector Optimization in Partially Ordered Linear Spaces. Peter Lang, Frankfurt(1986)

42.

Jahn, J.: Vector Optimization - Theory, Applications, and Extensions. Springer, Berlin (2004)

43.

Jahn, J., Khan, A.A.: Generalized contingent epiderivatives in set-valued optimization: optimality conditions. Numer. Funct. Anal. Optim. 23, 807–831 (2002)CrossRef

44.

Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Meth. Oper. Res. 46, 193–211 (1997)CrossRef

45.

Kasimbeily, R.: A nonlinear cone separation theorem and scalarization in nonconvex vector optimization. SIAM J. Optim. 20, 1591–1619 (2010)CrossRef

46.

Krasnoselski, M.A.: Positive Solutions of Operator Equations. Nordhoff, Groningen (1964)

47.

Kruger, A.Y., Mordukhovich, B. S.: Extremal points and the Euler equations in nonsmooth optimization. Dokl. Akad Nauk BSSR 24, 684–687 (1980)

48.

Luc, D.T.: Theory of Vector Optimization. Springer-Verlag, Berlin (1989)

49.

Makarov, E.K., Rachkovski, N.N. Unified representation of proper efficiency by means of dilating cones. J. Optim. Theory Appl. 101, 141–165 (1999)CrossRef

50.

Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth contraints. J. Appl. Math. Mech. 40, 960–969 (1976)CrossRef

51.

Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems. Soviet Math. Dokl. 22(1980), 526–530.

52.

Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183(1994), 250–288.CrossRef

53.

Mordukhovich, B.S.: Coderivatives of set-valued mappings: Calculus and applications. Nonlinear Anal. 30(1997), 3059–3070.CrossRef

54.

Mordukhovich, B.S.: Variational Analysis and Generalized differentiation, I: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences) 330, Springer, Berlin (2006)

55.

Mordukhovich, B.S.: Variational Analysis and Generalized differentiation, II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences) 331, Springer, Berlin (2006)

56.

Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996)CrossRef

57.

Mordukhovich, B.S., Shao, Y.: Nonconvex differential calculus for infinite-dimensional multifunctions. Set-valued Anal. 4, 205–236 (1996)CrossRef

58.

Mordukhovich, B.S., Shao, Y.: Stability of multifunctions in infinite dimensions: Point criteria and applications. SIAM J. Control Optim. 35, 285–314 (1997)CrossRef

59.

Mordukhovich, B.S., Treiman, J.S., Zhu, Q.J.: An extended extremal principle with applications to multiobjective optimization. SIAM J. Optim. 14, 359–379 (2003)CrossRef

60.

Pareto, V.: Cours d’economie politique. F. Rouge, Lausanne (1896)

61.

Qiu, J.-H.: Dual characterization and scalarization for Benson proper efficiency. SIAM J. Optim. 19, 144–162 (2008)CrossRef

62.

Rodríguez-Marín, L. and Sama, M.: (Λ, C)-contingent derivatives of set-valued maps. J. Math. Anal. Appl. 335, 974–989 (2007)CrossRef

63.

Rockafellar, R.T.: The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions. Hedelman Verlag, Berlin (1981)

64.

Rockafellar, R.T.: Lipschitz property of multifunctions. Nonlinear Anal: Theory Meth. Appl. 9, 867–885 (1985)CrossRef

65.

Sach, P.H.: Nearly subconvexlike set-valued maps and vector optimization problems. J. Optim. Theory Appl., 119, 335–356 (2003)CrossRef

66.

Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Func. Anal. and Optim. 19, 121–140 (1998)CrossRef

67.

Taa, A.: Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization. J. Math. Anal. Appl. 283, 398–415 (2003)CrossRef

68.

Thibault, L.: On subdifferentials of optimal value functions. SIAM J. Control Optim. 29, 1019–1036 (1991)CrossRef

69.

Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)CrossRef

70.

Zheng, X.Y.: Proper efficiency in local convex topological vector spaces. J. Optim. Theory Appl. 94, 469–486 (1997)CrossRef

71.

Zheng, X.Y.: Scalarization of Henig properly efficient points in a normed space. J. Optim. Theory Appl. 105, 233–247 (2000)CrossRef

72.

Zheng, X.Y, Ng, K. F.: Fermat rule for multifunctions in Banach spaces. Math. Prog. 104, 69–90 (2005)

73.

Zheng, X.Y, Ng, K. F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim., 17, 1154–1175 (2006)

74.

Zhuang, D.: Density results for proper efficiency. SIAM J. Control Optim. 32, 51–58 (1994)CrossRef

75.

Yang, X. M., Li, D., Wang, S.Y.: Near subconvexlikeness in vector optimization with set- valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)CrossRef

Titel: The Fermat Rule and Lagrange Multiplier Rule for Various Efficient Solutions of Set-Valued Optimization Problems Expressed in Terms of Coderivatives
verfasst von: Truong Xuan Duc Ha
Verlag: Springer Berlin Heidelberg
Buch: Recent Developments in Vector Optimization
Print ISBN: 978-3-642-21113-3

Electronic ISBN: 978-3-642-21114-0

Copyright-Jahr: 2012
DOI: https://doi.org/10.1007/978-3-642-21114-0_12

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