Abstract
A simple unified framework is presented for the study of strong efficient solutions, weak efficient solutions, positive proper efficient solutions, Henig global proper efficient solutions, Henig proper efficient solutions, super efficient solutions, Benson proper efficient solutions, Hartley proper efficient solutions, Hurwicz proper efficient solutions and Borwein proper efficient solutions of set-valued optimization problem with/or without constraints. Some versions of the Lagrange claim, the Fermat rule and the Lagrange multiplier rule are formulated in terms of the first- and second-order radial derivatives, the Ioffe approximate coderivative and the Clarke coderivative.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.
T.Q. Bao, B.S. Mordukhovich, Necessary conditions for super minimizers in constrained multiobjective optimization, manuscript (2008).
H.P. Benson, An improved definition of proper efficiency for vector minimization with respect to cones, J. Math. Anal. Appl., 71 (1978) 232–241.
J.M. Borwein, The geometry of Pareto efficiency over cones, Mathematische Operationforschung and Statistik, Serie Optimization, 11(1980), 235–248.
J.M. Borwein, D. Zhuang, Super efficiency in vector optimization, Trans. Amer. Math. Society, 338(1993), 105–122.
G. Bouligand, Sur l’existence des demi-tangentes à une courbe de Jordan, Fundamenta Mathematicae 15 (1030), 215–218.
G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (1998) 1187–1200.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
H.W. Corley, Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl. 58 (1988) 1–10.
G.P. Crepsi, I. Ginchev, M. Rocca, First-order optimality conditions in set-valued optimization, Math. Meth. Oper. Res. 63, (2006), 87–106.
M. Durea, First and second-order Lagrange claims for set-valued maps, J. Optim. Theory Appl., 133 (2007), 111–116.
B. El Abdouni, L. Thibault, Optimality conditions for problems with set-valued objectives, J. Applied Analysis 2 (1996) 183–201.
F. Flores-Bazán, Optimality conditions in non-convex set-valued optimization, Math. Meth. Oper. Res. 53 (2001), 403–417.
X.H. Gong, Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior, J. Math. Anal. Appl. 307 (2005) 12–37.
X.H. Gong, H.B. Dong, S.Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization, J. Math. Anal. Appl. 284 (2003) 332–350.
A. Gotz, J. Jahn, The Lagrange multiplier rule in set-valued optimization, SIAM J. Optim. 10 (1999), 331–344.
E. Goursat, Courses d’Analyse Mathématiques, Courcier, Paris (1813).
A. Guerraggio. E. Molno, A. Zaffaroni, On the notion of proper efficiency in vector optimization, J. Optim. Theory Appl., 82 (1994), 1–21.
T. X. D. Ha, Lagrange multipliers for set-valued optimization problems associated with coderivatives, J. Math. Anal. Appl. 311 (2005) 647–663.
R. Hartley, On cone efficiency, cone convexity, and cone compactness, SIAM Journal on Applied Mathematics 34 (1978), 211–222.
M.I. Henig, Proper efficiency with respect to the cones, J. Optim. Theory Appl. 36 (1982), 387–407.
J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France 60 (1979) 57–85.
H. Huang, The Lagrange multiplier rule for super efficiency in vector optimization, J. Math. Anal. Appl. 342 (2008), no. 1, 503–513.
L. Hurwicz, Programming in linear spaces, Edited by K.J. Arrow, L. Hurwicz, H. Uzawa, Standford University Press, Stanford, CA, 1958.
A.D. Ioffe, Approximate subdifferentials and applications 3: Metric theory, Mathematica 36 (1989), 1–38.
G. Isac, A.A. Khan, Dubovitskii-Milyutin approach in set-valued optimization, SIAM J. Control Optim. 47 (2008), no. 1, 144–162.
J. Jahn, Mathematical vector optimization in partially ordered linear spaces, Peter Lang, Frankfurt, 1986.
J. Jahn, Vector Optimization, Springer-Verlag, Berlin, 2004.
J. Jahn, A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: optimality conditions, Numer. Funct. Anal. Optim. 23 (2002) 807–831.
J. Jahn, R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res. 46 (1997) 193–211.
M.A. Krasnoselski, Positive Solutions of Operator Equations, Nordhoff, Groningen, 1964.
J.L. Lagrange, Théorie des Fonctions Analytiques, Impr.de la République Paris, 1797.
S. J. Li, X. Q. Yang, G.Y. Chen, Nonconvex vector optimization of set-valued mappings, J. Math. Anal. Appl. 283 (2003) 337–350.
D.T. Luc, Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.
E.K. Makarov, N.N. Rachkovski, Unified representation of proper efficiency by means of dilating cones, J. Optim. Theory Appl., 101 (1999),141–165.
J.-H. Qiu, Dual characterization and scalarization for Benson proper efficiency, SIAM J. Optim., 19 (2008), 144–162.
P.H. Sach, Nearly subconvexlike set-valued maps and vector optimization problems, J. Optim. Theory Appl., 119 (2003), 335–356.
A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numer. Func. Anal. Optimiz. 19 (1998), 121–140.
A. Taa, Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization, J. Math. Anal. Appl. 283 (2003) 398–415.
X.M. Yang, D. Li, S.Y. Wang, Nearly subconvexlikeness in vector optimization with set- valued functions, J. Optim. Theory Appl., 110 (2001), 413–427.
A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM J. Control Optim. 42 (2003), 1071–1086.
X.Y. Zheng, Scalarization of Henig proper efficient points in a normed space, J. Optim. Theory Appl., 105 (2000), 233–247.
X.Y. Zheng, Proper efficiency in local convex topological vector spaces, J. Optim. Theory Appl., 94 (1997), 469–486.
X.Y. Zheng, K.F. Ng, Fermat rule for multifunctions in Banach spaces, Math. Program, 104 (2005) 69–90.
X.Y. Zheng, K.F. Ng, The Lagrange multiplier rule for multifunctions in Banach spaces, SIAM J. Optim., 17 (2006), 1154–1175.
D. Zhuang, Density results for proper efficiency, SIAM J. Control Optim., 32 (1994), 51–58.
Acknowledgment
This research was initiated during the author’s stay at the Institute of Applied Mathematics of the University of Erlangen-Nürnberg under the Georg Forster grant of the Alexander von Humboldt Foundation. The author thanks Professor J. Jahn for hospitality, advice, and help in her work.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Professor George Isac
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Ha, T. (2010). Optimality Conditions for Several Types of Efficient Solutions of Set-Valued Optimization Problems. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_21
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0158-3_21
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0157-6
Online ISBN: 978-1-4419-0158-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)