Abstract
This paper deals with the Ekeland variational principle (EVP) for a set-valued map F with values in a vector space E. Using the concept of cone extension and the Mordukhovich coderivative, we formulate some variants of the EVP for F under various continuity assumptions. We investigate also the stability of a set-valued EVP. Our approach is motivated by the set approach proposed by Kuroiwa for minimizing set-valued maps.
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References
I. Ekeland (1974) ArticleTitleOn the Variational Principle Journal of Mathematical Analysis and Applications 47 324–353
H. Attouch H. Riahi (1993) ArticleTitleStability Results for Ekeland’s ε-Variational Principle and Cone Extremal Solutions Mathematics of Operation Research 18 173–201
G.Y. Chen X.X. Huang (1998) ArticleTitleEkeland’s ε-Variational Principle for Set-Valued Mapping Mathematical Methods of Operations Research 48 181–186
G.Y. Chen X.X. Huang S.H. Hou (2000) ArticleTitleA General Approximate Variational Principle for Set-Valued Maps Journal of Optimization Theory and Applications 106 151–164
X.X. Huang (2001) ArticleTitleNew Stability Results for Ekeland’s ε-Variational Principle for Vector-Valued and Set-Valued Maps Journal of Mathematical Analysis and Applications 262 12–23
X.X. Huang (2002) ArticleTitleStability Results for Ekeland’s ε-Variational Principle for Set-Valued Mappings Optimization 51 31–45
Kuroiwa, D., Some Duality Theorems for Set-Valued Optimization with Natural Criteria, Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, Singapore, Republic of Singapore, pp. 221–228, 2001.
D. Kuroiwa (2001) ArticleTitleOn Set-Valued Optimization Nonlinear Analysis 47 1395–1400
H.W. Corley (1988) ArticleTitleExistence and Lagrangian Duality for Maximizations of Set-Valued Maps Journal of Optimization Theory and Applications 54 489–501
J. Jahn (1986) Mathematical Vector Optimization in Partially-Ordered Linear Spaces Verlag Peter Lang Frankfurt, Germany
J.P. Aubin I. Ekeland (1984) Applied Nonlinear Analysis John Wiley and Sons New York, NY
F. Ferro (1996) ArticleTitleAn Optimization Result for Set-Valued Mappings and a Stability Property in Vector Problems with Constraints Journal of Optimization Theory and Applications 90 63–77
Fan, K., A Minimax Inequality and Applications, Inequalities III, Proceeding of the 3rd Symposium Dedicated to the Memory of Theodore S. Motzkin, University of California, Los Angeles, California 1969; Academic Press, New York, NY, pp. 103–113, 1972.
D.T. Luc (1989) Theory of Vector Optimization Springer Verlag New York, NY
S. Dancs M. Hegedus P. Medvegyev (1983) ArticleTitleA General Ordering and Fixed-Point Principle in Complete Metric Space Acta Scientiarum Mathematicarum (Szeged) 46 381–388
M.A. Krasnoselskii (1964) Positive Solutions of Operator Equations Noordhoff Groningen, Holland
B.S. Mordukhovich Y. Shao (1996) ArticleTitleNonsmooth Sequential Analysis in Asplund Spaces Transactions of the American Mathematical Society 348 1235–1280
B.S. Mordukhovich Y. Shao (2000) ArticleTitleOn Variational Characterizations of Asplund Spaces Canadian Mathematical Society, Conference Proceedings 27 245–254
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This research was supported by a Georg Forster Grant administered by the Alexander von Humboldt Foundation. The author thanks Professor J. Jahn, University of Erlangen-Nürnberg, for comments on the manuscript. The author thanks the referee for suggestions which improved the paper.
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Ha, T.X.D. Some Variants of the Ekeland Variational Principle for a Set-Valued Map. J Optim Theory Appl 124, 187–206 (2005). https://doi.org/10.1007/s10957-004-6472-y
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DOI: https://doi.org/10.1007/s10957-004-6472-y
Keywords
- Ekeland variational principle
- set-valued maps
- Mordukhovich coderivatives
- stability