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Some variants of the Ekeland variational principle for a set-valued map. (English) Zbl 1064.49018
Summary: 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.

##### MSC:
 49J53 Set-valued and variational analysis 49J52 Nonsmooth analysis (other weak concepts of optimality)
##### References:
 [1] · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 [2] · Zbl 0779.49015 · doi:10.1287/moor.18.1.173 [3] · Zbl 0936.49012 · doi:10.1007/s001860050020 [4] · Zbl 1042.90036 · doi:10.1023/A:1004663208905 [5] · Zbl 1053.49019 · doi:10.1006/jmaa.2000.7357 [6] · Zbl 1029.49020 · doi:10.1080/02331930211982 [7] 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. [8] · Zbl 1042.49524 · doi:10.1016/S0362-546X(01)00274-7 [9] · Zbl 0595.90085 · doi:10.1007/BF00940198 [10] [11] [12] · Zbl 0866.90103 · doi:10.1007/BF02192246 [13] 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. [14] [15] [16] [17] · Zbl 0881.49009 · doi:10.1090/S0002-9947-96-01543-7 [18]