The Fermat rule for multifunctions on Banach spaces

Mathematical Programming

Volumn 104, Issue 1, 2005, Pages 69-90

  Zheng, Xi Yin ^a   Ng, Kung Fu ^b  

^a YUNNAN UNIVERSITY   (China)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Coderivative; Multifunction; Normal cone; Pareto efficient point; Pareto solution

Indexed keywords

OPTIMIZATION; PARETO PRINCIPLE; PROBLEM SOLVING;

CODERIVATIVES; MULTIFUNCTIONS; NORMAL CONE; PARETO SOLUTION; PARETO SUFFICIENT SOLUTIONS;

MATHEMATICAL PROGRAMMING;

EID: 24044548650     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10107-004-0569-9     Document Type: Article

References (30)

1. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston, 1990
2. Borwein, J.M.: On the existence of Pareto efficient points. Math. Oper. Res. 18, 64-73 (1983)
3. Borwein, J.M.: Epi-lipschitz-like sets in Banach spaces: theorems and examples. Nonlinear Anal. 11, 1207-1217 (1987)
4. Borwein, J.M., Treiman, J.S., Zhu, Q.J.: Necessary conditions for constrained optimization problems with semicontinuous and continuous data. Trans. Amer. Math. Soc. 350, 2409-2429 (1998)
5. Borwein, J.M., Zhu, Q.J.: A survey of subdifferential calculus with applications. Nonlinear Anal. 38, 687-773 (1999)
6. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York, 1983
7. Cesari, L., Suryanarayana, M.B.: Existence Theorems for Pareto Optimization: Multivalued and Banach space-valued functionals Trans. Amer. Math. Soc. 244, 37-65 (1978)
8. Edwards, D.A.: On the homeomorphic affine embedding of a locally compact cone into a Banach dual space endowed with the vague topology. Proc. London Math. Soc. 14, 399-414 (1964)
9. Flores-Bazan, F.: Ideal, weakly efficient solutions for vector optimization problems. Math. Program. 93, 453-475 (2002)
10. Gotz, A., Jahn, J.: The Lagrange multiplier rule in set-valued optimization. SIAM J. Optim. 10, 331-344 (1999)
11. Henig, M.I.: Existence and characterization of efficient decisions with respect to cones. Math. Program. 23, 111-116 (1982)
12. Isac, G.: Pareto optimization in infinite-dimensional spaces: The importance of nuclear cones. J. Math. Anal. Appl. 182, 393-404 (1994)
13. Jameson, G.: Ordered Linear Spaces. Springer-Verlag, Berlin, 1970
14. Jimenez, B., Novo, V.: First and second sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284, 496-520 (2003)
15. Loewen, P.D.: Limits of Frechet normals in nonsmooth analysis. Optimization and nonlinear analysis (Haifa, 1990), Pitman Res. Notes Math. Ser., 244, pp. 178-188
16. Minami, M.: Weak Pareto-optimal necessary conditions in a nondifferential multiobjective program on a Banach space. J. Optim. Theory Appl. 41, 451-461 (1983)
17. Modukhovich, B.S.: Coderivative of set-valued mappings: calculus and application. Nonlinear Anal. 30, 3059-3070 (1997)
18. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235-1280 (1996)
19. Mordukhovich, B.S., Shao, Y.: Nonconvex differential calculus for infinite-dimensional multifunctions. Set-valued Anal. 4, 205-236 (1996)
20. Modukhovich, B.S., Treiman, J.S., Zhu, Q.J.: An extended extremal principle wih applications to multi-objective optimization. SIAM J. Optim. 14, 359-379 (2003)
21. Phelps, R.R.: Convex Functions. Monotone Operators and Differentiability, 2nd edn, Springer-Verlag, Berlin, 1993
22. Rudin, W.: Functional Analysis. McGraw-Hill, New York, 1973
23. Song, W.: Characterization of some remarkable classes of cones. J. Matm. ANal. Appl. 279, 308-316 (2003)
24. Sonntag, Y., Zalinescu, C.: Comparison of existence results for efficient points. J. Optim. Theory Appl. 105, 161-188 (2000)
25. Sterna-Karwat, A.: On existence of cone maximal points in real topological linear spaces. Israel J. Math. 54, 33-41 (1986)
26. Yang, X.Q., Jeyakumar, V.: First and second-order optimality conditions for composite multiobjective optimization. J. Optim. Theory Appl. 95, 209-224 (1997)
27. Ye, J.J., Zhu, Q.J.: Multiobjective optimization problem with variational inequality constraints. Math. Program. 96, 139-160 (2003)
28. Zaffaroni, A.: Degrees of efficiency and degree of minimality. SIAM J. Control Optim. 42, 1071-1086 (2003)
29. Zhu, Q.J.: Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints. SIAM J. Control Optim. 39, 97-112 (2000)
30. Zhu, Q.J.: The equivalence of several basic theorems for subdifferentials, Set-Valued Anal. 6, 171-185 (1998)