Characterization of scalar quasiconvexity and convexity of locally Lipschitz vector-valued maps

Optimization

Volumn 46, Issue 3, 1999, Pages 283-310

  Sach, Pham Huu ^a  

^a INSTITUTE OF MATHEMATICS   (Viet Nam)

Author keywords

Convexity; Monotonicity; Quasimonotonicity; Scalar quasiconvexity

Indexed keywords

EID: 0002980199     PISSN: 02331934     EISSN: None     Source Type: Journal    
DOI: 10.1080/02331939908844457     Document Type: Article

References (32)

1. Aubin, J. P. and Frankowska, H. (1990). Set-valued Analysis, Birkhäuser, Basel.
2. Aussel, D., Corvellec, J. N. and Lassonde, M. (1995). Mean value property and subdifferential criteria for lower semicontinuous functions. Transaction AMS, 347, 4147-4161.
3. Aussel, D., Corvellec, J. N. and Lassonde, M. (1994). Subdifferential characterization of quasiconvexity and convexity. J. Convex Analysis, 1, 1-7.
4. Carabini, R. and Komlosi, S., Polar quasiconcavity and polar quasimonotonicity concepts for vector-valued functions. Manuscript.
5. Cambini, R. and Komlosi, S. (1996). Generalized concavity and generalized monotonicity concepts for vector-valued functions. University of Pisa, Dept. of Statistics and Applied Mathematics, Preprint No. 98.
6. Cambini, R. and Komlosi, S. (1996). On the scalarization of pseudoconcavity and pseudomonotonicity concepts for vector-valued functions. Proceedings of the 5th Symposium on Generalized Convexity, held in Luminy-Marseille (France), June 17-21.
7. Clarke, F. H. (1983). Optimization and Nonsmooth Analysis, Wiley, New York.
8. Clarke, F. H., Stern, R. J. and Wolenski, P. R. (1993). Subgradient criteria for monotonicity, the Lipschitz condition and convexity. Can. J. Math., 45(6), 1167-1183.
9. Correa, R., Jofre, A. and Thibault, L. (1992). Characterization of lower semi-continuous convex function, Proc. Amer. Math. Soc., 116, 67-72.
10. Correa, R., Jofre, A. and Thibault, L. (1994). Subdifferential monotonicity as a characterization of convex functions. Numerical Functional Analysis and Optimization, 15, 531-535.
11. Correa, R., Jofre, A. and Thibault, L. (1995). Subdifferential characterization of convexity. In: Du, D. Z., Qi, L. and Womersley R. S. (Eds.), Recent Advances in Nonsmooth Optimization. World Scientific, pp. 18-23.
12. Crouzeix, J. P. (1977). Contributions à l'étude des fonctions quasiconvexes. Thèse d'État, Université de Clermont-Ferrand II.
13. Ellaia, E. and Hassouni, H. (1991). Characterization of nonsmooth functions through their generalized gradients. Optimization, 22, 401-416.
14. Loffe, A. D., Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps. Nonlinear Analysis, Theory, Methods and Applications, 8, 517-539.
15. Jeyakumar, V., Oettli, W., and Natividad, M. (1993). A solvability theorem for a class of quasiconvex mapping with applications to optimization. Journal of Mathematical Analysis and Applications, 179, 537-546.
16. Karamardian, S. and Schaible, S. (1990). Seven kinds of monotone maps. J. Optim. Theory Appl., 66, 37-46.
17. Komlosi, S. (1992). On generalized upper quasidifferentiability. In: Giannessi F. (Ed.) "Nonsmooth Optimization: Methods and Applications", Gordon and Breach, London, pp. 189-201.
18. Komlosi, S. (1995). Generalized monotonicity and generalized convexity. J. Optim Theory Appl., 84, 361-376.
19. Komlosi, S. (1995). Monotonicity and quasimonotonicity in Nonsmooth Analysis. In: Du, D. Z., Qi, L. and Womersley R. S. (Eds.), Recent Advances in Nonsmooth Optimization. World Scientific, pp. 193-214.
20. Luc, D. T. (1993). Characterisations of quasiconvex functions. Bull. Austral. Math. Soc., 48, 393-405.
21. Luc, D.T. and Swaminathan, S. (1993). A characterization of convex functions. Nonlinear Anal. Theory, Method, Appl., 30, 697-701.
22. Penot, J. P. (1995). Generalized convexity in the light of Nonsmooth Analysis. Proceedings of the 7th French-German Conference on Optimization, Dijon, France, Edited by Duriez R. and Michelot. C., Lecture Notes in Mathematical Systems and Economics, No. 429, Springer Verlag, Berlin, Germany, pp. 269-290.
23. Penot, J. P. (1996). Are generalized derivatives useful for generalized convexity? Proceedings of the 5th International Symposium on Generalized Convexity, held in Luminy-Marseille, France, June 17-21.
24. Penot, J. P. and Quang, P. H. (1997). On generalized convex functions and generalized monotonicity of set-valued maps. J. Optim. Theory Appl., 92, 343-356.
25. Penot, J. P. and Sach, P. H. (1997). Generalized monotonicity of subdifferentials and generalized convexity. J. Optim. Theory Appl., 64, 251-262.
26. Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press, Princeton, New Jersey.
27. Sach, P. H. (1996). Sufficient conditions for generalized convex set-valued maps. Optimization, 37, 293-304.
28. Sach, P. H., Another characterization of convexity for set-valued maps. Preprint, Hanoi Inst. of Math. (to appear in Numer. Funct. Analysis and Opt.).
29. Sach, P. H. and Craven, B. D. (1991). Invexity in multifunction optimization. Numerical Functional Analysis and Optimization, 12, 383-394.
30. Sach, P. H. and Penot, J. P. (1998). Characterizations of generalized convexities via generalized directional derivatives. Numer. Fund. Anal. and Optim., 19, 615-634.
31. Sach, P. H. and Yen, N. D. (1997). Convexity criteria for set-valued maps. Set-valued Analysis, 5, 37-45.
32. Zagrodny, D. (1992). General sufficient conditions for the convexity of a function. Zeitschrift für Analysis und ihre Anwendungen, 11, 277-283.