Characterizations of generalized convexities via generalized directional derivatives

Numerical Functional Analysis and Optimization

Volumn 19, Issue 5-6, 1998, Pages 615-634

  Sach, Pham Huu ^a   Penot, Jean Paul ^b  

^a INSTITUTE OF MATHEMATICS   (Viet Nam)

^b UNIVERSITÉ DE PAU ET DES PAYS DE L'ADOUR   (France)

Author keywords

Convexity; Derivatives; Generalized convexity; Generalized derivatives; Invexity; Monotone mapping; Pseudo convexity; Quasiconvexity; Quasimonotone operator

Indexed keywords

FUNCTIONS; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; MATHEMATICAL TECHNIQUES; TOPOLOGY;

BIFUNCTION; CONVEXITIES; DIRECTIONAL DERIVATIVES; INVEXITY; MONOTONE MAPPING; QUASIMONOTONE OPERATOR; TOPOLOGICAL VECTOR SPACE; UPPER SEMICONTINUOUS FUNCTION;

VECTORS;

EID: 0032117192     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1080/01630569808816849     Document Type: Article

References (33)

1. J.-P. Aubin and H. Frankowska, "Set-valued Analysis", Birkhäuser, Basel, 1990
2. D. Aussel, J-N. Corvellec and M. Lassonde, Sub differential characterization of quasi convexity and convexity, J. Convex Anal. 1 (2) (1994), 195-202.
3. F.H. Clarke, "Optimization and Nonsmooth Analysis", Wiley-Interscience, New York, 1983.
4. R. Correa, A. Jofre and L. Thibault, Characterization of lower semicontinuous convex functions, Proc. Am. Math. Soc. 116 (1992), 67-72.
5. B.D. Craven and B. Glover, Invex functions and duality, J. Austral. Math. Soc., Ser. A 41 (1986), 64-78.
6. B.D. Craven, P.M. Sach, N.D. Yen and T.D. Phuong, A new class of invex multifunctions, in : F. Giannessi (ed.) "Nonsmooth Optimization : Methods and Applications", Gordon and Breach, London, (1992), 52-69.
7. J.-P. Crouzeix and J.A. Ferland, Criteria for quasiconvexity and pseudoconvexity: relationships and comparisons, Math. Programming 23 (1982), 193-205.
8. W.E. Diewert, Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming, in: S. Schaible and "W.T. Ziemba (eds.) " Generalized Concavity in Optimization and Economics", Academic Press, New-York, 1981.
9. R. Ellaia and H. Hassouni, Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991), 401-416.
10. G. Giorgi and S. Komlosi, Dini derivatives in optimization, Revista di Mat. per le se. econ. e sociali 15 (1), (1993), 3-30.
11. M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545-550.
12. A.D. Ioffe, Approximate subdifferentials and applications I. The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 289-316.
13. A.D. Ioffe, Approximate subdifferentials and applications I. The metric theory, Mathematika 36 (1989), 1-38.
14. A.D. Ioffe, Non-smooth subdifferentials: Their calculus and applications, in World Congress of Nonlinear Analysts '92, Proc. Symposium Tampa, August 1992, V. Lakhmikantham, ed. W. de Gruyter (1996), 60-87.
15. S. Karamardian, Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), 445-454.
16. S. Komlosi, Some properties of nondifferentiable pseudoconvex functions, Math. Programming 26 (1983), 232-237.
17. S. Karamardian and S. Schaible, Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990), 37-46.
18. S. Komlosi, On generalized upper quasidifferentiability, in: F. Giannessi (ed.) "Nonsmooth Optimization : Methods and Applications", Gordon and Breach, London, (1992), 189-200.
19. S. Komlosi, Generalized monotonicity and generalized convexity, J. Opt. Theory Appl. 84 (1995), 361-376.
20. D.T. Luc, Characterizations of quasi convex functions, Bull. Austral. Math. Soc. 48 (1993), 393-405.
21. D.T. Luc and S. Swaminathan, A characterization of convex functions, Nonlinear Analysis, Theory, Methods & Appl., 30 (1993), 697-701.
22. O.L. Mangasarian, Pseudoconvex functions, SIAM J. Control 3 (1965), 281-290.
23. O.L. Mangasarian, "Nonlinear Programming", Me Graw-Hill, New-York, 1969.
24. J.-J. Moreau, Inf-convolution, sous-additivitë, convexité des fonctions numériques, J. Math. Pures et Appl. 49(1970), 109-154.
25. Ph. Michel and J.-P. Penot, A generalized derivative for calm and stable functions, Differential and Integral Equations, 5 (2) (1992), 433-454.
26. J.-P. Penot, Variations on the theme of nonsmooth analysis : another subdifferential, in V.F. Demyanov and D. Pallaschke eds., " Nondifferentiable optimization : Motivations and Applications", Proc. Sopron, 1984, Lect. Notes in Econ. and Math. Systems 255, Springer Verlag, Berlin 1985, pp 41-54.
27. J.-P. Penot and P. Terpolilli, Cônes tangents et singularités, C.R. Acad. Sc. Paris 296 (1983), 721-724.
28. J.-P. Penot and P.H. Quang, On generalized convex functions and generalized monotonicity of set-valued maps, J. Optim. Th. Appl. 92 (2) (1997), 343-356.
29. R.A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Analysis, Theory, Meth. and Appl. 14 (1990), 305-317.
30. B.N. Pshenichnyi, Necessary conditions for an extremum, Dekker, New York, 1971.
31. T.W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc. 42 (1990), 437-446.
32. R.T. Rockafellar, The theory of sub gradients and its applications to problems of optimization of convex and nonconvex functions, Presses de l'Université de Montrëal and Helderman Verlag, Berlin 1981.
33. D. Ward, The quantificational tangent cones, Canad. J. Math. 40 (3) (1988), 666-694.