Normal Characterization of the Main Classes of Quasiconvex Functions

Set-Valued Analysis

Volumn 8, Issue 3, 2000, Pages 219-236

  Aussel, D ^a   Daniilidis, A ^b  

^a UNIVERSITÉ DE PERPIGNAN   (France)

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

Author keywords

Normal characterization; Normal cone; Quasiconvexity

Indexed keywords

EID: 0034560546     PISSN: 09276947     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008728926224     Document Type: Article

References (31)

1. Ashton, D. and Atkins, D.: Multicriteria programming for financial planning, J. Oper. Res. Soc. 3 (1979), 259-270.
2. Aussel, D.: Subdifferential properties of quasiconvex and pseudoconvex functions: A unified approach, J. Optim. Theory Appl. 97 (1998), 29-45.
3. Aussel, D., Corvellec, J.-N. and Lassonde, M.: Subdifferential characterization of quasiconvexity and convexity, J. Convex Anal. 1 (1994), 195-201.
4. Aussel, D., Corvellec, J.-N. and Lassonde, M.: Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347 (1995), 4147-4161.
5. Aussel, D., Corvellec, J.-N. and Lassonde, M.: Nonsmooth constrained optimization and multidirectional mean value inequalities, SIAM J. Optim. 9 (1999), 690-706.
6. Avriel, M., Diewert, W.T., Schaible, S. and Zang, I.: Generalized Concavity, Plenum, New York, 1988.
7. t + H(u, Du) = 0, J. Differential Equations 126 (1996), 48-61.
8. ∞, Appl. Math. Optim. 35 (1997), 237-263.
9. Benoist, J.: Connectedness of the efficient set for strictly quasiconcave sets, J. Optim. Theory Appl. 96 (1998), 627-654.
10. Borde, J. and Crouzeix, J.-P.: Continuity properties of the normal cone to the level sets of a quasiconvex function, J. Optim. Theory Appl. 66 (1990), 415-429.
11. Clarke, F. H.: Optimization and Nonsmooth Analysis, 2nd edn, Classics Appl. Math. 5, SIAM, Philadelphia, PA, 1990 (originally published by Wiley-Interscience, New York, 1983).
12. Crouzeix, J.-P.: Contribution à l'étude des fonctions quasiconvexes, Thesis, 231 p., 1977.
13. n, In: Generalized Concavity in Optimization and Economics, Proc. NATO Adv. Study Inst., Vancouver/Can. 1980, 1981, pp. 109-130.
14. Crouzeix, J.-P.: Some properties of Dini derivatives of quasiconvex and pseudoconvex functions, In: Gianessi et al. (eds), New Trends in Mathematical Programming, Kluwer Acad. Publ., Dordrecht, 1998, pp. 41-57.
15. Crouzeix, J.-P. and Ferland, J. A.: Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons, Math. Program. Ser. A 23 (1992), 193-205.
16. Daniilidis, A. and Hadjisavvas, N.: Characterization of nonsmooth semistrictly quasiconvex and strictly quasiconvex functions, J. Optim. Theory Appl. 102 (1999), 525-536.
17. Daniilidis, A. and Hadjisavvas N.: On the subdifferentials of generalized convex functions and cyclic monotonicity, J. Math. Anal. Appl. 237 (1999), 30-42.
18. Hadjisavvas, N.: Continuity properties of quasiconvex functions in infinite-dimensional spaces, Working Paper No.94-03, Graduate School of Management, Univ. of California, April 1994, 10 p.
19. Hassouni, A.: Sous-differentiels des fonctions quasiconvexes, Thèse, Toulouse, 1983.
20. Karamardian, S.: Complementarity over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), 445-454.
21. Luc, D. T.: Characterization of quasiconvex functions, Bull. Austral. Math. Soc. 48 (1993), 393-406.
22. Luc, D. T.: Generalized monotone set valued maps and support bifunctions, Acta Math. Vietnam. 21 (1996), 213-252.
23. Martinez-Legaz, J. E.: Quasiconvex duality theory by generalized conjugation methods, Optimization 19 (1988), 603-652.
24. Penot, J.-P.: Generalized convexity in the light of nonsmooth analysis, In: R. Duvier and C. Michelot (eds), Recent Developments in Optimization, Lecture Notes in Econom. Math. Systems 425, Springer-Verlag, Berlin, 1995, pp. 3-60.
25. Penot, J.-P.: Are generalized derivatives useful for generalized convex functions?, In: J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle (eds), Generalized Convexity, Generalized Monotonicity, Kluwer Acad. Publ., Dordrecht, 1998, pp. 3-59.
26. Penot, J. P. and Quang, P.H.: Generalized convexity of functions and generalized monotonicity of set-valued maps, J. Optim. Theory Appl. 92 (1997), 343-356.
27. Penot, J.-P. and Volle, M.: Dualité de Fenchel et quasiconvexité, C.R. Acad. Sci. Paris, Serie I 304 (1987), 371-374.
28. Rockafellar, R. T.: On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216.
29. Rockafellar, R. T.: Generalized directional derivatives and subgradients of nonconvex functions, Canad. J. Math. 32 (1980), 257-280.
30. Schaible, S.: Fractional programming: Applications and algorithms, Europ. J. Oper. Res. 7 (1981), 111-120.
31. Schaible, S.: Generalized Monotonicity - Concepts and uses, In: Variational Inequalities and Network Equilibrium Problems, Proc. 19th course in the International School of Mathematics 'G. Stampacchia', Erice/Italy, June 19-25, 1994, Plenum, New York, 1995, pp. 289-299.