Normality and modulability indices. Part I: Convex cones in normed spaces

Journal of Mathematical Analysis and Applications

Volumn 338, Issue 1, 2008, Pages 365-391

  Iusem, A ^a   Seeger, A ^b  

^a IMPA   (Brazil)

^b UNIVERSITY OF AVIGNON   (France)

Author keywords

Convex cones; Modulability index; Normality index

Indexed keywords

EID: 34548814994     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2007.05.030     Document Type: Article

References (42)

1. Ando T. On fundamental properties of a Banach space with a cone. Pacific J. Math. 12 (1962) 1163-1169
2. Attouch H., and Wets R.J.B. Quantitative stability of variational systems. I. The epigraphical distance. Trans. Amer. Math. Soc. 328 (1991) 695-729
3. Aubin J.P., and Frankowska H. Set-Valued Analysis (1990), Birkhäuser, Boston
4. Beer G. Topologies on Closed and Closed Convex Sets (1993), Kluwer Acad. Publ., Dordrecht
5. Benabdellah H. Denting points in Bochner Banach ideal spaces X (E). J. Convex Anal. 6 (1999) 183-194
6. Berkson E. Some metrics on the subspaces of a Banach space. Pacific J. Math. 13 (1963) 7-22
7. Bonsall F.F. The decomposition of continuous linear functionals into non-negative components. Proc. Univ. Durham Philos. Soc. Ser. A 13 (1957) 6-11
8. Castaing C., and Valadier M. Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. vol. 580 (1977), Springer-Verlag, Berlin
9. Epelman M., and Freund R.M. Condition number complexity of an elementary algorithm for computing a reliable solution of a conic linear system. Math. Program. 88 (2000) 451-485
10. Epelman M., and Freund R.M. A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systems. SIAM J. Optim. 12 3 (2002) 627-655
11. Freund R.M. On the primal-dual geometry of level sets in linear and conic optimization. SIAM J. Optim. 13 (2003) 1004-1013
12. Freund R.M., and Vera R.M. Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm. SIAM J. Optim. 10 (1999) 155-176
13. Goffin J.L. The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5 (1980) 388-414
14. Goullet de Rugy A. Un théorème du genre "Ando-Edwards" pour les Fréchet ordonnés normaux. Pacific J. Math. 46 (1973) 155-166
15. Grosberg J., and Krein M. Sur la décomposition des fonctionnelles en composantes positives. C. R. Dokl. Acad. Sci. URSS (N.S.) 25 (1939) 723-726 (in French)
16. Han Z.Q. Remarks on the angle property and solid cones. J. Optim. Theory Appl. 82 (1994) 149-157
17. Hyers D.H., Isac G., and Rassias T.M. Topics in Nonlinear Analysis and Applications (1997), World Scientific, Singapore
18. Iusem A., and Seeger A. Pointedness, connectedness, and convergence results in the space of closed convex cones. J. Convex Anal. 11 (2004) 267-284
19. Iusem A., and Seeger A. Measuring the degree of pointedness of a closed convex cone: A metric approach. Math. Nachr. 279 (2006) 599-618
20. Iusem A., and Seeger A. Axiomatization of the index of pointedness for closed convex cones. Comput. Appl. Math. 24 (2005) 245-285
21. Iusem A., and Seeger A. On vectors achieving the maximal angle of a convex cone. Math. Program. 104 (2005) 501-523
22. A. Iusem, A. Seeger, Searching for critical angles in a convex cone, Math. Program. (2007), in press
23. A. Iusem, A. Seeger, Computing the radius of pointedness of a convex cone, Math. Program. (2007), doi:10.1007/s10107-006-0069-1, in press
24. A. Iusem, A. Seeger, Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones (May 2006), submitted for publication
25. A. Iusem, A. Seeger, Normality and modulability indices. Part II: Convex cones in Hilbert spaces, J. Math. Anal. Appl. (2007), doi:10.1016/j.jmaa.2007.05.109, in press
26. Kalauch A. On positive-off-diagonal operators on ordered normed spaces. Z. Anal. Anwendungen 22 (2003) 229-238
27. Kist J. Locally o-convex spaces. Duke Math. J. 25 (1958) 569-582
28. Krasnosel'skii M.A. Positive Solutions of Operator Equations. translated from the Russian by R.E. Flaherty, edited by L.F. Boron (1964), Noordhoff Ltd., Groningen
29. Krasnosel'skii M.A., and Zabreiko P.P. Geometrical Methods of Nonlinear Analysis (1984), Springer-Verlag, Berlin
30. Krein M. Propriétés fondamentales des ensembles coniques normaux dans l'espace de Banach. C. R. Dokl. Acad. Sci. URSS (N.S.) 28 (1940) 13-17 (in French)
31. Kusraev A.G., and Kutateladze S.S. Subdifferentials: Theory and Applications (1995), Kluwer Acad. Publ., Dordrecht
32. Lajara S. ALUR dual renormings of Banach spaces. J. Math. Anal. Appl. 299 (2004) 221-226
33. Lin B.L., Lin P.K., and Troyanski S.L. Characterizations of denting points. Proc. Amer. Math. Soc. 102 (1988) 526-528
34. Ng K.F. On a computation rule for polars. Math. Scand. 26 (1970) 14-16
35. Peressini A. Ordered Topological Vector Spaces (1967), Harper and Row Publishers, New York
36. Qiu J.H. On solidness of polar cones. J. Optim. Theory Appl. 109 (2001) 199-214
37. Rockafellar R.T., and Wets R.J.B. Variational Analysis (1998), Springer-Verlag, Berlin
38. Schaefer H.H. Topological Vector Spaces (1966), Macmillan Company, New York
39. Sion M. On general minimax theorems. Pacific J. Math. 8 (1958) 171-176
40. Troyanski S.L. On a property of the norm which is close to local uniform rotundity. Math. Ann. 271 (1985) 305-313
41. Walkup D.W., and Wets R.J.B. Continuity of some convex-cone-valued mappings. Proc. Amer. Math. Soc. 18 (1967) 229-235
42. Weston J.D. The decomposition of a continuous linear functional into non-negative components. Math. Scand. 5 (1957) 54-56