Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T16:57:41.605Z Has data issue: false hasContentIssue false
  • English
  • Français

Generic well posedness of supinf problems

Published online by Cambridge University Press:  17 April 2009

P.S. Kenderov  and
R.E. Lucchetti
P.S. Kenderov
Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, 1113 Sofia, Bulgaria
R.E. Lucchetti
Affiliation:
Department of Mathematics, University of Milano, via Saldini 50, 20133 Milano, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider two notions of well posedness for problems of the type and give conditions under which the majority (in Baire category sense) of bounded functions f defined in X × Y give rise to problems which are well posed. As a corollary we get that the problem is well posed for the majority of bounded lsc real valued functions f if, and only if, X contains a dense completely metrisable subset.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 1 , August 1996 , pp. 5 - 25
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Basar, T. and Olsder, G., Dynamic noncooperative game theory (Academic Press, New York, 1982).Google Scholar
[2]Beer, G. and Lucchetti, R., ‘Convex optimization and the epidistance topology’, Trans. Amer. Math. Soc. 327 (1991), 795–814.CrossRefGoogle Scholar
[3]Beer, G. and Lucchetti, R., ‘The epidistance topology: continuity and stability results with applications to convex optimization problems’, Math. Oper. Res. 17 (1992), 715–728.CrossRefGoogle Scholar
[4]Čoban, M.M. and Kenderov, P.S., ‘Dense Gâteaux differentiability of the sup norm in C(T) and topological properties of T’, C.R. Acad. Bulg. Sci. 38 (1985), 1603–1604.Google Scholar
[5]Čoban, M.M. and Kenderov, P.S., ‘Generic Gâteaux differentiability of convex functionals in C(T) and topological properties of T’, in Mathematics and Education in Mathematics, Proceedings of the XV-th Spring Conference of the Union of Bulgarian Mathematicians, 1986, pp. 141–149.Google Scholar
[6]Čoban, M.M., Kenderov, P.S. and Revalski, J.P., ‘Generic well-posedness of optimization problems in topological spaces’, Mathematika 36 (1989), 301–324.CrossRefGoogle Scholar
[7]De Blasi, F. and Myjak, J., ‘On the minimum distance to a closed convex set in a Banach space’, Bull. Acad. Sci. Pol 29 (1981), 373–376.Google Scholar
[8]De Blasi, F. and Myjak, J., ‘Some generic properties in convex and nonconvex optimization theory’, Ann. Soc. Math. Polon. 24 (1984), 1–14.Google Scholar
[9]Dontchev, A. and Zolezzi, T., Well-posed optimization problems, Lecture Notes in Mathematics 1543 (Springer-Verlag, Berlin, Heidelberg, New York, 1993).CrossRefGoogle Scholar
[10]Holá, L., ‘Most of the optimization problems have unique solution’, C.R. Acad. Bulg. Sci. 42 (1989), 5–8.Google Scholar
[11]Jane, J.E. and Rogers, C.A., ‘Borel electors for upper semicontinuous set–valued maps’, Acta Math. 155 (1985), 41–79.CrossRefGoogle Scholar
[12]Kenderov, P.S., ‘Most of optimization problems have unique solution’, C.R. Acad. Bulg. Sci. 37 (1984), 297–299.Google Scholar
[13]Kenderov, P.S. and Revalski, J.P., ‘The Banach–Mazur game and generic existence of solution to optimization problems’, Proc. Amer. Math. Soc. 118 (1993), 911–917.CrossRefGoogle Scholar
[14]Kenderov, P.S. and Ribarska, N.K., ‘Generic uniqueness of the solution of “min–max” problems’, in Lecture Notes in Economics and Mathematical Systems 304 (Springer-Verlag, Berlin, Heidelberg, New York, 1988), pp. 41–48.Google Scholar
[15]Lignola, M.B. and Morgan, J., ‘Topological existence and stability for minsup problems’, J. Math. Anal. Appl. (1990), 165–180.Google Scholar
[16]Loridan, P., ‘An application of Ekeland's variational principle to generalized Stackelberg problems’, (Written version of a lecture presented to the Workshop on Well-Posedness in Optimization, Santa Margherita, Ligure, 1991).Google Scholar
[17]Loridan, P. and Morgan, J., ‘ε–regularized two–level optimization problems; approximation and existence results’, in Lecture Notes in Mathematics 1405 (Springer-Verlag, Berlin, Heidelberg, New York, 1989), pp. 99–113.Google Scholar
[18]Lucchetti, R. and Patrone, F., ‘Sulla densitàe genericità di alcuni problemi di minimimo ben posti’, Boll. Un. Mat. Ital. B 15 (1978), 225–240.Google Scholar
[19]Morgan, J., ‘Constrained well–posed two–level optimization problems’, in Nonsmooth optimization and related topics, (Clarke, F.H., Dem'yanov, V. F. and Giannessi, F., Editors) (Plenum Press, New York and London, 1989).Google Scholar
[20]Namioka, I., ‘Radon–Nikodým compact spaces and fragmentability’, Mathematika 34 (1987), 258–281.CrossRefGoogle Scholar
[21]Patrone, F., ‘Most convex functions are nice’, Numer. Funct. Anal. Optimi. 9 (1987), 359–369.CrossRefGoogle Scholar
[22]Revalski, J.P., ‘Generic well–posedness in some classes of optimization problems’, Acta Univ. Carolin. Math. Phys. 28 (1987), 117–125.Google Scholar
[23]Ribarska, N.K., ‘Interval characterization of fragmentable spaces’, Mathematika 34 (1987), 243–257.CrossRefGoogle Scholar
[24]Von Stackelberg, H., The theory of market economy (Oxford University Press, Oxford, 1952).Google Scholar
[25]Tyhonov, A., ‘On the stability of the functional optimization problem’, USSR Comput. Math, and Math. Phys. 6 (1966), 28–33.CrossRefGoogle Scholar
[26]Zolezzi, T., Well posed optimal control problems: A perturbation approach, IMA Proceedings (Springer-Verlag, Berlin, Heidelberg, New York, to appear).CrossRefGoogle Scholar
[27]Zolezzi, T., ‘Well posedness of optimal control problems’, (Preprint No 236, Dipartimento di Matematica Universita di Genova, 1993).Google Scholar