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Generic well posedness of supinf problems

Published online by Cambridge University Press:  17 April 2009

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
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Abstract

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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
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), 795814.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), 715728.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), 16031604.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. 141149.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), 301324.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), 373376.Google Scholar
[8]De Blasi, F. and Myjak, J., ‘Some generic properties in convex and nonconvex optimization theory’, Ann. Soc. Math. Polon. 24 (1984), 114.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), 58.Google Scholar
[11]Jane, J.E. and Rogers, C.A., ‘Borel electors for upper semicontinuous set–valued maps’, Acta Math. 155 (1985), 4179.CrossRefGoogle Scholar
[12]Kenderov, P.S., ‘Most of optimization problems have unique solution’, C.R. Acad. Bulg. Sci. 37 (1984), 297299.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), 911917.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. 4148.Google Scholar
[15]Lignola, M.B. and Morgan, J., ‘Topological existence and stability for minsup problems’, J. Math. Anal. Appl. (1990), 165180.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. 99113.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), 225240.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), 258281.CrossRefGoogle Scholar
[21]Patrone, F., ‘Most convex functions are nice’, Numer. Funct. Anal. Optimi. 9 (1987), 359369.CrossRefGoogle Scholar
[22]Revalski, J.P., ‘Generic well–posedness in some classes of optimization problems’, Acta Univ. Carolin. Math. Phys. 28 (1987), 117125.Google Scholar
[23]Ribarska, N.K., ‘Interval characterization of fragmentable spaces’, Mathematika 34 (1987), 243257.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), 2833.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
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