Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Porosity of ill-posed problems

Authors: Robert Deville and Julian P. Revalski
Journal: Proc. Amer. Math. Soc. 128 (2000), 1117-1124
MSC (1991): Primary 46B20, 49J45
Published electronically: August 5, 1999
MathSciNet review: 1636942
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is $\sigma$-porous.

References [Enhancements On Off] (What's this?)

  • [BP] J. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Am. Math. Soc. 303(1987), 517-527. MR 88k:49013
  • [CK] M.M. \v{C}oban and P.S. Kenderov, Dense Gâteaux differentiability of the $\sup$-norm in $C(T)$ and the topological properties of $T$, Compt. rend. Acad. bulg. Sci. 38(1985), 1603-1604. MR 87h:46070
  • [CKR1] M.M. \v{C}oban, P.S. Kenderov and J.P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika 36(1989), 301-324. MR 91c:90119
  • [CKR2] M.M. \v{C}oban, P.S. Kenderov and J.P. Revalski, Topological spaces related to the Banach-Mazur game and the generic properties of optimization problems, Set-valued Analysis 3(1995), 263-279. MR 96h:90172
  • [DBMP] F.S. De Blasi, J. Myjak and P. Papini, Porous sets in best approximation theory, J. London Math. Soc. 44(1991), 135-142. MR 92h:41066
  • [DGZ1] R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Functional Analysis 111 (1993), 197-212. MR 94b:49010
  • [DGZ2] R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman monographs and Surveys in Pure and Appl. Math., Longman Scientific & Technical, 1993. MR 94d:46012
  • [DoZo] A. Dontchev, T. Zolezzi, Well-posed optimization problems, Lect. Notes in Math. # 1543, Springer Verlag, Berlin, 1993. MR 95a:49002
  • [Ek1] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47(1974), 324-353. MR 49:11344
  • [Ek2] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., Vol. 1(1979), 443-474. MR 80h:49007
  • [FV] M. Furi and A. Vignoli, About well-posed minimization problems for functionals in metric spaces, J. Opt. Theory Appl. 5(1970), 225-229. MR 41:9075
  • [Ge] P.Gr. Georgiev, Strong Ekeland's variational principle, Strong Drop theorem and applications, J. Math. Anal. Appl. 131(1988), 1-21. MR 89c:46019
  • [KR1] P.S. Kenderov and J.P. Revalski, The Banach-Mazur game and generic existence of solutions to optimization problems, Proc. Amer. Math. Soc. 118(1993), 911-917. MR 93i:49012
  • [KR2] P.S. Kenderov and J.P. Revalski, Generic well-posedness of optimization problems and the Banach-Mazur game, in Recent developments in well-posed variational problems, (R. Lucchetti and J.P. Revalski, eds), Mathematics and its Applications # 331, Kluwer Academic Publishers, Dordrecht, 1995, pp. 117-136. MR 96g:49004
  • [LuPa] R. Lucchetti and F. Patrone, Sulla densità e genericità di alcuni problemi di minimo ben posti, Bollettino U.M.I. 15-B(1978), 225-240. MR 58:13711
  • [Ph] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math. #1364, Springer Verlag, Berlin, 1989. MR 94f:46055
  • [PZ] D. Preiss and L. Zají\v{c}ek, Fréchet differentiation of convex functions in a Banach space with a separable dual, Proc. Amer. Math. Soc. 91(1984), 202-204. MR 87f:46072
  • [Ty] A.N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comp. Math. Math. Phys. 6(1966), 631-634. MR 33:6467
  • [Za] L. Zají\v{c}ek, Porosity and $\sigma$-porosity, Real Anal. Exchange, 13(1987-88), 314-350. MR 89e:26009

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46B20, 49J45

Retrieve articles in all journals with MSC (1991): 46B20, 49J45

Additional Information

Robert Deville
Affiliation: Laboratoire de Mathématiques, Université de Bordeaux, 351, cours de la Libération, 33 400 Talence, France

Julian P. Revalski
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, block 8, 1113 Sofia, Bulgaria

Keywords: Variational principles, well-posed optimization problems, ill-posed problems, porous sets, porosity
Received by editor(s): March 24, 1998
Received by editor(s) in revised form: June 1, 1998
Published electronically: August 5, 1999
Additional Notes: This paper was initiated during a short visit of the second named author in November 1997, in the University of Bordeaux
The second author was partially supported by the Bulgarian National Fund for Scientific Research under contract No. MM-701/97
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society