PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Existence of solutions in parametric optimization and porosity

Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this work we study the parametric family of the minimization problems f (b, x) → min, x ∈ X on a complete metric space X with a parameter b which belongs to a Hausdorff compact space Β. Here f(•,•) belongs to a space of functions on Β x X, say Μ, endowed with an appropriate metric. We study the set of all functions f(•,•) ∈ Μ for which the corresponding parametric family of the minimization problems has solutions for all parameters b ∈ Β. We show that the complement of this set is not only of the first category but also a σ-porous set. This result and its extensions are obtained as realizations of a variational principle.
Rocznik
Tom
Strony
161--184
Opis fizyczny
Bibliogr. 27 poz.
Twórcy
Bibliografia
  • [1] Benyamini Y. and Lindenstrauss J. Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Providence, RI, 2000.
  • [2] Coban M.M., Kenderov P.S. and Revalski J.P. Topological spaces related to the Banach-Mazur game and the. generic properties of optimization problems, Setvalued Analysis, vol 3 pp 263-279, 1995.
  • [3] De Blasi F.S. and Myjak J. On a generalized best, approximation problem, J. Approximation Theory, vol 94, pp 54-72, 1998.
  • [4] Deville R., Godefroy R. and Zizler V. Smoothness and Renormings in Banach Spaces Longman, 1993.
  • [5] Deville R. and Revalski J. Porosity of ill-posed problems, Proc. Amer. Math. Soc., vol 128, pp 1117-1124, 2000.
  • [6] Dontchev A.L. and Jongen H.Th. On the regularity of the Kuhn-Tucker curve, SIAM Journal on Control and Optimization, vol 24, pp 169-176, 1986.
  • [7] Dontchev A. L. and Zolezzi T. Well-Posed Optimization Problems, Lect. Notes in Math. 1543, Springer Berlin-Heidelberg-New York, 1993.
  • [8] Furi M. and Vignoli A. About well-posed minimization problems for functionals in metric spaces, J. Optim. Theory Appl., vol 5, pp 225-229, 1970.
  • [9] Ioffe A.D. and Zaslavski A. J. Variational principles and well-posedness in optimization and calculus of variations, SIAM Journal on Control and Optimization, vol 38, pp 566-581, 2000.
  • [10] Jongen H.Th. and Ruckmann J. -J. On stability and deformation in semi-infinite Optimization, Semi-Infinite Programming (Reemtsen R. and Ruckmann J. J., eds), Kluwer Academic Publishers, Netherlands, pp 29-67, 1998.
  • [11] Jongen H.Th. and Stein O. On generic one-parametric semi-infinite optimization, SIAM J. Optim., vol 7, pp 1103-1137, 1997.
  • [12] Kenderow P.S, and Revalski J.P. The Banach-Mazur game and the, generic existence of solutions to optimization problems, Proc. Amer. Math, Soc., vol 118, pp 911-917, 1993.
  • [13] Marcus M. and Zaslavski A.J. The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincare, Anal. non lineare, vol 16, pp 593-629, 1999.
  • [14] Mizel V.J. New developments concerning the Lavrentiev phenomenon, Calculus of Variations and Differential Equations. Chapman & Hall/CRC Research Notes in Mathematics Series, vol 410, CRC Press, Boca Raton, FL, pp 185-191, 1999.
  • [15] Reich S. and Zaslavski A.J. Convergence of generic infinite products of nonexpansive and uniformly continuous operators, Nonlinear Analysis, vol 36, pp 1049-1065, 1999.
  • [16] Reich S. and Zaslavski A.J. Generic convergence of descent methods in Banach spaces, Math. Oper. Research, vol 25, pp 231-242, 2000.
  • [17] Reich S. and Zaslavski A.J. Well-posedness and porosity in best approximation problems, Topological Methods in Nonlinear Analysis, vol, 18, pp 395-408, 2001.
  • [18] Revalski J. Generic properties concerning well posed optimization problems, C.R. Acad. Bulg. Sci., vol 38, pp 1431-1434, 1985.
  • [19] Zajicek L. Sets of ?-porosity and sets of ?-porosity (q), Casopis Pest. Mat., vol 101, pp 350-359, 1976.
  • [20] Zajicek L. Porosity and ?-porosity , Real Analysis Exchange, vol 13, pp 314-350, 1987.
  • [21] Zajicek L. Small non-?-porous sets in topologically complete metric spaces, Colloq. Math., vol 77, pp 293-304, 1998.
  • [22] Zaslavski A.J. Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis: Theory, Methods and Applications, vol 27, pp 895-932,1996.
  • [23] Zaslavski A.J. On a generic. existence result in optimization, SIAM Journal on Optimization, vol 11, pp 189-198, 2000.
  • [24] Zaslavski A.J. Generic well-posedness of optimal control problems without convexity assumptions, SIAM J. Control Optim., vol 39, pp 250-280, 2000.
  • [25] Zaslavski A.J. Existcnce of solutions of optimal control problems without convexity assumptions, Nonlinear Analysis: Thoory, Methods and Applications, vol 43, pp 339-361, 2001.
  • [26] Zaslavski A.J. Well-posedness and porosity in optimal control without convexity assumptions, Calculus of Variations and Partial Differential Equations, vol 13, pp 265-293, 2001.
  • [27] Zaslavski A.J. On a generic existence result in parametric optimization, Journal of Nonlinear abd Convex Analysis, vol 3, pp 177-190, 2002.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA7-0007-0018
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.