Abstract

The differentiability properties of the metric projections onto the cones of nonnegative functions in Lp(0,1)L^p(0,1) and in W1,p(0,1)W^{1,p}(0,1) are considered. It is shown that the metric projection mapping is Bouligand differentiable in Lp(0,1)L^p(0,1), but it is not Bouligand differentiable in W1,p(0,1)W^{1,p}(0,1).

Author information

Contact details are reproduced from the original publication and may be historical.

K. Malanowski

Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

kmalan@ibspan.waw.pl

Keywords

  • Metric projection
  • cones of nonnegative functions
  • Bouligand differentiability

References

  1. E. Bednarczuk, M. Pierre, E. Rouy, J. Sokolowski: Tangent sets in some functional spaces, Nonlinear Analysis 42 (2000) 871–886.https://doi.org/10.1016/S0362-546X(99)00134-0Google Scholar
  2. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  3. T. Grund: Forthcomming doctoral thesis, Technical University of Chemnitz-Zwickau.Google Scholar
  4. A. Haraux: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29 (1977) 615–631.https://doi.org/10.2969/jmsj/02940615Google Scholar
  5. K. Ito, K. Kunisch: Sensitivity analysis of solutions to optimization problems in Hilbert spaces with applications to optimal control and estimation, J. Diff. Equations 99 (1992) 1–40.https://doi.org/10.1016/0022-0396(92)90133-8Google Scholar
  6. J. Kruskal: Two convex counterexamples: a discontinuous envelope function and a nondifferentiable nearest-point mapping, Proc. Amer. Math. Soc. 23 (1969) 697–703.https://doi.org/10.1090/S0002-9939-1969-0259752-9Google Scholar
  7. A. B. Levy: Sensitivity of solutions to variational inequalities, SIAM J. Contr. Optim. 38 (1999) 50–60.https://doi.org/10.1137/S036301299833985XGoogle Scholar
  8. K. Malanowski: Second order conditions and constraint qualifications in stability and sensitivity analysis of solutions in optimization problems in Hilbert spaces, Appl. Math. Optim. 25 (1992) 51–79.https://doi.org/10.1007/BF01184156Google Scholar
  9. K. Malanowski: Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems, Adv. Math. Sci. Appl. 2 (1993) 397–443.Google Scholar
  10. K. Malanowski: Stability and sensitivity of solutions to nonlinear optimal control problems, Appl. Math. Optim. 32 (1995) 111–141.https://doi.org/10.1007/BF01185227Google Scholar
  11. K. Malanowski: Bouligand differentiability of solutions to parametric optimal control problems, Numer. Funct. Anal. Optim. 22 (2001) 973–990.https://doi.org/10.1081/NFA-100108318Google Scholar
  12. F. Mignot: Contrôle dans les inéquations variationelles, J. Funct. Anal. 22 (1976) 130–185.https://doi.org/10.1016/0022-1236(76)90017-3Google Scholar
  13. J. V. Outrata, Z. Schindler: An augmented Lagrangian method for a class of convex optimal control problems, Problems Contr. Inform. Theory 10 (1981) 67–81.Google Scholar
  14. M. Rao, J. Soklowski: Tangent sets in Banach spaces and applications to variational inequalities, Les Prépublications de l’Institut Élie Cartan 42/00 (2000).Google Scholar
  15. S. M. Robinson: Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity, Math. Program. Study 30 (1987, 97–116.https://doi.org/10.1007/BFb0121154Google Scholar
  16. S. M. Robinson: An implicit-function theorem for a class of nonsmooth functions, Math. Oper. Res. 16 (1991) 292–309.https://doi.org/10.1287/moor.16.2.292Google Scholar
  17. A. Shapiro: On concepts of directional differentiability, J. Math. Anal. Appl. 66 (1990) 477–487.https://doi.org/10.1007/BF00940933Google Scholar
  18. A. Shapiro: Perturbation analysis of optimization problems in Banach spaces, Numer. Funct. Anal. Optim. 13 (1992) 97–116.https://doi.org/10.1080/01630569208816463Google Scholar
  19. A. Shapiro: Directinally nondifferentiable metric projection, J. Optim. Theory Appl. 81 (1994) 203–204.https://doi.org/10.1007/BF02190320Google Scholar
  20. J. Sokolowski: Sensitivity analysis of control constrained optimal control problems for distributed parameter systems, SIAM J. Contr. Optim. 26 (1987) 1542–1556.https://doi.org/10.1137/0325085Google Scholar

Suggested citation

K. Malanowski. “Remarks on Differentiability of Metric Projections onto Cones of Nonnegative Functions.” Journal of Convex Analysis 10 (2003), No. 1, 285–294. https://doi.org/10.68381/jca1017

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press.