Set-Valued and Variational Analysis

, Volume 26, Issue 3, pp 449–467 | Cite as

The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces

  • Patrick MehlitzEmail author
  • Gerd Wachsmuth


We consider subsets of Lebesgue spaces which are defined by pointwise constraints. We provide formulas for corresponding variational objects (tangent and normal cones). Our main result shows that the limiting normal cone is always dense in the Clarke normal cone and contains the convex hull of the pointwise limiting normal cone. A crucial assumption for this result is that the underlying measure is non-atomic, and this is satisfied in many important applications (Lebesgue measure on subsets of \(\mathbb {R}^{d}\) or the surface measure on hypersurfaces in \(\mathbb {R}^{d}\)). Finally, we apply our findings to an optimization problem with complementarity constraints in Lebesgue spaces.


Decomposable set Lebesgue spaces Limiting normal cone Mathematical program with complementarity constraint Measurability 

Mathematics Subject Classification 2010

49J53 90C30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubin, J.-P., Frankowska, H.: Set-valued analysis. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2009. ISBN 978-0-8176-4847-3. Reprint of the 1990 editionGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ding, C., Sun, D., Ye, J.J.: First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints. Math. Program. 147 (1-2), 539–579 (2014). doi: 10.1007/s10107-013-0735-z MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Flegel, M., Kanzow, C.: A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints. In: Dempe, S, Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, volume 2 of Springer Optimization and Its Applications, pp 111–122. Springer, New York (2006),  10.1007/0-387-34221-4_6
  5. 5.
    Geremew, W., Mordukhovich, B.S., Nam, N.M.: Coderivative calculus and metric regularity for constraint and variational systems. Nonlinear Anal. Theory Methods Appl. 70(1), 529–552 (2009). doi: 10.1016/ MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hiai, F., Umegaki, H. : Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 7(1), 149–182 (1977). doi: 10.1016/0047-259X(77)90037-9 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mehlitz, P., Wachsmuth, G.: Weak and strong stationarity in generalized bilevel programming and bilevel optimal control. Optimization 65(5), 907–935 (2016). doi: 10.1080/02331934.2015.1122007 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Springer-Verlag, Berlin (2006)Google Scholar
  9. 9.
    Mordukhovich, B.S., Sagara, N.: Subdifferentials of nonconvex integral functionals in Banach spaces with applications to stochastic dynamic programming. to appear in J. Convex Anal. (2016). arXiv:1508.02239
  10. 10.
    Olech, C.: The Lyapunov Theorem: Its extensions and applications, volume 1446 of Lecture Notes in Mathematics, pp 84–103. Springer, Berlin (1990)Google Scholar
  11. 11.
    Papageorgiou, N.S., Kyritsi-Yiallourou, S.T.: Handbook of applied analysis, volume 19 of Advances in Mechanics and Mathematics. Springer, New York (2009). ISBN 978-0-387-78906-4. doi: 10.1007/b120946
  12. 12.
    Rockafellar, R. T.: Integrals which are convex functionals. Pacific J. Math. 24 (3), 525–539 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wachsmuth, G.: Mathematical Programs with Complementarity Constraints in Banach Spaces. J. Optim. Theory Appl. 166, 480–507 (2015). doi: 10.1007/s10957-014-0695-3 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wachsmuth, G.: Strong stationarity for optimization problems with complementarity constraints in absence of polyhedricity. Set-Valued and Variational Analysis (2016). doi: 10.1007/s11228-016-0370-y zbMATHGoogle Scholar
  15. 15.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005). doi: 10.1016/j.jmaa.2004.10.032 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceTechnische Universität Bergakademie FreibergFreibergGermany
  2. 2.Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations)Technische Universität ChemnitzChemnitzGermany

Personalised recommendations