Set-Valued and Variational Analysis

, Volume 24, Issue 2, pp 207–229 | Cite as

Normally Admissible Stratifications and Calculation of Normal Cones to a Finite Union of Polyhedral Sets

  • Lukáš Adam
  • Michal Červinka
  • Miroslav Pištěk


This paper considers computation of Fréchet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to those already known in the literature. Finally, we apply this approach to a class of time dependent problems and provide an illustration on a special structure arising in delamination modeling.


Union of polyhedral sets Tangent cone Fréchet normal cone Limiting normal cone Normally admissible stratification Time dependent problems Delamination model 

Mathematics Subject Classifications (2010)

90C31 90C33 65K10 34D20 37C75 


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  1. 1.
    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems, vol. 35. Springer (2008)Google Scholar
  2. 2.
    Achtziger, W., Kanzow, C.: Mathematical programs with vanishing constraints: Optimality conditions and constraint qualifications. Math. Program. 114, 69–99 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adam, L., Outrata, J., Roubíček, T.: Identification of some rate-independent systems with illustration on cohesive contacts at small strains. SubmittedGoogle Scholar
  4. 4.
    Bounkhel, M.: Regularity Concepts in Nonsmooth Analysis: Theory and Applications. Springer (2012)Google Scholar
  5. 5.
    Červinka, M., Outrata, J., Pištěk, M.: On stability of M-stationary points in MPCCs. Set-Valued Variational Anal. 22(3), 575–595 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dempe, S.: Foundations of Bilevel Programming. Springer (2002)Google Scholar
  7. 7.
    Dontchev, A., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6(4), 1087–1105 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dontchev, A., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer (2009)Google Scholar
  9. 9.
    Ewald, G.: Combinatorial Convexity and Algebraic Geometry. Springer (1996)Google Scholar
  10. 10.
    Flegel, M.L., Kanzow, C., Outrata, J.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15(2), 139–162 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goresky, M., MacPherson, R.: Stratified Morse Theory. Springer (1988)Google Scholar
  12. 12.
    Henrion, R., Outrata, J.: On calculating the normal cone to a finite union of convex polyhedra. Optim. A J. Math. Program. Oper. Res. 57(1), 57–78 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Henrion, R., Römisch, W.: On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling. Appl. Math. 52(6), 473–494 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ioffe, A.D., Outrata, J.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16(2–3), 199–227 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lu, S., Budhiraja, A.: Confidence regions for stochastic variational inequalities. Math. Oper. Res. 38(3), 545–568 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lu, S., Robinson, S.: Normal fans of polyhedral convex sets. Set-Valued Anal. 16(2), 281–305 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press (1996)Google Scholar
  18. 18.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer (2006)Google Scholar
  19. 19.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation II. Springer (2006)Google Scholar
  20. 20.
    Mordukhovich, B.S., Outrata, J.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18(2), 389–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Boston (1998)CrossRefzbMATHGoogle Scholar
  22. 22.
    Pflaum, M.: Analytic and Geometric Study of Stratified Spaces. Springer (2001)Google Scholar
  23. 23.
    Robinson, S.: Some continuity properties of polyhedral multifunctions. Math. Programm. Study 14, 206–214 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
  25. 25.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer (1998)Google Scholar
  26. 26.
    Roubíček, T., Panagiotopoulos, C.G., Mantic, V.: Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity. ZAMM J. Appl. Math. Mech. 93(10–11), 823–840 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Lukáš Adam
    • 1
  • Michal Červinka
    • 1
    • 2
  • Miroslav Pištěk
    • 1
  1. 1.Institute of Information Theory and AutomationCzech Academy of SciencesPrague 8Czech Republic
  2. 2.Faculty of Social SciencesCharles University in PraguePrague 1Czech Republic

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