Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 821–841 | Cite as

Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization

  • M. H. Farshbaf-Shaker
  • R. HenrionEmail author
  • D. Hömberg


Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.


Chance constraints Probabilistic constraints PDE constrained optimization 

Mathematics Subject Classifications (2010)

90C15 49J20 


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The authors express their gratitude to two anonymous referees whose very careful reading and critical comments led to a substantially improved presentation of this paper.


  1. 1.
    Van Ackooij, W., de Oliveira, W.: Convexity and optimization with copulae structured probabilistic constraints. Optimization 65, 1349–1376 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Van Ackooij, W., Henrion, R.: (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution. SIAM/ASA J. Uncertain. Quantif. 5, 63–87 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Cambridge (2003)zbMATHGoogle Scholar
  4. 4.
    Allaire, G., Dapogny, C.: A deterministic approximation method in shape optimization under random uncertainties. SMAI J. Comput. Math. 1, 83–143 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Van Ackooij, W., Henrion, R.: Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM J. Optim. 24, 1864–1889 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory. Springer, Berlin (1984)CrossRefGoogle Scholar
  7. 7.
    Babuska, I., Nobile, F., Tempone, R.: A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Babuska, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Engrg. 191, 4093–4122 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Babuska, I., Liu, K., Tempone, R.: Solving stochastic partial differential equations based on the experimental data. Math. Models Methods Appl. Sci. 13, 415–444 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borzi, A.: Multigrid and sparse-grid schemes for elliptic control problems with random coefficients. Comput. Vis. Sci. 13, 153–160 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics (2011)Google Scholar
  12. 12.
    Chang, K.-C.: Methods in Nonlinear Analysis. Springer, Berlin (2005)zbMATHGoogle Scholar
  13. 13.
    Conti, S., Held, H., Pach, M., Rumpf, M., Schultz, R.: Risk averse shape optimization. SIAM J. Control Optim. 49, 927–947 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Druet, P.E., Klein, O., Sprekels, J., Tröltzsch, F., Yousept, I.: Optimal control of 3D state-constrained induction heating problems with nonlocal radiation effects. SIAM J. Control Optim. 49, 1707–1736 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics Volume: 19 (2010)Google Scholar
  16. 16.
    Geihe, B., Lenz, M., Rumpf, M., Schultz, R.: Risk averse elastic shape optimization with parametrized fine scale geometry. Math. Program. 141, 383–403 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ghanem, R., Spanos, P.: Stochastic finite element expansion for random media. J. Eng. Mech. 115, 1035–1053 (1989)CrossRefGoogle Scholar
  18. 18.
    Gunzburger, M.D., Lee, H., Lee, J.: Error Estimates of Stochastic Optimal Neumann Boundary Control Problems. SIAM J. Numer. Anal. 49, 1532–1552 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Henrion, R.: Qualitative stability of convex programs with probabilistic constraints. In: Nguyen, V.H., Strodiot, J.-J., Tossings, P. (eds.) Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 481, pp. 164–180. Springer, Berlin (2000)Google Scholar
  20. 20.
    Henrion, R., Strugarek, C.: Convexity of Chance Constraints with Independent Random Variables. Comput. Optim. Appl. 41, 263–276 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Henrion, R., Möller, A.: A gradient formula for linear chance constraints under Gaussian distribution. Math. Oper. Res. 37, 475–488 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kouri, D.P., Heinkenschloss, M., Ridzal, D., van Bloemen Waanders, B.G.: A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM J. Sci. Comput. 35, A1847—A1879 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kouri, D.P., Surowiec, T.M.: Risk averse PDE-constrained optimization using the conditional value-at-risk. SIAM J. Optim. 26, 365–396 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kunoth, A., Schwab, C.: Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs. SIAM J. Control Optim. 51, 2442–2471 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lord, G.J., Powell, C.E., Sahrdlow, T.: An introduction to computational stochastic PDEs. Cambridge University Press, New York (2014)CrossRefGoogle Scholar
  26. 26.
    Marti, K.: Differentiation of probability functions: the transformation method. Comput. Math. Appl. 30, 361–382 (1995)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Prėkopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34, 335–343 (1973)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Prėkopa, A.: Programming under probabilistic constraints with a random technology matrix. Optimization 5, 109–116 (1974)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Prėkopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)CrossRefGoogle Scholar
  30. 30.
    Römisch, W., Schultz, R.: Stability analysis for stochastic programs. Ann. Oper. Res. 30, 241–266 (1991)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, MPS-SIAM series on optimization 9 (2009)Google Scholar
  32. 32.
    Sichau, A., Ulbrich, S.: A second order approximation technique for robust shape optimization. Appl. Mech. Mater. 104, 13–22 (2012)CrossRefGoogle Scholar
  33. 33.
    Uryasev, S.: Derivatives of probability functions and some applications. Ann. Oper. Res. 56, 287–311 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • M. H. Farshbaf-Shaker
    • 1
  • R. Henrion
    • 1
    Email author
  • D. Hömberg
    • 1
    • 2
  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Department of Mathematical SciencesNTNUTrondheimNorway

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