Perturbation Analysis of Chance-constrained Programs under Variation of all Constraint Data

  • René Henrion
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 532)

Abstract

A fairly general shape of chance constraint programs is
$$ (P) \min \{ g(x)|x \in X, \mu (H(x)) \ge p\} $$
where g: ℝ m → ℝ is a continuous objective function, \( X \subseteq \mathbb{R}^m \) is a closed subset of deterministic constraints, and the inequality defines a probabilistic constraint with H : ℝ m ⇉ ℝ s being a multifunction with closed graph, µ is a probability measure on s and p ∈ (0, 1) is some probability level. In the simplest case of linear chance constraints, g is linear, X is a polyhedron and H(x) = {z ∈ ℝ s |Axz} , where A is a matrix of order (s, m) and the inequality sign has to be understood component-wise.

Keywords

Rene 

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References

  1. 1.
    Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K. (1982): Non-Linear Parametric Optimization. Akademie-Verlag, BerlinGoogle Scholar
  2. 2.
    Henrion, R., Römisch, W. (1999): Metric Regularity and Quantitative Stability in Stochastic Programs with Probabilistic Constraints. Math. Programming, 84, 55–88Google Scholar
  3. 3.
    Henrion, R. (2000): Qualitative Stability of Convex Programs with Probabilistic Constraints. In: Nguyen, V.H. et al. (ed): Optimization. Lecture Notes in Economics and Mathematical Systems, Vol. 481, pp. 164–180. Springer, BerlinGoogle Scholar
  4. 4.
    Klatte, D. (1985): On the Stability of Local and Global Optimal Solutions in Parametric Problems of Nonlinear Programming. Seminarbericht 75, Humboldt-University BerlinGoogle Scholar
  5. 5.
    Klatte, D. (1994): On Quantitative Stability for Non-isolated Minima. Contr. Cybernetics, 23, 183–200Google Scholar
  6. 6.
    Lucchetti, R., Salinetti, G., Wets, R.J.-B. (1994): Uniform Convergence of Probability Measures: Topological Criteria. J. Multiv. Anal., 51, 252–264CrossRefGoogle Scholar
  7. 7.
    Mordukhovich, B.S. (1994): Generalized Differential Calculus for Nonsmooth and Setvalued Mappings. J. Math. Anal. Appl., 183, 250–288CrossRefGoogle Scholar
  8. 8.
    Prékopa, A. (1995): Stochastic Programming. Kluwer, DordrechtGoogle Scholar
  9. 9.
    Rockafellar, R.T., Wets, R.J.-B. (1997): Variational Analysis. Springer, New YorkGoogle Scholar
  10. 10.
    Römisch, W., Schultz, R. (1991): Stability Analysis for Stochastic Programs. Ann. Oper. Res., 30, 241–266CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • René Henrion
    • 1
  1. 1.Weierstrass InstituteBerlinGermany

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