Robustness Analysis of Stochastic Programs with Joint Probabilistic Constraints

  • Jitka Dupačová
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 391)

Abstract

Due to their frequently observed lack of convexity and/or smoothness, stochastic programs with joint probabilistic constraints have been considered as a hard type of constrained optimization problems, which are rather demanding both from the computational and robustness point of view. Dependence of the set of solutions on the probability distribution rules out the straightforward construction of the convexity-based global contamination bounds for the optimal value; at least local results for probabilistic programs of a special structure will be derived. Several alternative approaches to output analysis will be mentioned.

Keywords

Joint probabilistic constraints contamination technique output analysis 

References

  1. 1.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)MATHGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Nondegeneracy and quantitative stability of parametrized optimization problems with multiple solutions. SIAM J. Optim. 8, 940–946 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Branda, M., Dupačová, J.: Approximation and contamination bounds for probabilistic programs. Ann. Oper. Res. 193, 3–19 (2012)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dupačová, J.: Stability in stochastic programming with recourse – contaminated distributions. Math. Program. Study 27, 133–144 (1986)MATHCrossRefGoogle Scholar
  5. 5.
    Dupačová, J.: Stability in stochastic programming – probabilistic constraints. In: Arkin, V.I., Shiraev, A., Wets, R. (eds.) Stochastic Optimization. LNCIS, vol. 81, pp. 314–325. Springer, Berlin (1986)CrossRefGoogle Scholar
  6. 6.
    Dupačová, J.: Stochastic programming with incomplete information: A survey of results on postoptimization and sensitivity analysis. Optimization 18, 507–532 (1987)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dupačová, J.: Stability and sensitivity analysis in stochastic programming. Ann. Oper. Res. 27, 115–142 (1990)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dupačová, J.: Scenario based stochastic programs: Resistance with respect to sample. Ann. Oper. Res. 64, 21–38 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dupačová, J.: Reflections on robust optimization. In: Marti, K., Kall, P. (eds.) Stochastic Programming Methods and Technical Applications. LNEMS, vol. 437, pp. 111–127. Springer, Berlin (1998)CrossRefGoogle Scholar
  10. 10.
    Dupačová, J.: Uncertainties in minimax stochastic programs. Optimization 60, 1235–1250 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dupačová, J., Kopa, M.: Robustness in stochastic programs with risk constraints. Ann. Oper. Res. 200, 55–77 (2012), doi:10.1007/s10479-010-0824-9MATHCrossRefGoogle Scholar
  12. 12.
    Dupačová, J., Polívka, J.: Stress testing for VaR and CVaR. Quantitiative Finance 7, 411–421 (2007)MATHCrossRefGoogle Scholar
  13. 13.
    Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)MATHGoogle Scholar
  14. 14.
    Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Study 19, 101–119 (1982)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Henrion, R.: Perturbation analysis of chance-constrained programs under variation of all constraint data. In: Marti, K., et al. (eds.) Dynamic Stochastic Optimization. LNEMS, vol. 532, pp. 257–274. Springer, Berlin (2004)CrossRefGoogle Scholar
  16. 16.
    Henrion, R., Römisch, W.: Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints. Math. Program. 100, 589–611 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kyparisis, J., Fiacco, A.: Generalized convexity and concavity of the optimal value function in nonlinear programming. Math. Program. 39, 285–304 (1987)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Pagoncelli, B.K., Ahmed, S., Shapiro, A.: Sample average approximation method for chance constrained programming: Theory and applications. J. Optim. Theory Appl. 142, 399–416 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pflug, G., Wozabal, D.: Ambiguity in portfolio selection. Quant. Fin. 7, 435–442 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32, 301–316 (1971)MathSciNetMATHGoogle Scholar
  21. 21.
    Prékopa, A.: Stochastic Programming. Kluwer Acad. Publ., Dordrecht (1995)Google Scholar
  22. 22.
    Prékopa, A.: Probabilistic Programming. In: [27], ch. 5, pp. 267–351Google Scholar
  23. 23.
    Robinson, S.M.: Local structure of feasible sets in nonlinear programming, part II: Nondegeneracy. Math. Program. Study 22, 217–230 (1984)MATHCrossRefGoogle Scholar
  24. 24.
    Robinson, S.M.: Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity. Math. Program. Study 30, 45–66 (1987)MATHCrossRefGoogle Scholar
  25. 25.
    Römisch, W.: Stability of stochastic programming problems. In: [27], ch. 8, pp. 483–554Google Scholar
  26. 26.
    Römisch, W., Schultz, R.: Stability analysis for stochastic programs. Ann. Oper. Res. 30, 241–266 (1991)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming. Handbooks in OR & MS, vol. 10. Elsevier, Amsterdam (2003)MATHGoogle Scholar
  28. 28.
    Shapiro, A.: Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM J. Control and Optimization 26, 628–645 (1988)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Shapiro, A.: On differential stability in stochastic programming. Math. Program. 47, 107–116 (1990)MATHCrossRefGoogle Scholar
  30. 30.
    Shapiro, A.: Monte Carlo sampling methods. In: [27], ch. 6, pp. 353–425Google Scholar
  31. 31.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming. SIAM and MPS, Philadelphia (2009)MATHCrossRefGoogle Scholar
  32. 32.
    van Ackooij, W., Henrion, R., Möller, A., Zorgati, R.: On joint probabilistic constraints with Gaussian coefficient matrix. Operations Research Letters 39, 99–102 (2011)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program., Ser. A (published online November 10, 2011)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2013

Authors and Affiliations

  • Jitka Dupačová
    • 1
  1. 1.Faculty of Mathematics and Physics, Department of Probability and Mathematical StatisticsCharles University in PraguePragueCzech Republic

Personalised recommendations