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.
Chapter PDF
Similar content being viewed by others
References
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Bonnans, J.F., Shapiro, A.: Nondegeneracy and quantitative stability of parametrized optimization problems with multiple solutions. SIAM J. Optim. 8, 940–946 (1998)
Branda, M., Dupačová, J.: Approximation and contamination bounds for probabilistic programs. Ann. Oper. Res. 193, 3–19 (2012)
Dupačová, J.: Stability in stochastic programming with recourse – contaminated distributions. Math. Program. Study 27, 133–144 (1986)
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)
Dupačová, J.: Stochastic programming with incomplete information: A survey of results on postoptimization and sensitivity analysis. Optimization 18, 507–532 (1987)
Dupačová, J.: Stability and sensitivity analysis in stochastic programming. Ann. Oper. Res. 27, 115–142 (1990)
Dupačová, J.: Scenario based stochastic programs: Resistance with respect to sample. Ann. Oper. Res. 64, 21–38 (1996)
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)
Dupačová, J.: Uncertainties in minimax stochastic programs. Optimization 60, 1235–1250 (2011)
Dupačová, J., Kopa, M.: Robustness in stochastic programs with risk constraints. Ann. Oper. Res. 200, 55–77 (2012), doi:10.1007/s10479-010-0824-9
Dupačová, J., Polívka, J.: Stress testing for VaR and CVaR. Quantitiative Finance 7, 411–421 (2007)
Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)
Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Study 19, 101–119 (1982)
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)
Henrion, R., Römisch, W.: Hölder and Lipschitz stability of solution sets in programs with probabilistic constraints. Math. Program. 100, 589–611 (2004)
Kyparisis, J., Fiacco, A.: Generalized convexity and concavity of the optimal value function in nonlinear programming. Math. Program. 39, 285–304 (1987)
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)
Pflug, G., Wozabal, D.: Ambiguity in portfolio selection. Quant. Fin. 7, 435–442 (2007)
Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. (Szeged) 32, 301–316 (1971)
Prékopa, A.: Stochastic Programming. Kluwer Acad. Publ., Dordrecht (1995)
Prékopa, A.: Probabilistic Programming. In: [27], ch. 5, pp. 267–351
Robinson, S.M.: Local structure of feasible sets in nonlinear programming, part II: Nondegeneracy. Math. Program. Study 22, 217–230 (1984)
Robinson, S.M.: Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity. Math. Program. Study 30, 45–66 (1987)
Römisch, W.: Stability of stochastic programming problems. In: [27], ch. 8, pp. 483–554
Römisch, W., Schultz, R.: Stability analysis for stochastic programs. Ann. Oper. Res. 30, 241–266 (1991)
Ruszczyński, A., Shapiro, A. (eds.): Stochastic Programming. Handbooks in OR & MS, vol. 10. Elsevier, Amsterdam (2003)
Shapiro, A.: Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM J. Control and Optimization 26, 628–645 (1988)
Shapiro, A.: On differential stability in stochastic programming. Math. Program. 47, 107–116 (1990)
Shapiro, A.: Monte Carlo sampling methods. In: [27], ch. 6, pp. 353–425
Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming. SIAM and MPS, Philadelphia (2009)
van Ackooij, W., Henrion, R., Möller, A., Zorgati, R.: On joint probabilistic constraints with Gaussian coefficient matrix. Operations Research Letters 39, 99–102 (2011)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 IFIP International Federation for Information Processing
About this paper
Cite this paper
Dupačová, J. (2013). Robustness Analysis of Stochastic Programs with Joint Probabilistic Constraints. In: Hömberg, D., Tröltzsch, F. (eds) System Modeling and Optimization. CSMO 2011. IFIP Advances in Information and Communication Technology, vol 391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36062-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-36062-6_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36061-9
Online ISBN: 978-3-642-36062-6
eBook Packages: Computer ScienceComputer Science (R0)