Quantitative Stability in Stochastic Programming: The Method of Probability Metrics

References

  • Artstein Z. Sensitivity with respect to the underlying information in stochastic programs. J. Comput. Appl. Math. (1994) 56:127–136CrossrefGoogle Scholar
  • Artstein Z., Wets R. J.-B. Stability results for stochastic programs and sensors, allowing for discontinuous objective functions. SIAM J. Optim. (1994) 4:537–550CrossrefGoogle Scholar
  • Artstein Z., Wets R. J.-B. Consistency of minimizers and the SLLN for stochastic programs. J. Convex Anal. (1995) 2:1–17Google Scholar
  • Attouch H., Wets R. J.-B. Quantitative stability of variational systems II. A framework for nonlinear conditioning. SIAM J. Optim. (1993) 3:359–381CrossrefGoogle Scholar
  • Bank B., Guddat J., Klatte D., Kummer B., Tammer K.Non-Linear Parametric Optimization (1982) (Akademie-Verlag, Berlin) CrossrefGoogle Scholar
  • Blair C. E., Jeroslow R. G. The value function of a mixed integer program. Discrete Math. (1977) 19:121–138CrossrefGoogle Scholar
  • Bonnans J. F., Shapiro A.Perturbation Analysis of Optimization Problems (2000) (Springer-Verlag, New York) CrossrefGoogle Scholar
  • Cheng B. N., Rachev S. T. Multivariate stable commodities in the futures market. Math. Finance (1995) 5:133–153CrossrefGoogle Scholar
  • Dentcheva D., Römisch W. Differential stability of two-stage stochastic programs. SIAM J. Optim. (2000) 11:87–112CrossrefGoogle Scholar
  • Dudley R. M., Hennequin P. L. A course on empirical processes. École d'Été de Probabilities de Saint-Flour XII—1982 (1984) (Springer-Verlag, Berlin) 2–142Lecture Notes in Mathematics 1097CrossrefGoogle Scholar
  • Dudley R. M.Real Analysis and Probability (1989) (Wadsworth & Brooks/Cole, Pacific Grove) Google Scholar
  • Dupačová J. Stability and sensitivity analysis for stochastic programming. Ann. Oper. Res. (1990) 27:115–142CrossrefGoogle Scholar
  • Dupačová J. Applications of stochastic programming under incomplete information. J. Comput. Appl. Math. (1994) 56:113–125CrossrefGoogle Scholar
  • Dupačová J., Wets R. J.-B. Asymptotic behaviour of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. (1988) 16:1517–1549CrossrefGoogle Scholar
  • Fortet R., Mourier E. Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Norm. Sup. (1953) 70:266–285Google Scholar
  • Gröwe N. Estimated stochastic programs with chance constraints. Eur. J. Oper. Res. (1997) 101:285–305CrossrefGoogle Scholar
  • Henrion R., Römisch W. Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Math. Programming (1999) 84:55–88CrossrefGoogle Scholar
  • Kall P.Stochastic Linear Programming (1976) (Springer-Verlag, Berlin) CrossrefGoogle Scholar
  • Kall P., Guddat J., Jongen H. Th., Kummer B., Nožička F. On approximations and stability in stochastic programming. Parametric Optimization and Related Topics (1987) (Akademie-Verlag, Berlin) 387–407Google Scholar
  • Kaniovski Y. M., King A. J., Wets R. J.-B. Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann. Oper. Res. (1995) 56:189–208CrossrefGoogle Scholar
  • Kaňková V. A note on estimates in stochastic programming. J. Comput. Appl. Math. (1994) 56:97–112CrossrefGoogle Scholar
  • King A. J., Rockafellar R. T. Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. (1993) 18:148–162LinkGoogle Scholar
  • King A. J., Wets R. J.-B. Epi-consistency of convex stochastic programs. Stochastics and Stochastics Rep. (1991) 34:83–92CrossrefGoogle Scholar
  • Klatte D., Lommatzsch K. A note on quantitative stability results in nonlinear optimization. Proceedings 19. Jahrestagung Mathematische Optimierung (1987) (Humboldt-Universität, Berlin) 77–86Sektion Mathematik, Semi-narbericht Nr. 90Google Scholar
  • Klatte D. On quantitative stability for non-isolated minima. Control Cybernetics (1994) 23:183–200Google Scholar
  • Mandelbrot B. B. New methods in statistical economics. J. Political Econom. (1963) 71:421–440CrossrefGoogle Scholar
  • Mittnik S., Rachev S. T. Modeling asset returns with alternative stable distributions. Econometric Rev. (1993) 12:261–330CrossrefGoogle Scholar
  • Mordukhovich B. S. Lipschitzian stability of constraint systems and generalized equations. Nonlinear Anal., Theory, Methods Appl. (1994) 22:173–206CrossrefGoogle Scholar
  • Nožička F., Guddat J., Hollatz H., Hollatz B.Theorie der linearen parametrischen Optimierung (1974) (Akademie-Verlag, Berlin) Google Scholar
  • Pflug G., Ruszczyński A., Schultz R. On the Glivenko-Cantelli problem in stochastic programming: Linear recourse and extensions. Math. Oper. Res. (1998) 23:204–220LinkGoogle Scholar
  • Pflug G. Stochastic programs and statistical data. Ann. Oper. Res. (1999) 85:59–78CrossrefGoogle Scholar
  • Rachev S. T.Probability Metrics and the Stability of Stochastic Models (1991) (Wiley, Chichester, U.K.) Google Scholar
  • Rachev S. T., Mittnik S.Stable Paretian Models in Finance (2000) (Wiley, Chichester, U.K.) Google Scholar
  • Rachev S. T., Rüschendorf L.Mass Transportation Problems, Vol. I: Theory (1998) (Springer, New York) Google Scholar
  • Robinson S. M. Local epi-continuity and local optimization. Math. Programming (1987) 37:208–223CrossrefGoogle Scholar
  • Robinson S. M. Analysis of sample-path optimization. Math. Oper. Res. (1996) 21:513–528LinkGoogle Scholar
  • Robinson S. M., Wets R. J.-B. Stability in two-stage stochastic programming. SIAM J. Control Optim. (1987) 25:1409–1416CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B.Variational Analysis (1997) (Springer, Berlin) Google Scholar
  • Römisch W., Schultz R. Distribution sensitivity in stochastic programming. Math. Programming (1991a) 50:197–226CrossrefGoogle Scholar
  • Römisch W., Schultz R. Stability analysis for stochastic programs. Ann. Oper. Res. (1991b) 30:241–266CrossrefGoogle Scholar
  • Römisch W., Schultz R. Lipschitz stability for stochastic programs with complete recourse. SIAM J. Optim. (1996) 6:531–547CrossrefGoogle Scholar
  • Römisch W., Wakolbinger A., Guddat J., Jongen H. T., Kummer B., Nožička F. Obtaining convergence rates for approximations in stochastic programming. Parametric Optimization and Related Topics (1987) (Akademie-Verlag, Berlin) 327–343Google Scholar
  • Samorodnitsky G., Taqqu M. S.Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (1994) (Chapman & Hall, New York) Google Scholar
  • Schultz R. Strong convexity in stochastic programs with complete recourse. J. Comput. Appl. Math. (1994) 56:3–22CrossrefGoogle Scholar
  • Schultz R. On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Programming (1995) 70:73–89CrossrefGoogle Scholar
  • Schultz R. Rates of convergence in stochastic programs with complete integer recourse. SIAM J. Optim. (1996) 6:1138–1152CrossrefGoogle Scholar
  • Schultz R. Some aspects of stability in stochastic programming. Ann. Oper. Res. (2000) 100:55–84CrossrefGoogle Scholar
  • Shapiro A. Quantitative stability in stochastic programming. Math. Programming (1994) 67:99–108CrossrefGoogle Scholar
  • Shapiro A. Simulation-based optimization—Concergence analysis and statistical inference. Comm. Statist.—Stochastic Models (1996) 12:425–454CrossrefGoogle Scholar
  • Shapiro A., Homem-de-Mello T. On rate of convergence of optimal solutions of Monte Carlo approximations of stochastic programs. SIAM J. Optim. (2000) 11:70–86CrossrefGoogle Scholar
  • Talagrand M. Sharper bounds for Gaussian and empirical processes. Ann. Probab. (1994) 22:28–76CrossrefGoogle Scholar
  • Talagrand M. The Glivenko-Cantelli problem, ten years later. J. Theoret. Probab. (1996) 9:371–384CrossrefGoogle Scholar
  • van der Vaart A. W., Wellner J. A.Weak Convergence and Empirical Processes (1996) (Springer, New York) CrossrefGoogle Scholar
  • Vogel S. A stochastic approach to stability in stochastic programming. J. Comput. Appl. Math. (1994) 56:65–96CrossrefGoogle Scholar
  • Walkup D. W., Wets R. J.-B. Lifting projections of convex polyhedra. Pacific J. Math. (1969) 28:465–475CrossrefGoogle Scholar
  • Wets R. J.-B. Stochastic programs with fixed recourse: The equivalent deterministic program. SIAM Rev. (1994) 16:309–339CrossrefGoogle Scholar
  • Wets R. J.-B., Nemhauser G. L., Rinnooy Kan A. H. G., Todd M. J. Stochastic programming. Handbooks in Operations Research and Management Science, Vol. 1, Optimization (1989) (North-Holland, Amsterdam) 573–629Google Scholar
  • Ziemba W. T., Hammer P. L., Zoutendijk G. Choosing investment portfolios when the returns have stable distributions. Mathematical Programming in Theory and Practice (1974) (North-Holland)443–482Google Scholar
  • Zolotarev V. M. Probability metrics. Theory Probab. Appl. (1983) 28:278–302CrossrefGoogle Scholar
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