Previous |  Up |  Next

Article

Keywords:
continuous convergence; epi-convergence; stochastic programming; stability; estimates
Summary:
Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.
References:
[1] Artstein Z., Wets R. J.-B.: Stability results for stochastic programs and sensors, allowing for discontinuous objective functions. SIAM J. Optim. 4 (1994), 537–550 DOI 10.1137/0804030 | MR 1287815 | Zbl 0830.90111
[2] Bank B., Guddat J., Klatte D., Kummer, B., Tammer K.: Non-Linear Parametric Optimization. Akademie Verlag, Berlin 1982 Zbl 0502.49002
[3] Billingsley P.: Convergence of Probability Measures. Wiley, New York 1968 MR 0233396 | Zbl 0944.60003
[4] Billingsley P.: Probability and Measure. Wiley, New York 1979 MR 0534323 | Zbl 0822.60002
[5] Devroye L., Györfi L.: Nonparametric Density Estimation. The L$_1$-View. Wiley, 1985 MR 0780746 | Zbl 0546.62015
[6] Dupačová J., Wets R. J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic problems. Ann. Statist. 16 (1988), 1517–1549 DOI 10.1214/aos/1176351052 | MR 0964937
[7] Embrechts P., Klüppelberg, C., Mikosch T.: Modelling Extremal Events. Springer–Verlag, Berlin 1997 MR 1458613 | Zbl 0873.62116
[8] Györfi L., Härdle W., Sarda, P., Vieu P.: Nonparametric Curve Estimation from Time Series (Lecture Notes in Statistics 60). Springer–Verlag, Berlin 1989 MR 1027837
[9] Kall P.: On approximations and stability in stochastic programming. In: Parametric Programming and Related Topics (J. Guddat, H. Th. Jongen, B. Kummer, and F. Nožička, eds.), Akademie Verlag, Berlin 1987, pp. 86–103 MR 0909741 | Zbl 0636.90066
[10] Kaniovski Y. M., King A. J., Wets R. J.-B.: Probabilistic bounds (via large deviations) for the solution of stochastic programming problems. Ann. Oper. Res. 56 (1995), 189–208 DOI 10.1007/BF02031707 | MR 1339792
[11] Kaňková V.: A note on estimates in stochastic programming. J. Comput. Appl. Math. 56 (1994), 97–112 DOI 10.1016/0377-0427(94)90381-6 | MR 1338638
[12] Kaňková V., Lachout P.: Convergence rate of empirical estimates in stochastic programming. Informatica 3 (1992), 497–523 MR 1243755 | Zbl 0906.90133
[13] King A. J., Wets R. J.-B.: Epi-consistency of convex stochastic programs. Stochastics and Stochastics Reports 34(1991),83–92 MR 1104423 | Zbl 0733.90049
[14] Korf L. A., Wets R. J.-B.: Random lsc functions: an ergodic theorem. Math. Oper. Research 26 (2001), 421–445 DOI 10.1287/moor.26.2.421.10548 | MR 1895837 | Zbl 1082.90552
[15] Lachout P., Vogel S.: On continuous convergence and epi-convergence of random functions. Part I: Theory and relations. Kybernetika 39 (2003), 1, 75–98 MR 1980125
[16] Langen H.-J.: Convergence of dynamic programming models. Math. Oper. Res. 6 (1981), 493–512 DOI 10.1287/moor.6.4.493 | MR 0703092 | Zbl 0496.90085
[17] Liebscher E.: Strong convergence of sums of $\alpha $-mixing random variables with applications to density estimations. Stoch. Process. Appl. 65 (1996), 69–80 DOI 10.1016/S0304-4149(96)00096-8 | MR 1422880
[18] Lucchetti R., Wets R. J.-B.: Convergence of minima of integral functionals, with applications to optimal control and stochastic optimization. Statist. Decisions 11 (1993), 69–84 MR 1220438 | Zbl 0779.49030
[19] Pflug G. Ch., Ruszczyňski, A., Schultz R.: On the Glivenko–Cantelli problem in stochastic programming: Linear recourse and extensions. Math. Oper. Res. 23 (1998), 204–220 DOI 10.1287/moor.23.1.204 | MR 1606474 | Zbl 0977.90031
[20] Rachev S. T.: The Monge–Kantorovich mass transference problem and its stochastic applications. Theory Probab. Appl. 29 (1984), 647–676 MR 0773434
[21] Robinson S. M.: Local epi-continuity and local optimization. Math. Programming 37 (1987), 208–222 DOI 10.1007/BF02591695 | MR 0883021 | Zbl 0623.90078
[22] Robinson S. M., Wets R. J.-B.: Stability in two-stage stochastic programming. SIAM J. Control Optim. 25 (1987), 1409–1416 DOI 10.1137/0325077 | MR 0912447 | Zbl 0639.90074
[23] Römisch W., Schultz R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18 (1993), 590–609 DOI 10.1287/moor.18.3.590 | MR 1250562 | Zbl 0797.90070
[24] Silverman B. W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London 1986 MR 0848134 | Zbl 0617.62042
[25] Vogel S.: Stochastische Stabilitätskonzepte. Habilitation, Ilmenau Technical University, 1991
[26] Vogel S.: On stability in multiobjective programming – A stochastic approach. Math. Programming 56 (1992), 91–119 DOI 10.1007/BF01580896 | MR 1175561 | Zbl 0770.90061
[27] Vogel S.: A stochastic approach to stability in stochastic programming. J. Comput. Appl. Mathematics, Series Appl. Analysis and Stochastics 56 (1994), 65–96 MR 1338637 | Zbl 0824.90107
[28] Vogel S.: On stability in stochastic programming – Sufficient conditions for continuous convergence and epi-convergence. Preprint of Ilmenau Technical University, 1994 MR 1338637
[29] Wang J.: Continuity of feasible solution sets of probabilistic constrained programs. J. Optim. Theory Appl. 63 (1989), 79–89 DOI 10.1007/BF00940733 | MR 1022368
[30] Wets R. J.-B.: Stochastic programming. In: Handbooks in Operations Research and Management Science, Vol. 1, Optimization (G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, eds.), North Holland, Amsterdam 1989, pp. 573–629 MR 1105107 | Zbl 0752.90052
[31] Zervos M.: On the epiconvergence of stochastic optimization problems. Math. Oper. Res. 24 (1999), 2, 495–508 DOI 10.1287/moor.24.2.495 | MR 1853885 | Zbl 1074.90552
Partner of
EuDML logo