On continuous convergence and epi-convergence of random functions. Part I: Theory and relations

Silvia Vogel; Petr Lachout

Kybernetika (2003)

  • Volume: 39, Issue: 1, page [75]-98
  • ISSN: 0023-5954

Abstract

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Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization.

How to cite

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Vogel, Silvia, and Lachout, Petr. "On continuous convergence and epi-convergence of random functions. Part I: Theory and relations." Kybernetika 39.1 (2003): [75]-98. <http://eudml.org/doc/33623>.

@article{Vogel2003,
abstract = {Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization.},
author = {Vogel, Silvia, Lachout, Petr},
journal = {Kybernetika},
keywords = {continuous convergence; epi-convergence; stochastic programming; stability; continuous convergence; epi-convergence; stochastic programming; stability},
language = {eng},
number = {1},
pages = {[75]-98},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On continuous convergence and epi-convergence of random functions. Part I: Theory and relations},
url = {http://eudml.org/doc/33623},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Vogel, Silvia
AU - Lachout, Petr
TI - On continuous convergence and epi-convergence of random functions. Part I: Theory and relations
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 1
SP - [75]
EP - 98
AB - Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization.
LA - eng
KW - continuous convergence; epi-convergence; stochastic programming; stability; continuous convergence; epi-convergence; stochastic programming; stability
UR - http://eudml.org/doc/33623
ER -

References

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  1. Attouch H., Variational Convergence for Functions and Operators, Pitman, London 1984 Zbl0561.49012MR0773850
  2. Artstein Z., Wets R. J.-B., 10.1137/0804030, SIAM J. Optim. 4 (1994), 537-550 (1994) Zbl0830.90111MR1287815DOI10.1137/0804030
  3. Artstein Z., Wets R. J.-B., Consistency of minimizers and the SLLN for stochastic programs, J. Convex Analysis 2 (1995), 1–17 (1995) Zbl0837.90093MR1363357
  4. Bank B., Guddat J., Klatte D., Kummer, B., Tammer K., Non-Linear Parametric Optimization, Akademie Verlag, Berlin 1982 Zbl0502.49002
  5. Beer G., Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht 1993 Zbl0792.54008MR1269778
  6. Cohn D. L., Measure Theory, Birkhäuser, Boston 1980 Zbl0860.28001MR0578344
  7. Dupačová J., 10.1007/BF02055193, Ann. Oper. Res. 27 (1990), 115–142 (1990) MR1088990DOI10.1007/BF02055193
  8. Dupačová J., Wets R. J.-B., 10.1214/aos/1176351052, Ann. Statist. 16 (1988), 1517–1549 (1988) MR0964937DOI10.1214/aos/1176351052
  9. Kall P., 10.1287/moor.11.1.9, Math. Oper. Res. 11 (1986), 9–18 (1986) MR0830103DOI10.1287/moor.11.1.9
  10. 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 (1987) Zbl0636.90066MR0909741
  11. Kaniovski Y. M., King A. J., Wets R. J.-B., 10.1007/BF02031707, Ann. Oper. Res. 56 (1995), 189–208 (1995) MR1339792DOI10.1007/BF02031707
  12. Kaňková V., Lachout P., Convergence rate of empirical estimates in stochastic programming, Informatica 3 (1992), 497–523 (1992) Zbl0906.90133MR1243755
  13. King A. J., Wets R. J.-B., 10.1080/17442509108833676, Stochastics and Stochastics Reports 34 (1991), 83–92 (1991) Zbl0733.90049MR1104423DOI10.1080/17442509108833676
  14. Langen H.-J., 10.1287/moor.6.4.493, Math. Oper. Res. 6 (1981), 493–512 (1981) Zbl0496.90085MR0703092DOI10.1287/moor.6.4.493
  15. Pflug G. Ch., Ruszczyňski, A., Schultz R., 10.1287/moor.23.1.204, Math. Oper. Res. 23 (1998), 204–220 (1998) Zbl0977.90031MR1606474DOI10.1287/moor.23.1.204
  16. Robinson S. M., 10.1007/BF02591695, Math. Programming 37 (1987), 208–222 (1987) Zbl0623.90078MR0883021DOI10.1007/BF02591695
  17. Robinson S. M., Wets R. J.-B., 10.1137/0325077, SIAM J. Control Optim. 25 (1987), 1409–1416 (1987) Zbl0639.90074MR0912447DOI10.1137/0325077
  18. Rockafellar R. T., Wets R. J.-B., Variational Analysis, Springer–Verlag, Berlin 1998 Zbl0888.49001MR1491362
  19. Römisch W., Schultz R., 10.1007/BF01594935, Math. Programming 50 (1991), 197–226 (1991) Zbl0743.90083MR1103933DOI10.1007/BF01594935
  20. Römisch W., Schultz R., 10.1287/moor.18.3.590, Math. Oper. Res. 18 (1993), 590–609 (1993) Zbl0797.90070MR1250562DOI10.1287/moor.18.3.590
  21. Römisch W., Schultz R., 10.1137/0806028, SIAM J. Optim. 6 (1996), 531–447 (1996) Zbl0854.90113MR1387338DOI10.1137/0806028
  22. Römisch W., Wakolbinger A., Obtaining convergence rates for approximations 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. 327–343 (1987) MR0909737
  23. Salinetti G., Wets R. J.-B., On the convergence of closed-valued measurable multifunctions, Trans. Amer. Math. Soc. 266 (1981), 275–289 (1981) Zbl0501.28005MR0613796
  24. Salinetti G., Wets R. J-B., 10.1287/moor.11.3.385, Math. Oper. Res. 11 (1986), 385–419 (1986) Zbl0611.60004MR0852332DOI10.1287/moor.11.3.385
  25. Vogel S., Stochastische Stabilitätskonzepte, Habilitation, Ilmenau Technical University, 1991 
  26. Vogel S., 10.1007/BF01580896, Math. Programming 56 (1992), 91–119 (1992) Zbl0770.90061MR1175561DOI10.1007/BF01580896
  27. Vogel S., A stochastic approach to stability in stochastic programming, J. Comput. Appl. Math., Series Appl. Analysis and Stochastics 56 (1994), 65–96 (1994) Zbl0824.90107MR1338637
  28. Vogel S., On stability in stochastic programming – Sufficient conditions for continuous convergence and epi-convergence, Preprint of Ilmenau Technical University, 1994 MR1338637
  29. Wang J., 10.1007/BF00940733, J. Optim. Theory Appl. 63 (1989), 79–89 (1989) MR1022368DOI10.1007/BF00940733
  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 (1989) Zbl0752.90052MR1105107
  31. Zervos M., 10.1287/moor.24.2.495, Math. Oper. Res. 24 (1999), 2, 495–508 (1999) Zbl1074.90552MR1853885DOI10.1287/moor.24.2.495

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