Stochastic Integer Programming: Limit Theorems and Confidence Intervals

References

• Ahmed S., Shapiro A. The sample average approximation method for stochastic programs with integer recourse. Optim. Online (2002) Google Scholar
• Apolloni B., Pezzella F. Confidence intervals in the solution of stochastic integer linear programming problems. Ann. Oper. Res. (1984) 1:67–78Crossref, Google Scholar
• Arcones M. A., Giné E., Lepage R., Billard L. On the bootstrap of M-estimators and other statistical functionals. Exploring the Limits of the Bootstrap, Wiley Series in Probability and Mathematical Statistics (1992) (Wiley, New York) 13–47Google Scholar
• Artstein Z., Wets R. J-B. Consistency of minimizers and the SLLN for stochastic programs. J. Convex Anal. (1995) 2:1–17Google Scholar
• Blair C. E., Jeroslow R. G. The value function of a mixed integer program. Discrete Math. (1977) 19:121–138Crossref, Google Scholar
• Dupačová J., Wets R. J-B. Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems. Ann. Statist. (1988) 16:1517–1549Crossref, Google Scholar
• Efron B. Bootstrap methods: Another look at the jackknife. Ann. Statist. (1979) 7:1–26Crossref, Google Scholar
• Ermoliev Y. M., Norkin V. I. Normalized convergence in stochastic optimization. Ann. Oper. Res. (1991) 30:187–198Crossref, Google Scholar
• Futschik A., Pflug G. C. Confidence sets for discrete stochastic optimization. Ann. Oper. Res. (1995) 56:95–108Crossref, Google Scholar
• Gill R. D. Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1). Scandinavian J. Statist. Theory Appl. (1989) 16:97–128Google Scholar
• Gill R. D., van der Vaart A. W. Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 2). Scandinavian J. Statist. Theory Appl. (1993) 20:271–288Google Scholar
• Giné E. Empirical processes and applications: An overview. Bernoulli (1996) 2:1–28Crossref, Google Scholar
• Giné E., Giné E., Grimmet G. R., Saloff-Coste L. Lectures on some aspects of the bootstrap. Lectures on Probability Theory and Statistics, Lecture Notes in Mathematics (1997) 1665(Springer, Berlin, Germany) 37–151Crossref, Google Scholar
• Giné E., Zinn J. Bootstrapping general empirical measures. Ann. Probab. (1990) 18:851–869Crossref, Google Scholar
• Hall P.The Bootstrap and Edgeworth Expansion. Springer Series in Statistics (1992) (Springer, New York) Crossref, Google 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) 56:189–208Crossref, Google Scholar
• King A. J., Rockafellar R. T. Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. (1993) 18:148–162Link, Google Scholar
• King A. J., Wets R. J-B. Epi-consistency of convex stochastic programs. Stochastics Stochastics Rep. (1991) 34:83–92Crossref, Google Scholar
• Klatte D., Lommatzsch K. A note on quantitative stability results in nonlinear optimization. Proc. 19. Jahrestagung Mathematische Optimierung (1987) Berlin, Germany:77–86Sektion Mathematik, Seminarbericht Nr. 90, Humboldt-Universität BerlinGoogle Scholar
• Kleywegt A. J., Shapiro A., Homem-de-Mello T. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. (2001) 12:479–502Crossref, Google Scholar
• Lachout P. Personal communication. (2004) Google Scholar
• Louveaux F., Schultz R.Stochastic Integer Programming. Handbooks in Operations Research and Management Science (2003) 10(Elsevier, Amsterdam, The Netherlands) 213–266Chapter 4Google Scholar
• Mammen E.When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statistics (1992) 77(Springer, New York) Crossref, Google Scholar
• Morita H., Ishii H., Nishida T. Confidence region method for a stochastic programming problem. J. Oper. Res. Soc. Japan (1987) 30:218–231Google Scholar
• Norkin V. I. Convergence of the empirical mean method in statistics and stochastic programming. Cybernetics Systems Anal. (1992) 28:253–264Crossref, Google Scholar
• Pflug G. C. Asymptotic stochastic programs. Math. Oper. Res. (1995) 20:769–789Link, Google Scholar
• Pflug G. C. Stochastic programs and statistical data. Ann. Oper. Res. (1999) 85:59–78Crossref, Google Scholar
• Pflug G. C., Ruszczyński A., Schultz R. On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse. Math. Methods Oper. Res. (1998) 47:39–49Crossref, Google Scholar
• Politis D. N., Romano J. P. Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. (1994) 22:2031–2050Crossref, Google Scholar
• Politis D. N., Romano J. P., Wolf M.Subsampling. Springer Series in Statistics (1999) (Springer, New York) Google Scholar
• Politis D. N., Romano J. P., Wolf M. Subsampling, symmetrization, and robust interpolation. Comm. Statist. Theory Methods (2000) 29:1741–1757Crossref, Google Scholar
• Rachev S. T.Probability Metrics and the Stability of Stochastic Models (1991) (Wiley, Chichester, UK) Google Scholar
• Rachev S. T., Römisch W. Quantitative stability in stochastic programming: The method of probability metrics. Math. Oper. Res. (2002) 27:792–818Link, Google Scholar
• Robinson S. M. Local epi-continuity and local optimization. Math. Programming (1987) 37:208–223Crossref, Google Scholar
• Römisch W. Stability of stochastic programming problems. Handbooks in Operations Research and Management Science (2003) 10(Elsevier, Amsterdam, The Netherlands) 483–554Chapter 8Google Scholar
• Römisch W., Kotz S., Read C. B., Balakrishnan N., Vidakovic B. Delta method, infinite dimensional. Encyclopedia of Statistical Sciences (2005) 2nd ed.(Wiley, New York) . Extended entryGoogle Scholar
• Rubinstein R. Y., Shapiro A.Discrete Event Systems, Sensitivity Analysis and Stochastic Optimization by the Score Function Method (1993) (Wiley, Chichester, UK) Google Scholar
• Ruszczyński A., Shapiro A.Stochastic Programming. Handbooks in Operations Research and Management Science (2003) 10(Elsevier, Amsterdam, The Netherlands) Google Scholar
• Schultz R. Rates of convergence in stochastic programs with complete integer recourse. SIAM J. Optim. (1996) 6:1138–1152Crossref, Google Scholar
• Schultz R., Stougie L., van der Vlerk M. H. Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis reductions. Math. Programming (1998) 83:229–252Crossref, Google Scholar
• Shapiro A. Asymptotic properties of statistical estimators in stochastic programming. Ann. Statist. (1989) 17:841–858Crossref, Google Scholar
• Shapiro A. On concepts of directional differentiability. J. Optim. Theory Appl. (1990) 66:477–487Crossref, Google Scholar
• Shapiro A., Uryasev S. Statistical inference of stochastic optimization problems. Probabilistic Constrained Optimization: Methodology and Applications (2000) (Kluwer Academic Publishers, Dordrecht, The Netherlands) 282–304Crossref, Google 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) 11:70–86Crossref, Google Scholar
• Talagrand M. Sharper bounds for Gaussian and empirical processes. Ann. Probab. (1994) 22:28–76Crossref, Google Scholar
• Talagrand M. The Glivenko-Cantelli problem, ten years later. J. Theoret. Probab. (1996) 9:371–384Crossref, Google Scholar
• van der Vaart A. W.Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (1998) (Cambridge University Press, Cambridge, UK) Google Scholar
• van der Vaart A. W., Wellner J. A.Weak Convergence and Empirical Processes. Springer Series in Statistics (1996) (Springer, New York) Crossref, Google Scholar
• Wets R. J-B. Stochastic programs with fixed recourse: The equivalent deterministic program. SIAM Rev. (1974) 16:309–339Crossref, Google Scholar