On the Glivenko-Cantelli Problem in Stochastic Programming: Linear Recourse and Extensions

Integrals of optimal values of random linear programming problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Uniform convergence of the approximations is proved under fairly broad conditions allowing non-convex or discontinuous dependence on the parameter value and random size of the linear programming problem.

[1]  Alexander Shapiro,et al.  Asymptotic Behavior of Optimal Solutions in Stochastic Programming , 1993, Math. Oper. Res..

[2]  Roger J.-B. Wets,et al.  Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems , 1995, Ann. Oper. Res..

[3]  C. Leake.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1994 .

[4]  Wansoo T. Rhee,et al.  A Concentration Inequality for the K-Median Problem , 1989, Math. Oper. Res..

[5]  R. Wets,et al.  Stochastic programming , 1989 .

[6]  E. Giné,et al.  Some Limit Theorems for Empirical Processes , 1984 .

[7]  R. Rao.  Relations between Weak and Uniform Convergence of Measures with Applications , 1962 .

[8]  Jason H. Goodfriend,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1995 .

[9]  Flemming Topsøe.  Uniformity in convergence of measures , 1977 .

[10]  R. Wets,et al.  Epi‐consistency of convex stochastic programs , 1991 .

[11]  J. Dupacová,et al.  ASYMPTOTIC BEHAVIOR OF STATISTICAL ESTIMATORS AND OF OPTIMAL SOLUTIONS OF STOCHASTIC OPTIMIZATION PROBLEMS , 1988 .

[12]  Yuri Ermoliev,et al.  Numerical techniques for stochastic optimization , 1988 .

[13]  M. Talagrand.  The Glivenko-Cantelli Problem , 1987 .

[14]  D. Pollard.  Convergence of stochastic processes , 1984 .

[15]  Y. Ermoliev,et al.  Normalized convergence in stochastic optimization , 1991, Ann. Oper. Res..

[16]  Stephen M. Robinson,et al.  Local epi-continuity and local optimization , 1987, Math. Program..

[17]  V. Vapnik,et al.  Necessary and Sufficient Conditions for the Uniform Convergence of Means to their Expectations , 1982 .

[18]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[19]  Silvia Vogel,et al.  On stability in multiobjective programming — A stochastic approach , 1992, Math. Program..

[20]  S. M. Robinson,et al.  Stability in two-stage stochastic programming , 1987 .

[21]  A. Shapiro.  Asymptotic Properties of Statistical Estimators in Stochastic Programming , 1989 .

[22]  Roberto Lucchetti,et al.  Uniform convergence of probability measures: topological criteria , 1994 .

[23]  Rüdiger Schultz.  Rates of Convergence in Stochastic Programs with Complete Integer Recourse , 1996, SIAM J. Optim..

[24]  Zvi Artstein,et al.  Stability Results for Stochastic Programs and Sensors, Allowing for Discontinuous Objective Functions , 1994, SIAM J. Optim..

[25]  Sara van de Geer,et al.  On rates of convergence and asymptotic normality in the multiknapsack problem , 1991, Math. Program..

[26]  Rüdiger Schultz.  On structure and stability in stochastic programs with random technology matrix and complete integer recourse , 1995, Math. Program..

[27]  R. Tyrrell Rockafellar,et al.  Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming , 1993, Math. Oper. Res..

[28]  S. Vogel.  A stochastic approach to stability in stochastic programming , 1994 .

[29]  Peter Kall,et al.  On approximations and stability in stochastic programming , 1987 .

[30]  Werner Römisch,et al.  Lipschitz Stability for Stochastic Programs with Complete Recourse , 1996, SIAM J. Optim..