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Consistency of statistical estimators of solutions to stochastic optimization problems. (English) Zbl 1495.90117

Summary: We consider the asymptotic behavior of the infimal values and the statistical estimators of the solutions to a general stochastic optimization problem. We establish the epi-convergence of the performance criteria of approximate problems when the approximate probability laws, obtained by sampling the values of the random variable, converge weakly and tightly. Based on this key convergence, consistency properties of the infimal values and the estimators of the solutions to the approximate problems are obtained. Applying these results and properties of epi/hypo-convergence of bifunctions to Lagrangians of stochastic mathematical programs, we obtain the consistency of the saddle points of approximate Lagrangians and hence the consistency of the optimal values and the estimators of the solutions of approximate mathematical programs and their dual programs.

MSC:

90C15 Stochastic programming
90C31 Sensitivity, stability, parametric optimization
49J53 Set-valued and variational analysis
47H04 Set-valued operators
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