Estimation of the Necessary Sample Size for Approximation of Stochastic Optimization Problems with Probabilistic Criteria
Stochastic optimization problems with probabilistic and quantile objective functions are considered. The probability objective function is defined as the probability that the value of losses does not exceed a fixed level. The quantile function is defined as the minimal value of losses that cannot be exceeded with a fixed probability. Sample approximations of the considered problems are formulated. A method to estimate the accuracy of the approximation of the probability maximization and quantile minimization is described for the case of a finite set of feasible strategies. Based on this method, we estimate the necessary sample size to obtain (with a given probability) an epsilon-optimal strategy to the original problems by solving their approximations in the cases of finite set of feasible strategies. Also, the necessary sample size is obtained for the probability maximization in the case of a bounded infinite set of feasible strategies and a Lipschitz continuous probability function.
KeywordsStochastic optimization Sample approximation Probability function Quantile function
The work is supported by the Russian Foundation for Basic Research (project 19-07-00436).
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