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- W2489237792 abstract "5.1 Statistical Properties of Sample Average Approximation EstimatorsConsider the following stochastic programming problem:Minx∈X{ƒ(x)≔E[F(x,ξ)]}.(5.1)Here X is a nonempty closed subset of ℝn, ξ is a random vector whose probability distribution P is supported on a set Ξ ⊂ ℝd, and F : X × Ξ → ℝ. In the framework of two-stage stochastic programming, the objective function E(x, ξ) is given by the optimal value of the corresponding second-stage problem. Unless stated otherwise, we assume in this chapter that the expectation function ƒ(x) is well defined and finite valued for all x ∈ X. This implies, of course, that for every x ∈ X the value F(x, ξ) is finite for a.e. ξ ∈ Ξ. In particular, for two-stage programming this implies that the recourse is relatively complete.Suppose that we have a sample ξ1,…, ξN of N realizations of the random vector ξ. This random sample can be viewed as historical data of N observations of ξ, or it can be generated in the computer by Monte Carlo sampling techniques. For any x ∈ X we can estimate the expected value ƒ(x) by averaging values F(x, ξj), j = 1,…, N." @default.
- W2489237792 created "2016-08-23" @default.
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- W2489237792 date "2009-01-01" @default.
- W2489237792 modified "2023-09-26" @default.
- W2489237792 title "5. Statistical Inference" @default.
- W2489237792 doi "https://doi.org/10.1137/1.9780898718751.ch5" @default.
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