Inventory models under uncertainty: an adaptive approach

Consider the following nonlinear programming (NLP) problem: min xg₀(x)= min x∫ψ(x, y)f_Y(y, x) dy=min E[ψ₀(x, Y)]s.t.g_j(x)=∫ψ_j(x, y)f{hook}_Y(x, y) dy=E[ψ_j(x, Y)] ≤ 0, j=1,...,M,where x ∈ X ⊂ Rⁿ,y∈ D ⊂ R^m, ψ_j(x,Y), j=0,1,...,M are given functions, and f_T(y, x)is a probability density function depending on a vector of parameters x. We assume that the pdf (probability density function) f_Y(y, x) is unknown but a sample Y₁,..., Y_N from it is available. To find the approximate solution of this NLP problem (the exact solution is not available since f_Y(y,'x) is unknown) we use the sample Y₁...Y_N directly in an adaptive procedure called stochastic approximation in which the optimal solution x^* of (1) is approximated iteratively, i.e., step by step. We consider several stochastic optimization models which can be fitted in the framework of the NLP problem (1) and present adaptive stochastic approximation procedures to approximate the optimal solution x*.