Scenario generation in stochastic programming

Stability-based methods for scenario generation in stochastic programming are reviewed. In particular, we briefly discuss Monte Carlo sampling, Quasi-Monte Carlo methods, quadrature rules based on sparse grids and optimal quantization. In addition, we provide some convergence results based on recent developments in multivariate integration. The method of optimal scenario reduction and techniques for scenario trees generation are also reviewed. Many stochastic programming models may be rewritten into the form min { ∫ Ξ f0(x, ξ)P (dξ) : x ∈ X, ∫ Ξ fk(x, ξ)P (dξ) ≤ 0, k = 1, ...,K } , (1) where X is a closed subset of R, Ξ a closed subset of R, the functions fk map from R ×Ξ to the extended real numbers R for k = 0, ...,K, and P is a probability distribution on Ξ. The set X is used to describe all constraints not depending on P , and the set Ξ to contain the support of P . The integrands fk are assumed to be lower semicontinuous jointly in (x, ξ) implying that all integrals in (1) are well defined (although possibly infinite). Classical examples are linear two-stage stochastic programs and optimization models with probabilistic constraints. Linear two-stage models (see Section 1.5.2.1) appear for K := 0 and f0 having the representation f0(x, ξ) := 〈c(ξ), x〉 + inf{〈q(ξ), y〉 : W (ξ)y = h(ξ) − T (ξ)x, y ≥ 0} (2) by means of the infimum of a second stage linear program where some of the coefficients are affine functions of the d-dimensional random vector ξ and the variable x is the first stage decision. Models with probabilistic constraints appear, for example, for K = 1, f0(x, ξ) = 〈c, x〉 and f1(x, ξ) = p− 1l{ξ∈Ξ:T (ξ)x≥h(ξ)}(ξ), where 1lB denotes the characteristic function of a set B in R d and p ∈ (0, 1) is a probability level (see also Section 1.5.6.1). 1 1 Approximation Stability results for (1) with respect to approximations Q of the original probability distribution P (see [30] for a survey) state that infimal values v(P ) and v(Q) and solution sets S(P ) and S(Q) of the stochastic programs (1) with distributions P and Q, respectively, get close if the (uniform) distance of P and Q dF (P,Q) = sup f∈F ∣