Scenario reduction in stochastic programming with respect to discrepancy distances

Abstract

Stochastic programming, Chance constraints, Two-stage, Mixed-integer, Scenario reduction, Discrepancy, Kolmogorov metric,

Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming

“…, m k , respectively. Then, due to its special form, every polyhedron B in B poly(W) can be written as Analogous to the concept of supporting cells in [9], we will show that it suffices to consider the following supporting polyhedra. Loosely speaking, a polyhedron B ∈ B poly(W) is called supporting if each of its facets contains an element of ξ 1 , .…”

“…In Section 2 we state the problem of optimal scenario reduction with respect to a discrepancy distance α B and decompose it into a combinatorial and a linear optimization problem. Extending our earlier work in [9], we discuss in Section 3 how the coefficients of the linear program may be computed in case of the polyhedral discrepancy α B poly(W) . Algorithms for determining the optimally reduced probability distribution (with respect to α B poly(W) ) are developed in Sections 4 and 5.…”

Solution sensitivity-based scenario reduction for stochastic unit commitment

“…While both [3] and [4] derived scenario reduction method according to the upper bound of Fortet-Mourier metrics instead of the metrics themselves, [28] refined a scenario reduction method for two-stage stochastic programs rigorously based on Fortet-Mourier metrics. These methods were further extended to chance constrained and mixed-integer two-stage stochastic programs in [29], which are stated with respected to cell discrepancy (or the Kolmogorov metric), while [30] extended the work in [29] with a certain polyhedral discrepancy. Further extensions to multi-stage stochastic programs were made in [31][32][33][34].…”

Scenario Tree Generation for Multi-stage Stochastic Programs

“…Originally, the concept of scenario reduction was developed in [DGKR03,HR03]. More recently, it has been improved for two-stage models in [HR07] and extended to mixed-integer and chance constrained models in [HKR08,HKR09] as well as to multi-stage models in [HR09b]. The concept does not impose special conditions on the underlying probability distribution except the existence of certain moments.…”