On Stability of Multistage Stochastic Programs

Abstract

We study the quantitative stability of linear multistage stochastic programs under perturbations of the underlying stochastic processes. It is shown that the optimal values behave Lipschitz continuously with respect to an L p -distance. In order to establish continuity of the recourse function with respect to the current state of the stochastic process, we assume continuity of the conditional distributions in terms of a Fortet-Mourier metric. The main stability result holds for nonanticipative approximations of the underlying process and thus represents a rigorous justification of established discretization techniques.AMS 2000 subject classification: 90C15, 90C31

Scenario Tree Generation for Multi-stage Stochastic Programs

“…We note that our condition (A4) is similar to assumption 2.6 in [Küc08] and Corollary 1 reminds of [Küc08, Theorem 3]. We also note that in case of finite supports Ξ t , t = 1, .…”

Numerical evaluation of approximation methods in stochastic programming

“…However, specific regularity assumptions on the problem (P ) and the processes ξ andξ are necessary to ensure certain approximation qualities, cf. [5], [11], and [13]. Indeed, without such conditionsξ may be close to ξ in some sense, but passing from (P ) to (P ) may lead to significant changes in the optimal value, e.g., by providing arbitrage possibilities, see [11,Example A.4].…”

“…[5], [11], and [13]. Indeed, without such conditionsξ may be close to ξ in some sense, but passing from (P ) to (P ) may lead to significant changes in the optimal value, e.g., by providing arbitrage possibilities, see [11,Example A.4]. Being interested in a good approximation of the unknown value v(ξ), it is thus reasonable rather to evaluate the approximate solutionx with regard to the original data process ξ, that is, to consider E[ϕ(ξ,x(ξ))].…”

“…is an optimal solution for (11). Observe that the algorithms we have used for the scenario tree construction do not maintain the martingale property, in general.…”

“…Stability analysis of stochastic programs yields further hints how approximations should look like, cf. [5,11,13,16].…”