On the Scenario-Tree Optimal-Value Error for Stochastic Programming Problems

Abstract

Stochastic programming problems generally lead to large-scale programs if the number of random outcomes is large or if the problem has many stages. A way to tackle them is provided by scenario-tree generation methods, which construct approximate problems from a reduced subset of outcomes. However, it is well known that the number of scenarios required to keep the approximation error within a given tolerance grows rapidly with the number of random parameters and stages. For this reason, to limit the fast growth of complexity, scenario-tree generation methods tailored to problems must be developed. These will use more information about the problem than just the underlying probability distributions; namely, they will also take into account the objective function and the constraints. In this paper, we develop a general framework to build problem-driven scenario trees. We do so by studying how the optimal-value error arises as a sum of lower-level errors made at each node of the tree. We show how these small but numerous node errors depend on the specific features of the problem and how they can be controlled by designing scenario trees with appropriate branching structures and discretization points and weights. We illustrate our approach on two examples.

Problem-Driven Scenario Clustering in Stochastic Optimization

“…The objective function is indeed important to consider since, for instance, a risk-averse problem featuring a logutility function might require a different scenario set than its risk-neutral version as it might be sensitive to different aspects of the distribution. Although the vast majority of methods used for scenario generation fall in the distribution-driven category, as it has been the only predominant one historically, approaches falling in the problem-driven category have been recently studied in [Henrion and Römisch, 2018, Fairbrother et al, 2019, Keutchayan et al, 2020 as a mean to provide more efficient procedures to generate scenarios.…”

Problem-Driven Scenario Generation for Stochastic Programming Problems: A Survey

“…It is a process of investigating the impact of variations in the model's input parameters on the resulting output. In the work [33], a concrete theoretical foundation for assessing the optimal-value error is proposed, which denotes the difference between the optimal value of the original SP problem and that of the approximated problem generated using different scenario-trees. The study highlights that sub-optimal discretization at a node in the scenario-tree generation process contributes to the optimal-value error, and suggests that suitable scenario-trees can be designed to numerically integrate specific classes of functions determined by the problem's structure.…”

Neural combinatorial optimization: from routing to integer programming

“…context) of an instance also influence the landscape of the objective function. In contrast, the approaches in the problem-oriented paradigm endeavor to involve problem-specific properties in scenario reduction [77][78][79][80]. For example, Keutchayan et al [80] attempt to search top-K scenarios with their objective values separately approximating the expected objective values derived from the K scenario subsets, in which the top-K scenarios exist respectively.…”

“…On the other hand, it can be found that the results delivered by CVAE-SIPA are generally superior or comparable to those by CVAE-SIP, suggesting a better cross-distribution generalization. In our training, we do not assume explicit dependencies between the context and scenarios, following the settings in [21,24,79]. However, we would like to show that the trained networks can well generalize to the dependent settings.…”