# Scenario reduction revisited: fundamental limits and guarantees

@article{Rujeerapaiboon2017ScenarioRR,
title={Scenario reduction revisited: fundamental limits and guarantees},
author={Napat Rujeerapaiboon and Kilian Schindler and Daniel Kuhn and Wolfram Wiesemann},
journal={Mathematical Programming},
year={2017},
pages={1-36}
}
The goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms. We distinguish continuous scenario reduction, where the new atoms may be chosen freely, and discrete scenario reduction, where the new atoms must be chosen from among the existing ones. Using the Wasserstein distance as measure of proximity between distributions, we identify those n-point distributions on the unit ball that are least susceptible to scenario…

## Figures and Tables from this paper

Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution
• Computer Science, Mathematics
ArXiv
• 2021
It is proved that computing the Wasserstein distance between a discrete probability measure supported on two points and the Lebesgue measure on the standard hypercube is already #P-hard, and it is shown that smoothing the dual objective function is equivalent to regularizing the primal objective function.
Problem-Driven Scenario Clustering in Stochastic Optimization
• Mathematics
• 2021
In stochastic optimisation, the large number of scenarios required to faithfully represent the underlying uncertainty is often a barrier to finding efficient numerical solutions. This motivates the
Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning
• Computer Science, Mathematics
Operations Research & Management Science in the Age of Analytics
• 2019
This tutorial argues that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.
An Information Theoretic Approach to Probability Mass Function Truncation
• Mathematics, Computer Science
2019 IEEE International Symposium on Information Theory (ISIT)
• 2019
This paper proposes and analyzes a few criteria to truncate pmf’s so that the truncated one is as much close as possible to the original pmf, under different information theoretic measures of distance.
Scenario Reduction for Stochastic Day-Ahead Scheduling: A Mixed Autoencoder Based Time-Series Clustering Approach
• Computer Science
IEEE Transactions on Smart Grid
• 2021
A mixed autoencoder based clustering approach to select a reduced scenario set from high-dimensional time series and shows that the model outperforms the state of the art, in terms of statistical metrics and through empirical analysis.
Upper and Lower Bounds for Large Scale Multistage Stochastic Optimization Problems: Application to Microgrid Management
• Computer Science, Mathematics
• 2019
The decomposition methods are much faster than the SDDP method in terms of computation time, thus allowing to tackle problem instances incorporating more than 60 state variables in a Dynamic Programming framework.
Decentralized Multistage Optimization of Large-Scale Microgrids under Stochasticity
• Mathematics
• 2021
Microgrids are recognized as a relevant tool to absorb decentralized renewable energies in the energy mix. However, the sequential handling of multiple stochastic productions and demands, and of
Distributionally Robust Optimization: A Review
• Computer Science, Mathematics
ArXiv
• 2019
Main concepts and contributions to DRO are surveyed, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization are surveyed.
Optimization-based Scenario Reduction for Data-Driven Two-stage Stochastic Optimization
Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has
Scenario generation by selection from historical data
• M. Kaut
• Computer Science
Comput. Manag. Sci.
• 2021
The methods range from standard sampling and k -means, through iterative sampling-based selection methods, to a new moment-based optimization approach, and are compared on a simple portfolio-optimization model.

## References

SHOWING 1-10 OF 43 REFERENCES
Distributionally Robust Stochastic Optimization with Wasserstein Distance
• Mathematics
• 2016
Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is an underlying probability distribution that is
Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations
• Mathematics, Computer Science
Math. Program.
• 2018
It is demonstrated that the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs—in many interesting cases even as tractable linear programs.
Scenario reduction in stochastic programming: An approach using probability metrics
• Mathematics
• 2000
Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of
Scenario reduction in stochastic programming
• Mathematics, Computer Science
Math. Program.
• 2003
Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics in a convex stochastic programming problem with a discrete initial probability distribution.
Scenario Reduction Algorithms in Stochastic Programming
• Mathematics, Computer Science
Comput. Optim. Appl.
• 2003
Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately for convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P.
Stability of Stochastic Programming Problems
Abstract The behaviour of stochastic programming problems is studied in case of the underlying probability distribution being perturbed and approximated, respectively. Most of the theoretical results
A Dependent LP-Rounding Approach for the k-Median Problem
• Computer Science, Mathematics
ICALP
• 2012
This paper revisits the classical k-median problem and gives an efficient algorithm to construct a probability distribution on sets of k centers that matches the marginals specified by the optimal LP solution.
Quantitative Stability in Stochastic Programming: The Method of Probability Metrics
• Mathematics, Computer Science
Math. Oper. Res.
• 2002
Stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution are studied and rates of convergence in probability are derived under metric entropy conditions.
Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls
• Mathematics, Computer Science
Oper. Res.
• 2018
It is shown that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions.
Approximating k-Median via Pseudo-Approximation
• Computer Science, Mathematics
SIAM J. Comput.
• 2016
A novel approximation algorithm for $k-median is presented that achieves an approximation guarantee of$1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of$3+\ epsilon\$ by exploiting the power of pseudo-approximation.