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… 

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References

SHOWING 1-10 OF 43 REFERENCES
Distributionally Robust Stochastic Optimization with Wasserstein Distance
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
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
...
1
2
3
4
5
...