On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems
Name
On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive.pdf
Size
325.43 KB
Format
Adobe PDF
Checksum (MD5)
9a0a843f2ed467496999a4c861604809
Author(s)
Goyal, Vineet
Bertsimas, Dimitris J
Date Issued
May 2010
Journal
Mathematics of Operations Research
Publisher
Institute for Operations Research and the Management Sciences
Citation
Bertsimas, D., and V. Goyal. “On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems.” Mathematics of Operations Research 35.2 (2010): 284–305.
Version
Author's final manuscript
Abstract
We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive.
If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1287/moor.1090.0440
