## On the Power and Limitations of Affine Policies in Two-Stage Adaptive Optimization

##### Author(s)

Bertsimas, Dimitris J.; Goyal, Vineet
DownloadOn the Power and Limitations of Affine Policies in Two-Stage Adaptive Optimization.pdf (290.6Kb)

OPEN_ACCESS_POLICY

# Open Access Policy

Creative Commons Attribution-Noncommercial-Share Alike

##### Terms of use

##### Metadata

Show full item record##### Abstract

We consider a two-stage adaptive linear optimization problem under right hand side uncertainty with a min–max objective and give a sharp characterization of the power and limitations of affine policies (where the second stage solution is an affine function of the right hand side uncertainty). In particular, we show that the worst-case cost of an optimal affine policy can be Omega(m12−) times the worst-case cost of an optimal fully-adaptable solution for any delta > 0, where m is the number of linear constraints. We also show that the worst-case cost of the best affine policy is O(m) times the optimal cost when the first-stage constraint matrix has non-negative coefficients. Moreover, if there are only k ≤ m uncertain parameters, we generalize the performance bound for affine policies to O(k) , which is particularly useful if only a few parameters are uncertain. We also provide an O(k) -approximation algorithm for the general case without any restriction on the constraint matrix but the solution is not an affine function of the uncertain parameters. We also give a tight characterization of the conditions under which an affine policy is optimal for the above model. In particular, we show that if the uncertainty set, R+m is a simplex, then an affine policy is optimal. However, an affine policy is suboptimal even if is a convex combination of only (m + 3) extreme points (only two more extreme points than a simplex) and the worst-case cost of an optimal affine policy can be a factor (2 − delta) worse than the worst-case cost of an optimal fully-adaptable solution for any delta > 0.

##### Date issued

2011-01##### Department

Massachusetts Institute of Technology. Operations Research Center; Sloan School of Management##### Journal

Mathematical Programming

##### Publisher

Springer and Mathematical Optimization Society

##### Citation

Bertsimas, Dimitris, and Vineet Goyal. “On the Power and Limitations of Affine Policies in Two-stage Adaptive Optimization.” Mathematical Programming, Ser. A (2011).

Version: Author's final manuscript

##### ISSN

0025-5610

1436-4646