# Robust Optimization

Robust Optimization is a relatively new approach to modeling uncertainty in optimization problems. Whereas stochastic programming assumes there is a probabilistic description of the uncertainty, robust optimization works with a deterministic, set-based description of the uncertainty. The robust optimization approach constructs a solution that is feasible for any realization of the uncertainty in a given set.

For a given optimization problem, there can be multiple robust versions depending on the structure of the uncertainty set. When formulating a robust counterpart of an optimization problem, maintaining tractability is an important issue. From Bertsimas, Brown and Caramanis (2011), the robust counterpart of a linear program can be written as:
$\begin{array}{ccccc} \min & c^T x & & & \\ \mbox{s.t.} & A x & \leq & b & \forall a_1 \in \mathcal{U}_1, \cdots, a_m \in \mathcal{U}_m \end{array}$ where $$a_i$$ represents the $$i$$th row of the uncertain matrix $$A$$ and takes values in the uncertainty set $$\mathcal{U}_i \in \mathbb{R}^n$$. Then $a_i^T x \leq b_i \iff \max_{\{a_i \in \mathcal{U}_i\} } a_i^T x \leq b_i, \forall i.$

The types of uncertainty sets include ellipsoidal, polyhedral, cardinality constrained, and norm uncertainty.

### References

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• Ben-Tal, A., El Ghaoui, L., and Nemirovski, A. 2009. Robust Optimization. Princeton University Press, Princeton, NJ.
• Bertsimas, D., Brown, D. B., and C. Caramanis. 2011. Theory and Applications of Robust Optimization. SIAM Review 53(3): 464-501.
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