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:
\[ \begin{array}{ccccc}
\min & c^T x & & & \\
\mbox{s.t.} & A x & \leq & b & \forall a_1 \in \mathcal{U}_1, \cdots, a_m \in \mathcal{U}_m
\] 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.


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