Adaptive robust optimization
Reactive decision making: wait and see
Adaptive robust optimization is a powerful approach that combines the benefits of robust optimization with the ability to adapt to changing circumstances. It allows decision-makers to dynamically adjust their strategies in response to new information or evolving uncertainties, improving the robustness and flexibility of their decisions.$$
\begin{equation}
\begin{aligned}
\min_{x_t(\cdot)}~~& c_1^{\top}x_1 + \rho\left[ \sum_{t=2}^Tc_t(\xi_{[t]})^{\top} x_t(\xi_{[t]}) \right] && \\
\text{s.t. }~~ & {A}_1{x}_1 \geq {b}_1 && \\
& \sum_{s=2}^t {A}_{s}(\xi_{[s]}) {x}_s(\xi_{[s]}) \geq {b}_t(\xi_{[t]}) & & \qquad \forall \xi \in \Xi,\ t \in T_{-1}
\end{aligned}
\end{equation}
$$
We made several contributions in the field of multistage adaptive optimization. We proposed hybrid methods that combine scenario-based and decision rule approaches to address the computational challenges posed by uncertain parameters and constraints involving multiplication. We explored the trade-off between solution quality and computational time by comparing different types of decision rules, including linear, nonlinear, and hybrid rules. We also introduced solution frameworks based on robust optimization techniques, uncertainty set partitioning, and lifting methods to handle endogenous and exogenous uncertainty in multistage adaptive stochastic optimization problems. Those methods were applied to various real-world problems, demonstrating improved solution quality and computational efficiency. Additionally, we presented a novel method using a lifting network to generate flexible piecewise linear decision rules, offering superior quality and flexibility compared to traditional linear rules.
Relevant publications: