Machine Learning for K-adaptability in Two-stage Robust Optimization

TL;DR

A machine learning-based node selection strategy based on general two-stage robust optimization insights is proposed that outperforms a vanilla, random node selection strategy when tested on problems of the same type as the training problems as well as in cases when the K-value or the problem size differs from the training ones.

Abstract

Two-stage robust optimization problems constitute one of the hardest optimization problem classes. One of the solution approaches to this class of problems is K-adaptability. This approach simultaneously seeks the best partitioning of the uncertainty set of scenarios into K subsets, and optimizes decisions corresponding to each of these subsets. In general case, it is solved using the K-adaptability branch-and-bound algorithm, which requires exploration of exponentially-growing solution trees. To accelerate finding high-quality solutions in such trees, we propose a machine learning-based node selection strategy. In particular, we construct a feature engineering scheme based on general two-stage robust optimization insights that allows us to train our machine learning tool on a database of resolved B&B trees, and to apply it as-is to problems of different sizes and/or types. We experimentally show that using our learned node selection strategy outperforms a vanilla, random node selection strategy when tested on problems of the same type as the training problems, also in case the K-value or the problem size differs from the training ones.

Figures (33)

• Figure 1: A framework of the $K$-adaptability problem, where we split the uncertainty set (red box) in $K=2$ parts. Here, $\boldsymbol{x}$ represents the first-stage decisions, and $\boldsymbol{y}_1$ with $\boldsymbol{y}_2$ those of the second-stage.
• Figure 2: Search tree for $K$-adaptability branch-and-bound ($K = 2$).
• Figure 3: The scope of node selection
• Figure 4: Node selection with ML predictions. Node selections are decided by the prediction of the function $\mu$ with input features $F_n^k$, where $n$ is the node from which a selection is made and $k$ relates to its $k$-th child node.
• Figure 5: Example of a feature generation procedure for node $n$ and its two child nodes.

Theorems & Definitions (3)

• Definition 1
• Example 1
• Example 2