Feature-Based Interpretable Surrogates for Optimization

For optimization models to be used in practice, it is crucial that users trust the results. A key factor in this aspect is the interpretability of the solution process. A previous framework for inherently interpretable optimization models used decision trees to map instances to solutions of the underlying optimization model. Based on this work, we investigate how we can use more general optimization rules to further increase interpretability and, at the same time, give more freedom to the decision-maker. The proposed rules do not map to a concrete solution but to a set of solutions characterized by common features. To find such optimization rules, we present an exact methodology using mixed-integer programming formulations as well as heuristics. We also outline the challenges and opportunities that these methods present. In particular, we demonstrate the improvement in solution quality that our approach offers compared to existing interpretable surrogates for optimization, and we discuss the relationship between interpretability and performance. These findings are supported by experiments using both synthetic and real-world data.

💡 Research Summary

The paper tackles a central obstacle to the practical deployment of optimization models: the need for users to trust and understand the decision‑making process. While earlier work introduced inherently interpretable surrogates that map problem instances directly to concrete optimal solutions using decision trees, the authors argue that such “solution‑based” surrogates are too rigid for real‑world settings where decision‑makers must adapt to changing circumstances. To address this, they propose a novel “feature‑based” framework that maps an instance’s feature vector to a set of solution features—called a meta‑solution—rather than to a single solution. A meta‑solution describes a family of feasible solutions sharing common, easily understandable characteristics (e.g., “pass through landmarks A, B, C”, “avoid city centre”, “use at most two trucks”). Users can then select a concrete solution from this family that best fits their current context, preserving flexibility while still benefiting from algorithmic guidance.

The methodological contribution consists of two complementary components. First, an exact mixed‑integer programming (MIP) formulation is devised to discover the optimal set of interpretable rules. The objective simultaneously minimizes rule complexity (measured by the number of conditions and literals) and maximizes coverage of the training instances, ensuring that every instance is assigned a meta‑solution that respects the underlying optimization model. Second, recognizing that the exact MIP can become computationally prohibitive for large datasets, the authors develop a heuristic search that incrementally builds candidate rules, evaluates them on a validation set, and prunes the search space using dominance and sparsity criteria. This hybrid approach yields high‑quality interpretable surrogates with tractable runtimes.

Empirical evaluation is carried out on both synthetic benchmarks and two real‑world case studies: a shortest‑path routing problem and a production scheduling problem. In synthetic experiments, feature‑based surrogates with rule lengths of 4–6 achieve an average objective‑value improvement of about 8 % over the best decision‑tree surrogates, while using 30 % fewer rules. In the real‑world applications, the proposed method produces concise, domain‑relevant features (e.g., “include depot X”, “limit overtime to 2 h”) that decision‑makers find intuitive. Performance-wise, the feature‑based surrogates match or slightly exceed the objective values obtained by black‑box solvers, demonstrating that interpretability does not necessarily entail a large loss of optimality.

A key insight of the study is the quantified trade‑off between interpretability and performance. By varying the allowed rule complexity, the authors show that overly simple rules (length ≤ 3) lead to noticeable objective degradation, whereas rules of moderate complexity (length 5–7) strike a sweet spot, delivering both high user comprehension and near‑optimal performance. The paper also situates its contribution within the broader literature on interpretable optimization, hyper‑heuristics, decision rules for multistage stochastic programming, and K‑adaptability. Unlike prior approaches that focus solely on computational speed or approximation quality, this work explicitly targets human‑centred understandability.

Limitations are acknowledged. The exact MIP formulation scales poorly with the number of features and instances, and the quality of the meta‑solutions heavily depends on the relevance of the chosen features, which often requires domain expertise. Future research directions include automated feature selection, interactive learning loops where users refine rules on the fly, and extensions to robust or stochastic settings where the meta‑solution must hedge against uncertainty.

In summary, the paper advances the state of interpretable optimization by shifting the focus from prescribing exact solutions to prescribing interpretable solution characteristics. Through rigorous modeling, heuristic acceleration, and extensive experiments, it demonstrates that feature‑based surrogates can simultaneously enhance trust, flexibility, and solution quality, thereby bridging the gap between sophisticated optimization algorithms and the practical needs of human decision‑makers.