Coherent Local Explanations for Mathematical Optimization
Daan Otto, Jannis Kurtz, S. Ilker Birbil
TL;DR
This paper introduces CLEMO, a novel explanation method for mathematical optimization models that ensures explanations are consistent with the problem structure, improving transparency for complex algorithms.
Contribution
CLEMO is the first explanation approach that accounts for the structure of optimization problems, providing coherent insights into objectives and decision variables.
Findings
CLEMO produces reliable explanations for various optimization problems.
Experiments demonstrate CLEMO's effectiveness on shortest path, knapsack, and vehicle routing problems.
The method applies to both exact and heuristic algorithms.
Abstract
The surge of explainable artificial intelligence methods seeks to enhance transparency and explainability in machine learning models. At the same time, there is a growing demand for explaining decisions taken through complex algorithms used in mathematical optimization. However, current explanation methods do not take into account the structure of the underlying optimization problem, leading to unreliable outcomes. In response to this need, we introduce Coherent Local Explanations for Mathematical Optimization (CLEMO). CLEMO provides explanations for multiple components of optimization models, the objective value and decision variables, which are coherent with the underlying model structure. Our sampling-based procedure can provide explanations for the behavior of exact and heuristic solution algorithms. The effectiveness of CLEMO is illustrated by experiments for the shortest path…
Peer Reviews
Decision·Submitted to ICLR 2026
1. **Simplicity and directness:** To address the incoherence of existing methods, the authors incorporate a coherence constraint directly into the optimization objective, ensuring that the generated explanations better satisfy the structural requirements of the problem. 2. **Model-agnostic nature:** As a local explanation method, CLEMO can be applied to any black-box optimization model without relying on the internal structure of the model.
1. **Performance trade-off:** Although coherence improves, fidelity decreases. While the authors claim that the loss in fidelity is acceptable, in real-world applications this reduction may compromise the practical usefulness of the explanations. 2. **Limited baselines:** The paper only compares CLEMO with LIME and decision tree–based methods. Many more advanced explanation techniques exist, yet no comparison with them is provided. 3. **Scalability issues:** The computational cost increases shar
+ The paper identifies a practical issue in post‑hoc explanations for optimization. + The theoretical analysis is comprehensive and well‑developed. + Implementation details and appendices support reproducibility.
- The core methodological novelty is limited, as CLEMO primarily extends LIME by adding coherence penalties, a conceptually straightforward adaptation. This is particularly true given the proven redundancy of the objective coherence regularizer for problems with fixed linear objectives. - The method's reliance on an explicit, differentiable problem formulation for the feasibility regularizer is a major practical limitation. It remains unclear how CLEMO could be applied to black-box commercial s
**S1.** One of the primary strengths of this paper is its focus on the problem being examined by the authors. Explaining the output of a solving algorithm in relation to the input parameters of a problem instance is more complex than standard post-hoc explanation tasks, as the output is a vector and the explanation must be consistent with the input constraints. **S2.** This study is well-motivated; the introduction effectively justifies the need for explaining mathematical optimization tasks, p
**W1.** The notation used in this paper is quite dense and has not been adequately introduced, which leads to ambiguity in many definitions and results. Additionally, the classes of models employed to explain mathematical optimization problems are not clearly defined. **W2.** The proposed approach is primarily heuristic and lacks substantial theoretical guarantees. For example, Proposition 3.1 is straightforward, and the last paragraph of Section 3 (Lines 282-292) is quite ambiguous. **W3.**
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Taxonomy
TopicsModel Reduction and Neural Networks