Pith. sign in

REVIEW 3 major objections 4 minor 41 references

Coherent Local Explanations for Mathematical Optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that CLEMO, a sampling-based local explanation method for optimization models, can explain objective values and decision variables simultaneously while enforcing that the explanations are coherent with the model's…

desk verdict CLEMO is a useful idea with a solid core, but the binary-feasibility claim needs fixing and the evaluation leans on in-sample numbers. read the letter →

arxiv 2502.04840 v2 pith:VA3VOSJ4 submitted 2025-02-07 math.OC cs.LG

Daan Otto , Jannis Kurtz , S. Ilker Birbil This is my paper · ORCID
classification math.OCcs.LG MSC 90C3190C1090C27
keywords mathematicaloptimizationlocalexplanationsLIMEsensitivityanalysiscoherentknapsackproblemvehicleroutingshortestpath
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces CLEMO, a sampling-based method for locally explaining what drives the objective value and decision variables of a mathematical optimization model. Standard local explainers treat each component separately, so predicted solutions can violate constraints and predicted objectives can disagree with the objective of the predicted decisions. CLEMO fits all components together and adds a coherence penalty that pushes the surrogate toward feasible decisions whose objective matches the predicted objective. On shortest path, knapsack, and vehicle routing problems, it reports sharply reduced incoherence while keeping accuracy close to plain regression. If the claim holds, users can trust local sensitivity explanations to respect the model's own logic.

What carries the argument

The mechanism is a vector-valued surrogate $g(\theta) = (g_f(\theta), g_x(\theta))$ whose components are linear or logistic functions, trained on weighted samples $\theta_i$ near $\theta_0$. Coherence is enforced by the regularizer $R_C(g(\theta_i)) = \lambda_{C1}(g_f(\theta_i) - f(g_x(\theta_i); \theta_i))^2 + \lambda_{C2} \delta(g_x(\theta_i), X(\theta_i))$, where $\delta$ sums constraint violations of a point with respect to the feasible region. The paper proves that this regularizer is convex in the surrogate coefficients when the objective is affine in the decisions and the feasibility-distance is convex, so for such models every local minimum of the fitting problem is global.

What would settle it

Run CLEMO on a binary knapsack instance, round or threshold each predicted probability to a binary value, and then check the rounded decision vectors against the capacity constraint and the predicted objective across many sampled parameter vectors. If the rounded vectors are frequently infeasible while the probability vectors satisfy the penalty, the claimed coherence does not extend to the actual discrete decisions the explanations describe.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that CLEMO produces locally accurate surrogate explanations for an optimization model that are coherent with its structure: the predicted objective value equals the objective of the predicted decisions, and the predicted decisions satisfy the model constraints. The method samples neighboring parameter vectors, records the solver's solution for each, and fits linear models for continuous outputs and logistic models for binary outputs, minimizing a weighted sum of accuracy loss and a coherence regularizer. Experiments report that on the knapsack problem CLEMO reduces weighted objective incoherence by more than 50% and feasibility incoherence by more than 99% relative to a LIME-style benchmark, at the cost of roughly 20% more accuracy loss, and that analogous improvements hold for the shortest path and vehicle routing cases with exact and heuristic solvers.

Load-bearing premise

For binary and integer decision variables, the feasibility regularizer is evaluated on continuous logistic-regression probability outputs rather than on actual integer solutions, so the coherence guarantee for the true discrete decisions depends on penalizing probability-level constraint violations transferring to the integer outcomes.

Editorial extensions

If this is right

  • CLEMO can explain any exact or heuristic algorithm that returns feasible solutions, because the coherence penalty uses only a formulation of the model, not the solver's internals.
  • For linear objectives with fixed cost coefficients and unique least-squares fits, independent linear predictors automatically satisfy objective coherence, so the regularizer is needed mainly for feasibility and for nonlinear or binary cases.
  • Users can read a single set of feature contributions for objective and decisions that cannot contradict each other, removing the misleading explanations the paper illustrates with a two-variable example.
  • Because the surrogate fit converges quickly, early stopping can cut the runtime overhead while preserving most of the coherence gain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would round or threshold the logistic predictions for binary decisions before measuring feasibility; the paper does not do this, so the strength of the coherence guarantee for true integer solutions remains open.
  • The same sampling and penalty scheme could serve as an auditing tool for solver instability, since resampling stability is already measured and reported.
  • One could apply CLEMO to predict-then-optimize pipelines, treating forecasts as the parameters to explain how forecast error propagates into decisions.
  • Using decision trees as the surrogate class instead of linear and logistic models would yield rule-based explanations carrying the same coherence conditions, though fitting would become more complex.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces CLEMO, a LIME-style method for locally explaining the output of a solution algorithm for a parameterized optimization problem. It fits a vector of interpretable surrogate models for the objective value and the decision variables, and adds a coherence regularizer that penalizes (i) discrepancies between the predicted objective and the objective value of the predicted decision vector, and (ii) violations of the feasible-region constraints by the predicted decision vector. Experiments on shortest path, knapsack, and capacitated vehicle routing problems compare CLEMO with independently fitted linear/logistic regression and decision-tree baselines on accuracy, coherence, and stability. The paper also proves convexity of the coherence regularizer in the linear continuous case and shows that independently fitted linear predictors satisfy objective coherence when the objective is fixed and linear.

Significance. If the feasibility claim for integer problems could be substantiated, CLEMO would be a useful contribution to the growing literature on explainable optimization; the problem it addresses—incoherence of separately fitted surrogates—is real. The paper ships code, provides a formal coherence definition, and gives a nontrivial convexity and objective-coherence analysis for the linear continuous case; the stability analysis over resampled datasets is a nice addition. However, the evaluation for knapsack and vehicle routing currently relies on a continuous relaxation of feasibility, so the headline claim for integer problems is not yet established.

major comments (3)
  1. [3 (Binary Decision Variables), Eq. (8)–(9); Tables 2 and 4] The feasible-region coherence reported for the knapsack and CVRP experiments is computed on logistic probability predictions σ(β_c^T θ), not on the integer solutions these surrogates are meant to explain. Since δ in (9) is a sum of inequality violations and the integrality constraints x ∈ {0,1}^p cannot be expressed as such inequalities, the values in the 'Feasible region' columns of Tables 2 and 4 measure only violations of the LP relaxation. A probability vector can satisfy w^T x ≤ 1 while being far from every feasible integer vector, so the reported reductions (e.g., 0.01–0.04 in Table 2) do not establish condition (4) for the actual decisions. The paper needs either a genuinely discrete post-processing (e.g., rounding/thresholding) with coherence evaluated on the resulting integer vectors, or a reformulation of δ that accounts for integrality; without this, the central claim of coherence for KP and CVRP is unsupported.
  2. [4 (Setup); Tables 2 and 4] All coherence and accuracy numbers are reported on the same training set D that is used to fit the surrogate models, and the coherence regularizer directly penalizes the RC metric that is reported. The observed reduction in incoherence on D is therefore partly enforced by construction rather than demonstrated as a property of the explanation. To substantiate the claim that CLEMO yields coherent explanations without substantial accuracy loss, the authors should evaluate on a separate set of parameters sampled from the same local neighborhood, and report both accuracy and coherence out-of-sample.
  3. [3 (Prop. 3.1) and A.3 (Theorem A.1)] The convexity result and the objective-coherence guarantee are proven only for linear surrogates g(θ)=β^T θ. For the logistic surrogates used for binary decision variables, σ(β^T θ) is not affine in β, so Proposition 3.1 does not apply and the optimization problem solved in the KP and CVRP experiments has no convexity guarantee. Moreover, the 'coherence' guaranteed by Theorem A.1 concerns condition (3) only, not feasibility. The paper should state this scope limitation explicitly and either extend the theoretical analysis or temper the claims about guaranteed coherence for the binary experiments.
minor comments (4)
  1. [3 (Binary Decision Variables)] The displayed accuracy-loss formula appears to have a set error: the first sum should run over the non-binary components (the complement of B) rather than over B, and the second sum should run over B. Please correct the notation so that the squared loss applies to continuous components and the log loss to binary components.
  2. [4.2] The text says the accuracy loss increased by 'roughly 20%' when comparing CLEMO to LR, but the numbers in Table 2 show increases of roughly 10–12% (e.g., Type 2: 1076 to 1203). Please align the statement with the table.
  3. [4 (Setup)] The hyperparameter rule λ_j = 0.5 L_max/L_j for non-dominant loss terms introduces a dependence of the final objective on the benchmark solution; this should be mentioned as a possible source of bias, and a sensitivity analysis over λ would strengthen the empirical claims.
  4. [A.4.2] The FSI is defined as an average over pairs and then summed over k=1..5, and Table 2 reports 'mean stability measures over 10 instances per type'; it would be helpful to state explicitly whether the FSI entries are averaged over instances and over which components, since the description in the appendix is somewhat terse.

Circularity Check

1 steps flagged · score 4.0 of 10

Coherence gain is partly self-measured: the reported 'Incoherence (RC)' metric in Tables 1, 2 and 4 is the same regularizer that CLEMO minimizes, so the headline coherence result is enforced, not independently discovered.

  1. fitted input called prediction [Section 3, Eqs. (6) and (8); Section 4, Tables 1, 2, 4 and setup text]
    "To generate coherent explanations we solve the problem ... R_C corresponds to the coherence regularizer that punishes predictors which do not admit the coherence conditions (3) and (4) ... we use R_C(g(θ_i)) = λ_{C1}(g_f(θ_i) − f(g_x(θ_i); θ_i))^2 + λ_{C2} δ(g_x(θ_i), X(θ_i)) ... This way we can compare CLEMO to the benchmark on local accuracy (12) and incoherence (8)."

    CLEMO's coefficients are chosen as the minimizer of (10) whose objective includes Σ_i w_i R_C(g(θ_i)), and the tables' 'Incoherence (RC)' columns evaluate exactly the same R_C from (8), usually on the same sampled dataset D. Hence the result that CLEMO is more coherent than LR (which minimizes only ℓ_A) is not an independent measurement: the coherence gap is a direct consequence of adding R_C to the training loss. The split into 'Objective' and 'Feasible region' merely separates the two additive terms of R_C, so the headline coherence improvement reduces to the method's own objective. The accuracy and stability columns provide independent empirical support, which is why the circularity is only partial.

full rationale

The paper's derivation chain is largely self-contained. Theorem A.1 is a genuine linear-algebra proof that unique independent least-squares fits satisfy objective coherence for fixed linear objectives; Proposition 3.1 gives a standard convex-composition argument; no load-bearing self-citation or imported uniqueness theorem is used. The main circular element is the evaluation of coherence: Eq. (8) defines R_C, Eq. (6)/(10) minimizes it, and Tables 1, 2, and 4 report it as 'Incoherence (RC)'. Thus the qualitative claim that CLEMO finds significantly more coherent explanations than LIME-type LR is, for the coherence component, a restatement of the training objective rather than an empirical discovery. This is partial circularity, because the claims that accuracy is not substantially compromised and that stability is comparable are measured by loss terms and indices that are not minimized by CLEMO, and they support the method independently. Separately, and not counted as circularity, the feasible-region values for knapsack and CVRP are computed on logistic probability vectors via the constraint-violation distance (9), which omits integrality, so they do not establish membership in the integer feasible set required by condition (4); this is a soundness gap rather than a circular reduction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method relies on several hand-set hyperparameters and domain assumptions. The most important conceptual assumption is that continuous probability predictions can be used to evaluate discrete feasibility in the regularizer. No new physical or mathematical entities are introduced.

free parameters (4)
  • ν (RBF kernel bandwidth) = mean distance to θ0 over training set
    Controls the locality of the weights in Equation (11); set per dataset rather than derived.
  • λ_A1, λ_A2, λ_C1, λ_C2 = 0.5 L_max / L_j (or 1)
    Hyperparameters balancing accuracy and coherence terms, tuned using the LR benchmark loss values on the same training data.
  • Sampling scale (0.2θ0) = 0.2θ0
    Standard deviation of the normal perturbation used to generate training samples; chosen without justification.
  • Number of training samples N = 1000
    Dataset size used in all experiments; arbitrary.
assumptions (4)
  • standard math Convex composition rules for affine and convex functions (Boyd and Vandenberghe)
    Used in Proposition 3.1 to establish convexity of the coherence regularizer.
  • domain assumption δ(x, X(θ)) as sum of max{0, γ_t(x, θ)} is an appropriate feasibility distance
    Equation (9) assumes constraint violations measure distance to the feasible set; reasonable for continuous constraints but questionable for integer feasibility.
  • domain assumption Normal perturbation θ_i ~ N(θ0, 0.2θ0) describes the local neighborhood of θ0
    Used in Algorithm 2 to generate training data; no validation that this distribution matches the intended notion of locality.
  • ad hoc to paper Logistic regression predictions can be treated as decision-variable values for feasibility checking
    Binary decisions are predicted as probabilities, then substituted into the constraints in the regularizer; this is a relaxation, not a true integer solution, and no justification is given.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coherent Local Explanations for Mathematical Optimization." pith.science (2026). https://pith.science/paper/VA3VOSJ4

@misc{pith2026250204840,
  author       = {Pith},
  title        = {Pith review of: Coherent Local Explanations for Mathematical Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VA3VOSJ4}},
  note         = {Machine review of arXiv:2502.04840}
}
read the original 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 problem, the knapsack problem, and the vehicle routing problem.

Figures

Figures reproduced from arXiv: 2502.04840 by the authors.

Figure 1
Figure 1. Solution of shortest path of SPP-θ instance as determined by Dijkstra’s Algorithm and as predicted by CLEMO. 4.1 Shortest Path Problem As a first experiment, we explain an instance of the Shortest Path where possible cost-changes depend on one single parameter. An instance of the SPP is given by a connected graph G = (V, E, c), with nodes V , edges E and edge-costs c, and specified start and terminal nodes s, t ∈ V … view at source ↗
Figure 2
Figure 2. Convergence of CLEMO over SLSQP iterations for different sizes of KP. respectively, while the weighted accuracy loss increased only by roughly 20%. In Figures 5 to 8 in the appendix, we plotted the accuracy and incoherence of each instance to strengthen our conclusion. Besides, as datasets are randomly generated, we measure the stability of explanations over resampling. For the KP, we analyze the stability of CLEMO … view at source ↗
Figure 3
Figure 3. Explanation as found by CLEMO for the objective value visualized in the present problem [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Decision variables solution of shortest path of SPP- [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Scatter plot of the total incoherence (i.e., coherence loss) and total accuracy losses as found by the different methods on 10 distinct sample sets per instance of the knapsack problem of type 1. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Scatter plot of the total incoherence (i.e., coherence loss) and total accuracy losses as found by the different methods on 10 distinct sample sets per instance of the knapsack problem of type 2 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Scatter plot of the total incoherence (i.e., coherence loss) and total accuracy losses as found by the different methods on 10 distinct sample sets per instance of the knapsack problem of type 3. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Scatter plot of the total incoherence (i.e., coherence loss) and total accuracy losses as found by the different methods on 10 distinct sample sets per instance of the knapsack problem of type 4. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Explanation as found by CLEMO for the decision variable [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 26 canonical work pages

  1. [1]

    Peeking Inside the Black-Box: A Survey on Explain- able Artificial Intelligence (XAI)

    Amina Adadi and Mohammed Berrada. Peeking Inside the Black-Box: A Survey on Explain- able Artificial Intelligence (XAI). IEEE Access, 6:52138–52160, 2018. ISSN 2169-3536. doi: 10.1109/ACCESS.2018.2870052. URL https://ieeexplore.ieee.org/document/ 8466590/?arnumber=8466590. Conference Name: IEEE Access

    work page arXiv 2018

  2. [2]

    A Framework for Data-Driven Explainability in Mathematical Optimization

    Kevin-Martin Aigner, Marc Goerigk, Michael Hartisch, Frauke Liers, and Arthur Miehlich. A Framework for Data-Driven Explainability in Mathematical Optimization. Proceedings of the AAAI Conference on Artificial Intelligence, 38(19):20912–20920, March 2024. ISSN 2374-3468. doi: 10.1609/aaai.v38i19.30081. URL https://ojs.aaai.org/index.php/ AAAI/article/view...

    work page doi:10.1609/aaai.v38i19.30081 2024

  3. [3]

    Machine learning for combinatorial optimization: a methodological tour d’horizon

    Yoshua Bengio, Andrea Lodi, and Antoine Prouvost. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2):405–421, 2021

    2021

  4. [4]

    Introduction to linear optimization, volume 6

    Dimitris Bertsimas and John N Tsitsiklis. Introduction to linear optimization, volume 6. Athena scientific Belmont, MA, 1997

    work page 1997

  5. [5]

    Benchmarking and survey of explanation methods for black box models

    Francesco Bodria, Fosca Giannotti, Riccardo Guidotti, Francesca Naretto, Dino Pedreschi, and Salvatore Rinzivillo. Benchmarking and survey of explanation methods for black box models. Data Mining and Knowledge Discovery , 37(5):1719–1778, September 2023. ISSN 1573-756X. doi: 10.1007/s10618-023-00933-9. URL https://doi.org/10.1007/ s10618-023-00933-9

    doi:10.1007/s10618-023-00933-9 2023

  6. [6]

    Sensitivity analysis: A review of recent advances

    Emanuele Borgonovo and Elmar Plischke. Sensitivity analysis: A review of recent advances. European Journal of Operational Research , 248(3):869–887, February 2016. ISSN 0377-

    work page 2016

  7. [7]

    Convex optimization

    Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge University Press, 2004

    2004

  8. [8]

    Subgradient methods

    Stephen Boyd, Lin Xiao, and Almir Mutapcic. Subgradient methods. lecture notes of EE392o, Stanford University, Autumn Quarter, 2004(01), 2003

    2004

Show all 41 references
  1. [9]

    Opportunities and Challenges in Explainable Artificial Intelligence (XAI): A Survey, June 2020

    Arun Das and Paul Rad. Opportunities and Challenges in Explainable Artificial Intelligence (XAI): A Survey, June 2020. URLhttp://arxiv.org/abs/2006.11371. arXiv:2006.11371

    2020 arXiv

  2. [10]

    Why model why? Assessing the strengths and limitations of LIME, November 2020

    Jürgen Dieber and Sabrina Kirrane. Why model why? Assessing the strengths and limitations of LIME, November 2020. URL http://arxiv.org/abs/2012.00093. arXiv:2012.00093 [cs]

    2020 arXiv

  3. [11]

    Explainable AI (XAI): Core Ideas, Techniques, and Solutions

    Rudresh Dwivedi, Devam Dave, Het Naik, Smiti Singhal, Rana Omer, Pankesh Patel, Bin Qian, Zhenyu Wen, Tejal Shah, Graham Morgan, and Rajiv Ranjan. Explainable AI (XAI): Core Ideas, Techniques, and Solutions. ACM Computing Surveys, 55(9):194:1–194:33, January 2023. ISSN 0360-03...

    2023 doi

  4. [12]

    Explainable Data-Driven Optimization: From Context to Decision and Back Again

    Alexandre Forel, Axel Parmentier, and Thibaut Vidal. Explainable Data-Driven Optimization: From Context to Decision and Back Again. In Proceedings of the 40th International Conference on Machine Learning, pages 10170–10187. PMLR, July 2023. URL https://proceedings. mlr.press/v...

    2023

  5. [13]

    Or-tools routing library

    Vincent Furnon and Laurent Perron. Or-tools routing library. URL https://developers. google.com/optimization/routing/

  6. [14]

    Shortest path algorithms

    Giorgio Gallo and Stefano Pallottino. Shortest path algorithms. Annals of Operations Research, 13(1):1–79, December 1988. ISSN 1572-9338. doi: 10.1007/BF02288320. URL https: //doi.org/10.1007/BF02288320

    1988 doi

  7. [15]

    A framework for inherently interpretable optimization models

    Marc Goerigk and Michael Hartisch. A framework for inherently interpretable optimization models. European Journal of Operational Research, 310(3):1312–1324, November 2023. ISSN 0377-2217. doi: 10.1016/j.ejor.2023.04.013. URL https://www.sciencedirect.com/ science/article/pii/S...

    2023 doi

  8. [16]

    Counterfactual explanations and how to find them: literature review and benchmarking

    Riccardo Guidotti. Counterfactual explanations and how to find them: literature review and benchmarking. Data Mining and Knowledge Discovery, 38(5):2770–2824, September 2024. ISSN 1573-756X. doi: 10.1007/s10618-022-00831-6. URL https://doi.org/10.1007/ s10618-022-00831-6

    2024 doi

  9. [17]

    Gurobi Optimizer Reference Manual, 2024

    Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2024. URL https://www. gurobi.com

    2024

  10. [18]

    A review on global sensitivity analysis methods

    Bertrand Iooss and Paul Lemaître. A review on global sensitivity analysis methods. Uncertainty management in simulation-optimization of complex systems: algorithms and applications, pages 101–122, 2015

    2015

  11. [19]

    A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for the capacitated vehicle routing problem

    Imdat Kara, Gilbert Laporte, and Tolga Bektas. A note on the lifted Miller–Tucker–Zemlin subtour elimination constraints for the capacitated vehicle routing problem. European Journal of Operational Research, 158(3):793–795, November 2004. ISSN 0377-2217. doi: 10.1016/ S0377-22...

    2004

  12. [20]

    Christopher Beck

    Anton Korikov and J. Christopher Beck. Objective-Based Counterfactual Explanations for Linear Discrete Optimization. In Andre A. Cire, editor, Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 18–34, Cham, 2023. Springer Nature Swit...

    2023 doi

  13. [21]

    Christopher Beck

    Anton Korikov, Alexander Shleyfman, and J. Christopher Beck. Counterfactual Explana- tions for Optimization-Based Decisions in the Context of the GDPR. In Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence , pages 4097–4103, Montreal, Canada...

    2021 doi

  14. [22]

    Counterfactual Explanations for Linear Optimization, May 2024

    Jannis Kurtz, Ilker Birbil, and Dick den Hertog. Counterfactual Explanations for Linear Optimization, May 2024. URL http://arxiv.org/abs/2405.15431. arXiv:2405.15431

    2024 arXiv

  15. [23]

    Explainable AI: A Review of Machine Learning Interpretability Methods

    Pantelis Linardatos, Vasilis Papastefanopoulos, and Sotiris Kotsiantis. Explainable AI: A Review of Machine Learning Interpretability Methods. Entropy, 23(1):18, January 2021. ISSN 1099-4300. doi: 10.3390/e23010018. URL https://www.mdpi.com/1099-4300/23/1/18. Number: 1 Publish...

    2021 doi

  16. [24]

    A Unified Approach to Interpreting Model Predictions, Novem- ber 2017

    Scott Lundberg and Su-In Lee. A Unified Approach to Interpreting Model Predictions, Novem- ber 2017. URL http://arxiv.org/abs/1705.07874. arXiv:1705.07874

    2017 arXiv

  17. [25]

    Xiang Wang, Y

    Dang Minh, H. Xiang Wang, Y . Fen Li, and Tan N. Nguyen. Explainable artificial intelli- gence: a comprehensive review. Artificial Intelligence Review, 55(5):3503–3568, June 2022. ISSN 1573-7462. doi: 10.1007/s10462-021-10088-y. URL https://doi.org/10.1007/ s10462-021-10088-y

    2022 doi

  18. [26]

    Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer series in Operations Research and Financial Engineering. Springer, New York, NY , 2. ed. edition, 2006

    2006

  19. [27]

    Operational research: methods and applications

    Fotios Petropoulos, Gilbert Laporte, Emel Aktas, Sibel A Alumur, Claudia Archetti, Hayriye Ayhan, Maria Battarra, Julia A Bennell, Jean-Marie Bourjolly, John E Boylan, et al. Operational research: methods and applications. Journal of the Operational Research Society , 75(3): 4...

    2024

  20. [28]

    Where are the hard knapsack problems? Computers & Operations Research, 32(9):2271–2284, September 2005

    David Pisinger. Where are the hard knapsack problems? Computers & Operations Research, 32(9):2271–2284, September 2005. ISSN 0305-0548. doi: 10.1016/j.cor.2004.03.002. URL https://www.sciencedirect.com/science/article/pii/S030505480400036X

    2005 doi

  21. [29]

    Saman Razavi, Anthony Jakeman, Andrea Saltelli, Clémentine Prieur, Bertrand Iooss, Emanuele Borgonovo, Elmar Plischke, Samuele Lo Piano, Takuya Iwanaga, William Becker, Stefano Tarantola, Joseph H. A. Guillaume, John Jakeman, Hoshin Gupta, Nicola Melillo, Giovanni Rabitti, Vin...

    2021

  22. [30]

    Why Should I Trust You?

    Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. "Why Should I Trust You?": Explain- ing the Predictions of Any Classifier. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 1135–1144, San Francisco Cali- fornia...

    2016

  23. [31]

    Improving LIME Robustness with Smarter Locality Sampling, March 2021

    Sean Saito, Eugene Chua, Nicholas Capel, and Rocco Hu. Improving LIME Robustness with Smarter Locality Sampling, March 2021. URL http://arxiv.org/abs/2006.12302. arXiv:2006.12302 [cs, stat]

    2021 arXiv

  24. [32]

    Machine learning augmented branch and bound for mixed integer linear programming

    Lara Scavuzzo, Karen Aardal, Andrea Lodi, and Neil Yorke-Smith. Machine learning augmented branch and bound for mixed integer linear programming. Mathematical Programming, pages 1–44, 2024

    2024

  25. [33]

    Shankaranarayana and Davor Runje

    Sharath M. Shankaranarayana and Davor Runje. ALIME: Autoencoder Based Approach for Local Interpretability, September 2019. URL http://arxiv.org/abs/1909.02437. arXiv:1909.02437

    2019 arXiv

  26. [34]

    Lectures on parametric optimization: An introduction

    Georg Still. Lectures on parametric optimization: An introduction. Optimization Online, page 2, 2018

    2018

  27. [35]

    Statistical stability indices for LIME: obtaining reliable explanations for Machine Learning models

    Giorgio Visani, Enrico Bagli, Federico Chesani, Alessandro Poluzzi, and Davide Capuzzo. Statistical stability indices for LIME: obtaining reliable explanations for Machine Learning models. Journal of the Operational Research Society , 73(1):91–101, January 2022. ISSN 0160-5682...

    2022

  28. [36]

    Harvey M. Wagner. Global Sensitivity Analysis. Operations Research, 43(6):948–969, 1995. ISSN 0030-364X. URL https://www.jstor.org/stable/171637. Publisher: INFORMS

    1995

  29. [37]

    Integer programming

    Laurence A Wolsey. Integer programming. John Wiley & Sons, 2020

    2020

  30. [38]

    Deterministic Local Interpretable Model- Agnostic Explanations for Stable Explainability

    Muhammad Rehman Zafar and Naimul Khan. Deterministic Local Interpretable Model- Agnostic Explanations for Stable Explainability. Machine Learning and Knowledge Extrac- tion, 3(3):525–541, September 2021. ISSN 2504-4990. doi: 10.3390/make3030027. URL https://www.mdpi.com/2504-4...

    2021 doi

  31. [39]

    Why Should You Trust My Explanation?

    Yujia Zhang, Kuangyan Song, Yiming Sun, Sarah Tan, and Madeleine Udell. "Why Should You Trust My Explanation?" Understanding Uncertainty in LIME Explanations, June 2019. URL http://arxiv.org/abs/1904.12991. arXiv:1904.12991

    2019 arXiv

  32. [40]

    S-LIME: Stabilized-LIME for Model Explanation

    Zhengze Zhou, Giles Hooker, and Fei Wang. S-LIME: Stabilized-LIME for Model Explanation. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 2429–2438, Virtual Event Singapore, August 2021. ACM. ISBN 978-1-4503-8332-5. doi: 10.1145/3447...

    2021

  33. [2217]

    URL https://www.sciencedirect.com/science/ article/pii/S0377221715005469

    doi: 10.1016/j.ejor.2015.06.032. URL https://www.sciencedirect.com/science/ article/pii/S0377221715005469

    2015 doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.