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TACIT: Optimization Models that Learn from Their Mistakes

Maxime Bouscary, Marco Molinaro, Sirui Li, Saurabh Amin, Ishai Menache, Konstantina Mellou

arXiv:2609.38434v1math.OCcs.AI

TL;DR

Optimization models can omit tacit operational requirements, and sparse user corrections may fit many possible repairs. TACIT combines LLM-proposed structural changes with optimization-derived and calibrated repairs; across 38 scenarios, it repairs more cases than the best baseline.

  • Problem

    Tacit operational requirements can be missing from optimization formulations, while sparse corrections may be consistent with many possible repairs.

    Source: §1 · pp. 1–2

  • Method

    TACIT combines LLM-proposed structural templates calibrated by optimization with optimization-derived cuts that an LLM contextualizes into modeling constraints.

    Source: §1 · p. 2, §4.2 · p. 7

  • Results

    78.9% of 38 misspecification scenarios were repaired by TACIT, versus 60.5% for the best baseline.

    Source: §1 · p. 3

  • Takeaways & Limitations

    User corrections provide sufficient signal to systematically improve optimization models, a step toward models that learn from their mistakes.

    Source: §6 · p. 10

    The problem setup assumes observed target solutions are optimal under the unknown target formulation, though some experiments relax this assumption.

    Source: §3 · p. 4

Abstract

from arXiv · show

Real-world optimization problems are difficult to model accurately because many objectives and constraints reside in domain experts' tacit knowledge, making them hard to formalize. As a result, optimization models often contain miscalibrated objectives, missing constraints, or omitted decision variables, leading to solutions that fail to reflect operational realities. We address this challenge by automatically repairing misspecified formulations using historical data consisting of past solutions and subsequent user overrides. Traditional approaches such as inverse optimization and constraint learning tend to overfit sparse data and produce complex formulations. Our central idea is to combine the reasoning capabilities and prior knowledge of LLMs with the formal grounding provided by optimization. We realize this idea through two complementary paradigms. Top-down, an LLM proposes structural repairs, including new constraints and variables, whose numerical parameters are calibrated and validated through optimization. Bottom-up, optimization infers cuts from observed decisions, which the LLM contextualizes into interpretable, generalizable modeling constraints. We evaluate our approach on 38 misspecification scenarios spanning nine classes of optimization problems, several drawn from real-world applications, and show that TACIT can repair 78.9% of them (vs. 60.5% for the best baseline).

1 Introduction

Optimization formulations can miss tacit operational requirements, so user corrections may be consistent with many possible repairs. TACIT combines LLM reasoning and optimization to propose interpretable structural changes and calibrate them against feedback.

1 Introduction

  • Tacit domain knowledge can leave formulations with misaligned objectives, missing constraints, or omitted variables, producing decisions that fail to reflect operational realities.LLM-generated formulations can also remain incomplete when requirements are absent from the problem description.Source: §1 · pp. 1–2
  • Sparse solution corrections are indirect and non-identifying: users correct decisions rather than diagnose model errors, and many formulations can explain the same correction.Local fixes may fit a few corrections without capturing the underlying rule.Source: §1 · p. 2
  • TACIT combines optimization-derived cuts contextualized by an LLM with LLM-proposed structural templates calibrated through optimization.Repairs can add or remove constraints, introduce variables, and modify objectives jointly.Source: §1 · pp. 2–3
  • TACIT uses sparse offline user-corrected solutions without querying users or asking them to diagnose model errors.Its semantically meaningful repairs aim to remain interpretable and generalize beyond observed feedback.Source: §1 · p. 3
  • 78.9% of 38 misspecification scenarios were repaired by TACIT, versus 60.5% for the best baseline.The benchmark spans nine classes of optimization problems, including real-world applications; recovered formulations generalize to unseen instances.Source: §1 · p. 3

2 Related Work

Related work learns from decisions, preferences, or model specifications, but these approaches differ from TACIT in the kinds of repairs they represent and the evidence they require.

2 Related Work

  • Data-driven customization learns how users modify solutions, while preference models encode behavior in added variables and constraints that lack real-world meaning; TACIT instead recovers an explicit formulation for human inspection.Prior methods include affine mappings, regression and decision trees, and richer routing and scheduling predictors [Abdellaoui et al., 2026].Source: §2 · p. 3
  • Inverse optimization and model-learning methods estimate objectives or feasible regions from decisions, but do not naturally represent structural repairs requiring new variables coordinated with constraints or objective terms.This literature motivates TACIT’s numerical calibration components while highlighting identifiability challenges.Source: §2 · pp. 3–4
  • Autoformulation methods assume intended requirements appear in the specification, while diagnostic repair relies on explicit specifications, infeasibility, or predefined checks.TACIT targets requirements absent from both the formulation and its specification, using downstream behavior as evidence.Source: §2 · p. 4

3 Problem definition

The paper treats repair as finding a modified formulation that explains observed target decisions while avoiding the initial decisions. Its setup allows errors in objectives, constraints, and variables, including auxiliary structure.

3 Problem definition

  • A formulation maps instance data to an optimization problem defined by an objective and feasible set, and the target may differ from the initial model in objectives, constraints, variables, or any combination.The target formulation is the unknown ground truth governing decisions.Source: §3 · p. 4
  • In the vehicle-routing example, repair coordinates auxiliary variables, linking constraints, a new objective term, and estimation of its coefficient.Only primary variables—actionable decisions reported by practitioners—are observed in feedback; auxiliary variables may be introduced by the target formulation.Source: §3 · p. 4
  • Observed data consist of instances, initial solutions, and target solutions projected onto the primary variables; the setup assumes target optimality, an assumption relaxed in some experiments.It also assumes the target is strictly preferred whenever it differs from the initial solution.Source: §3 · pp. 4–5
  • Repair minimizes separation loss on observed data and assesses generalization on held-out instances, while favoring parsimonious, interpretable transformations.The loss measures the fraction of instances the candidate formulation fails to separate: target solutions should be feasible and optimal while initial solutions are not.Source: §3 · p. 5

4 Methodology

TACIT iteratively proposes and evaluates formulation transformations through two complementary pathways, then calibrates candidate objectives and incorporates improvements into an incumbent model.

4 Methodology

  • TACIT maintains an incumbent formulation, generates candidates through bottom-up and top-down pathways, evaluates them using separation loss before and after objective calibration, and adopts improving candidates.The process repeats until the computational budget is exhausted.Source: §4 · pp. 5–6

4.1 Bottom-up: cuts and contextualization

  • Bottom-up cut search restricts inequalities to affine functions of instance data and family-level weights, then favors cuts that eliminate incumbent solutions while remaining sparse.Sparsity regularization is intended to limit overfitting, improve interpretability, and promote generalization; an LLM polishes the cuts and extrapolates them into indexed constraint families.Source: §4.1 · p. 6

4.2 Top-down: templates and calibration

  • Top-down, an LLM proposes a structural template that may add auxiliary variables, constraints, or domain changes, leaving learnable parameters unspecified for optimization to calibrate.Calibration preserves observed targets while separating incumbent solutions; templates that cannot feasibly lift every target are discarded.Source: §4.2 · p. 7

4.3 Objective calibration

  • Objective calibration keeps the feasible set fixed while learning an objective that makes target solutions nearly optimal and improves their objective relative to the current model solution.It allows some target suboptimality to accommodate potentially suboptimal observed targets.Source: §4.3 · p. 7

5 Numerical Results

TACIT is evaluated on a benchmark of diverse formulation errors and improves repair across LLM backbones. Its gains are strongest when repairs require reconstructing missing model structure, and results also examine component contributions, data availability, imperfect feedback, and interpretability.

5.1 Experimental Setup

  • The benchmark contains 38 LP and MILP scenarios across nine problem classes, covering errors in objectives, constraints, variables, and domains.Scenarios combine classical problems and realistic applications; some deliberately combine multiple misspecifications.Source: §5.1 · pp. 7–8

5.2 Results

  • TACIT improves every test metric across all backbones; with GPT-5.6-sol, it adds 17.4 and 18.4 percentage points and reaches 64.2% recovery and 78.9% recoverable rate.Table 1 compares TACIT with LLM-only repair and AOP across backbones.Source: Table 1 · p. 8, §5.2 · p. 8

5.1 Experimental Setup

  • Each method is run five times on 128 training instances and evaluated on 256 disjoint test instances; TACIT uses five framework iterations per run.The compared methods include the initial formulation, AOP, and an LLM-only baseline.Source: §5.1 · p. 8, §5.2 · p. 8

5.2 Results

  • On missing variables, test optimality rises 0.30 →0.44 with DeepSeek-V4-Flash and 0.46 →0.74 with Kimi-K2.6; on global constraints, it rises 0.21 →0.47 and 0.35 →0.69.TACIT's gains are concentrated in repairs requiring reconstruction of structure absent from the initial formulation.Source: §5.2 · pp. 8–9, Figure 2 · p. 9
  • With DeepSeek-V4-Flash, adding bottom-up inference to the top-down pathway raises recovery and recoverability from 41.1% and 57.9% to 45.3% and 63.2%.The strongest pure approach reaches 21.1% recovery and 31.6% recoverability; adding optimization-based calibration to LLM-generated templates raises those rates to 41.1% and 57.9%.Source: Table 2 · p. 9, §5.2 · p. 9
  • Top-down inference is stable across training sizes, while bottom-up improves by 12.6 percentage points from 32 to 256 instances; TACIT maintains strong performance across sizes.The comparison is reported in Table B.2.Source: §5.2 · p. 9
  • Allowing up to 5% suboptimality in observed decisions leaves test feasibility nearly unchanged at 0.78 to 0.79, while test recovery decreases by 4.8 percentage points.This setting evaluates target-optimal behavior even when TACIT may never observe a target-optimal solution.Source: §5.2 · p. 9
  • TACIT returns explicit, inspectable formulation modifications, including recognizable constraints, auxiliary variables, and structured objective terms across examined examples.The examples include exact target recovery in TP-Scenario2 and an objective repair using auxiliary variables in MCF-Scenario2.Source: §5.2 · p. 10

6 Conclusion

The conclusion presents TACIT as an end-to-end approach to formulation repair that combines LLM-guided semantic reasoning with optimization-based search and validation. Across diverse errors, it improves recovery over baselines, especially for structurally demanding cases, and remains effective with limited and imperfect feedback.

6 Conclusion

  • TACIT combines LLM-guided semantic reasoning with optimization-based search and validation to address formulation repair when the error source is unknown and model logic may be absent.Sparse feedback can be consistent with many possible repairs.Source: §6 · p. 10
  • Across diverse formulation errors, TACIT improves recovery over baselines, especially on structurally demanding misspecifications, while remaining effective with limited and imperfect feedback.Source: §6 · p. 10
  • The results show that user corrections provide sufficient signal to systematically improve optimization models, a step toward models that learn from their mistakes.Source: §6 · p. 10

A.1 Limitation of embedded ML models for formulation correction

Embedded machine-learning correction models cannot represent every formulation repair, even with added constraints. The section illustrates this limitation through a missing nonlinear objective term and a family of instances.

A.1 Limitation of embedded ML models for formulation correction

  • Prior approaches embed a learned map from an incorrect solution to a target solution inside optimization [Hewitt and Frejinger, 2020, Bayani et al., 2024, Abdellaoui et al., 2026].The map is learned so the embedded model's optimal solutions ideally match those of the target formulation for every instance.Source: §A.1 · p. 13
  • Some approaches relax the fixed-point equality with a Lagrangian penalty, while this analysis retains the equality to avoid an extra parameter.The penalty is λ∥x − f(x)∥∞; the extra parameter is λ.Source: §A.1 · p. 13
  • The embedded approach cannot represent every correction, including a missing nonlinear objective term, even when auxiliary variables, constraints, and added constraints are allowed.The nonlinear term can be represented in a MILP, but restricting the original variables' feasible set cannot generally recover it; TACIT's templates can handle such corrections.Source: §A.1 · p. 13
  • Theorem 1 gives an initial and target formulation for which no correction function f and added constraints K reproduce the target's optimal solutions for every instance.The target formulation includes the optimal solutions of the initial formulation only after introducing auxiliary variables and constraints together with the missing objective term.Source: §A.1 · p. 13
  • The proof's target formulation is a MILP, obtained by adding auxiliary variables and constraints alongside the missing objective term.The passage also specifies y1 and y2 as binary variables.Source: §A.1 · p. 13
  • For the scalar instance family, the embedded model's feasible set S is independent of d, so its objective is min over S of dx.The proof defines S through the fixed-point condition x = f(x) and membership in K.Source: §A.1 · pp. 13–14
  • The target optima are {1}, {0}, and {−1} at d = −2, 0, and 2, respectively.These distinct singleton optimal sets are given for the three proof instances.Source: §A.1 · p. 14
  • Matching the target at d = −2 and d = 2 forces 1 and −1 into S, making every point in S optimal at d = 0, contrary to the target's singleton optimum.This contradiction establishes that no such f and K exist.Source: §A.1 · p. 14

B.1 Additional Results

The additional results describe tests of recovery as training data grows, performance across misspecification types, and TACIT on suboptimal targets, alongside an AOP penalty trade-off.

B.1 Additional Results

  • Table B.1 reports test optimality rates by misspecification type for the DeepSeek-V4-Flash backbone, but the supplied passage gives no cell values.The table's scope is per-type test optimality; no outcome comparison can be inferred from its caption alone.Source: Table B.1 · p. 14
  • Table B.2 compares final-formulation recovery rates across numbers of training instances per scenario, without supplying the rates here.The caption identifies the comparison but does not establish how recovery changes as the training set grows.Source: Table B.2 · p. 14
  • Table B.3 evaluates TACIT with DeepSeek-V4-Flash on datasets with suboptimal targets; targets are generated within a relative gap of up to 5%.When the initial solution has a feasible lifting, the target must still beat it by a relative margin of 0.005.Source: §B.1 · pp. 14–15, Table B.3 · p. 15
  • AOP feasibility rises from 0.51 to 0.62 and 0.52 to 0.64 as penalty c increases, while optimality falls from 0.13 to 0.01 and 0.15 to 0.04, with no ground-truth recovery.The passage attributes the trade-off to pulling solutions toward typical plans rather than recovering the rule.Source: §B.1 · p. 15

B.2 Example Repaired Formulation

The examples show bottom-up cut refinement and top-down templates recovering or approximating target formulations, while the appendix also outlines TACIT's iterative evaluation and search procedures.

B.2.1 Bottom-Up

  • Cut refinement recovers the missing constraint exactly in two cases: it rewrites a binary implication in GPU and generalizes a day-specific WA cut across days.The raw WA cut has test optimality 0, while the refined cut preserves employees 3 and 5 as anchors and recovers the ground-truth constraint.Source: §B.2.1 · p. 15

B.2.2 Top-Down

  • Three LLM-proposed templates are fitted into concrete corrections: a route-cost variable needs no constants, TP calibration finds batch size L = 2, and CFL calibration sets facility 4's cap to 0.5.Other CFL caps are set to 1, leaving the original constraint unchanged.Source: §B.2.2 · p. 15, Table B.6 · p. 16

B.2.3 Final formulation samples

  • In TP Scenario 2, TACIT recovers the target formulation up to a big-M constant of 24 rather than 1000.Source: §B.2.3 · p. 16
  • The objective combines the target objective scaled by 4.32 with a small tie-breaker based on the number of requests.Source: §B.2.3 · p. 17
  • In MCF Scenario 2, TACIT recovers an equivalent formulation; its final inequality is valid but redundant because feasible flows are nonnegative.Source: §B.2.3 · p. 17

C.1 Routine and Subroutines

  • The framework keeps one incumbent, evaluates candidate formulations before and after objective calibration, and searches cuts on the incumbent plus accumulated cuts while proposing templates from the initial model.An improving candidate becomes the incumbent; the two searches repeat until the budget is exhausted.Source: §C.1 · p. 17

C.2 Hyperparameters

  • Cut search varies the penalty, obtains candidate cuts, has the LLM polish them, then evaluates accumulated cuts and extrapolated constraint families.The implementation limits removed constraints and previously proposed templates to avoid repetitions, and solves MIPs with a 30-second per-instance limit.Source: §C.2 · p. 18

C.3 Compute and runtime

The compute and runtime section reports model roles, token use, and median run times; the accompanying template-search steps use mismatched solution pairs and optimization calibration.

C.3 Compute and runtime

  • For N = 128 instances, median runtime ranges from 2 minutes for calibration alone to 2.1 hours for the full framework, versus 0.2 hours for the LLM-only baseline.Template search takes 0.8 hours and cuts with refinement 1.3 hours.Source: §C.3 · p. 18
  • The large model generates templates and refines cuts while the small model handles simpler transformation tasks; averaged across three large models, runs use 154k large-model and 346k small-model tokens.The LLM-only baseline uses an average of 210k tokens per run, all from the large model.Source: §C.3 · p. 18
  • Template search samples mismatched model-target solution pairs, requests templates from the LLM, calibrates feasible constraints, and evaluates each instantiated template.Source: §C.3 · p. 19

D Benchmark Applications and Scenarios

The benchmark comprises 38 scenarios over nine base problems, each pairing an initial formulation with a target formulation that generated observed solutions.

D Benchmark Applications and Scenarios

D.1 Capacitated facility location (CFL)

The appendix specifies capacitated facility-location misspecifications and benchmark taxonomy, then describes cut search, template completion, and objective calibration procedures.

D.1 Capacitated facility location (CFL)

  • CFL scenarios test a 50-unit service radius and capacity targets of 0.8 or 1.2 times facility capacity on instances with 6–8 facilities and 8–14 customers.The initial model uses full capacity; the 0.8 target tightens it, while the 1.2 target allows more capacity than the initial constraint.Source: §D.1 · p. 19

D.2 Other problems

  • The benchmark taxonomy covers 38 scenarios across nine problem classes; extra marks identify constraints that cut off target-feasible or target-optimal solutions.The remaining eight problems and their scenarios are slated for release with the open-source benchmark.Source: Table D.1 · p. 20, §D.2 · p. 20

E.1 Cut Search

  • Cut Search groups variables into instance-independent families for shared weights and learns sparse cuts that are valid for targets while excluding current-model solutions.In the vehicle-routing example, each of five shared vehicles defines a family; the sparsity penalty trades separation for interpretability and reduced overfitting.Source: §E.1 · p. 21

E.2 Template completion

  • Template completion calibrates weights to preserve observed targets while separating current-model solutions, evaluating excluded solutions under their best new-variable completion.Constraint generation is used, and templates unable to feasibly lift every target are discarded.Source: §E.2 · pp. 21–22

E.3 Objective Calibration

  • Objective Calibration assigns affine, data-dependent coefficients per variable family and balances lifted-target near-optimality against separation from current-model solutions.The parameter λ controls this tradeoff, and the calibration uses constraint generation.Source: §E.3 · p. 22
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