TACIT: Optimization Models that Learn from Their Mistakes

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

math.OCSep 29, 2026

TACIT: Optimization Models that Learn from Their Mistakes

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

Organizations: Massachusetts Institute of Technology, Cambridge, MA, USA · Microsoft Research, Redmond, WA, USA

Abstract

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).

Figures & tables

		Training				Test
	Model	$\mathcal{L}$ $\downarrow$	sep. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	recovery $\uparrow$	recoverable $\uparrow$
AOP	AOP linear ( $c\!=\!\!16$ )	0.94	0.0%	0.57	0.07	0.57	0.04	0.0%	0.0%
AOP	AOP tree ( $c\!=\!\!16$ )	0.93	0.0%	0.58	0.10	0.57	0.07	0.0%	0.0%
LLM-only	DeepSeek-V4-F	0.64	18.9%	0.57	0.37	0.57	0.37	19.5%	31.6%
	Kimi-K2.6	0.56	29.5%	0.66	0.46	0.66	0.47	30.0%	36.8%
	GPT-5.6-sol	0.40	47.9%	0.74	0.59	0.75	0.59	46.8%	60.5%

Table 1: TACIT against LLM baselines under different LLM backbone models.

Figure 2: Change in optimality rate induced by TACIT under different LLM backbone models.

	Training				Test
	$\mathcal{L}$ $\downarrow$	sep. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	recovery $\uparrow$	recoverable $\uparrow$
Obj. calibration only	0.76	5.8%	0.41	0.23	0.41	0.23	5.3%	5.3%
Cuts	0.35	33.2%	0.57	0.33	0.54	0.27	14.7%	21.1%
Cuts + contextualize	0.35	33.2%	0.59	0.35	0.57	0.30	17.9%	26.3%
Template	0.60	22.6%	0.56	0.39	0.56	0.39	21.1%	31.6%
Template + complete	0.31	47.9%	0.77	0.59	0.77	0.58	41.1%	57.9%

Table 2: Ablation of TACIT components using DeepSeek-V4-Flash. All variants share preprocessing and objective calibration. Template only uses LLM-guessed constants.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

	Objective		Constraints			Variables		Constants
	Miscal.	Missing	Local	Global	Extra	Missing	Domain	Missing	All
Instances	7	6	10	15	5	10	2	16	38
Initial model	0.24	0.22	0.20	0.11	0.17	0.18	0.01	0.13	0.17
Calib. only	0.49	0.26	0.20	0.11	0.28	0.21	0.01	0.13	0.23
Cuts	0.52	0.20	0.41	0.07	0.40	0.12	0.00	0.06	0.27
Cuts + contextualize	0.52	0.22	0.43	0.11	0.41	0.13	0.00	0.08	0.30

Appendix

Table B.1: Test optimality rate per misspecification type, with DeepSeek-V4-Flash as LLM backbone.

Training instances	32	64	128	256
Calibration	4.7%	5.8%	5.3%	6.3%
Cuts + contextualize	7.9%	13.7%	17.9%	20.5%
Template + complete	36.8%	43.2%	41.1%	40.5%
All	36.3%	45.8%	45.3%	43.2%

Appendix

Table B.2: Recovery rate of the final formulation against the number of training instances per scenario.

	Training			Test
Dataset	$\mathcal{L}$ $\downarrow$	feas. $\uparrow$	opt. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	rec. % $\uparrow$	recoverable $\uparrow$
Optimal targets	0.19	0.79	0.64	0.78	0.61	45.3%	63.2%
Suboptimal targets	0.19	0.81	0.58	0.79	0.53	40.5%	52.6%

Appendix

Table B.3: Performance of TACIT with DeepSeek-V4-Flash backbone on datasets with suboptimal target solutions.

	Training				Test
Model	$\mathcal{L}$ $\downarrow$	sep. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	feas. $\uparrow$	opt. $\uparrow$	rec. % $\uparrow$	recoverable $\uparrow$
Initial	0.85	0.0%	0.46	0.17	0.46	0.17	-	-
AOP linear ( $c{=}1$ )	0.90	0.0%	0.50	0.15	0.51	0.13	0.0%	0.0%
AOP linear ( $c{=}16$ )	0.94	0.0%	0.57	0.07	0.57	0.04	0.0%	0.0%
AOP linear ( $c{=}256$ )	0.98	0.0%	0.61	0.02	0.62	0.01	0.0%	0.0%
AOP tree ( $c{=}1$ )	0.91	0.0%	0.52	0.17	0.52	0.15	0.0%	0.0%

Appendix

Table B.4: AOP methods across different values of the penalty parameter

c

GPU Scenario 3: demand 0 can only be satisfied whenever all other demands are satisfied.
Missing constraint:	$z_{0}\leq z_{i},\ \forall i\in I,\ i>0$
Raw cut from ( P cut ):	$0.9999\,z_{0}-0.0001\,z_{1}-0.0001\,z_{2}-0.9996\,z_{3}-0.0001\,z_{4}\leq 0$
LLM refined cut:	$z_{0}\leq z_{i},\ \forall i\in I,\ i>0$
WA Scenario 3: employees 3 and 5 can never work on the same day.
Missing constraint:	$x_{3,d}+x_{5,d}\leq 1,\ \forall d\in D$
Raw cut from ( P cut ):	$0.5\,x_{3,7}+0.5\,x_{5,7}-0.5\leq 0$

Appendix

Table B.5: Example of cut processing on two scenarios.

	Template (raw)	Template (fitted)
CVRP Scenario 2: the ground truth minimizes the longest route instead of the total distance.
Variables:	$z\in\mathbb{R}_{\geq 0}$	$z\in\mathbb{R}_{\geq 0}$
Constants:	–	–
Constraints:	$\displaystyle\sum_{i\in N}\sum_{\begin{subarray}{c}j\in N\\ j\neq i\end{subarray}}c_{i,j}\,x_{i,j,k}\leq z,\ \forall k\in K$	$\displaystyle\sum_{i\in N}\sum_{\begin{subarray}{c}j\in N\\ j\neq i\end{subarray}}c_{i,j}\,x_{i,j,k}\leq z,\ \forall k\in K$
TP Scenario 3: goods are shipped in batches of a fixed size.
Variables:	$y_{i,j}\in\mathbb{Z},\ \forall i\in I,j\in J$	$y_{i,j}\in\mathbb{Z},\ \forall i\in I,j\in J$

Appendix

Table B.6: Example of template processing on three scenarios. Templates proposed by the LLM, before (raw) and after fitting the new constants with ( P constr ). Fitted values are shown in blue.

		Objective		Constraints			Variables		Constants
	Scenario	Miscalibrated	Missing	Local	Global	Extra	Missing	Domain	Missing
CFL	S1	$\square$	$\square$	$\square$	$\blacksquare$	$\square$	$\blacksquare$	$\square$	$\blacksquare$
	S2	$\square$	$\square$	$\square$	$\blacksquare$	$\square$	$\square$	$\square$	$\blacksquare$
	S3	$\square$	$\square$	$\square$	$\blacksquare$	$\blacksquare$	$\square$	$\square$	$\blacksquare$
	S4	$\blacksquare$	$\square$	$\square$	$\square$	$\square$	$\square$	$\square$	$\square$
	S5	$\blacksquare$	$\blacksquare$	$\square$	$\square$	$\square$	$\blacksquare$	$\square$	$\square$

Appendix

Table D.1: Taxonomy of scenario misspecifications over the 38 benchmark scenarios spanning 9 problems: CFL (capacitated facility location), GPU (GPU allocation), MIS (maximum independent set), MCF (minimum-cost flow), MPP (multi-period production planning), SC (set cover), TP (transportation), CVRP (capacitated vehicle routing), WA (workforce assignment). A constraint is local when the restriction it adds involves fixed indices (e.g.

\sum_{j}\left(x_{3,j}+x_{5,j}\right)\leq 1

) and global when it is stated over generic indices (e.g.

\sum_{i,j}x_{i,j}\leq 1

). Extra marks a constraint of the initial model that cuts off target-feasible or target-optimal solutions. Constraints that only define an auxiliary variable are counted under missing variable rather than as a missing constraint, and big-

M

bounds are not counted as missing constants.

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