LLMOPT: Learning to Define and Solve General Optimization Problems from Scratch
Authors: Caigao JIANG
Xiang Shu
Hong Qian
Xingyu Lu
JUN ZHOU
Aimin Zhou
Yang Yu
Venue: ICLR 2025 Poster (2025)
Keywords: Optimization
Optimization Problem Formulation
Problem Definition
Foundation Model
Primary area: optimization

Abstract: Optimization problems are prevalent across various scenarios. Formulating and then solving optimization problems described by natural language often requires highly specialized human expertise, which could block the widespread application of optimization-based decision making. To automate problem formulation and solving, leveraging large language models (LLMs) has emerged as a potential way. However, this kind of approach suffers from the issue of optimization generalization. Namely, the accuracy of most current LLM-based methods and the generality of optimization problem types that they can model are still limited. In this paper, we propose a unified learning-based framework called LLMOPT to boost optimization generalization. Starting from the natural language descriptions of optimization problems and a pre-trained LLM, LLMOPT constructs the introduced five-element formulation as a universal model for learning to define diverse optimization problem types. Then, LLMOPT employs the multi-instruction tuning to enhance both problem formalization and solver code generation accuracy and generality. After that, to prevent hallucinations in LLMs, such as sacrificing solving accuracy to avoid execution errors, the model alignment and self-correction mechanism are adopted in LLMOPT. We evaluate the optimization generalization ability of LLMOPT and compared methods across six real-world datasets covering roughly 20 fields such as health, environment, energy and manufacturing, etc. Extensive experiment results show that LLMOPT is able to model various optimization problem types such as linear/nonlinear programming, mixed integer programming, and combinatorial optimization, and achieves a notable 11.08% average solving accuracy improvement compared with the state-of-the-art methods. The code is available at https://github.com/caigaojiang/LLMOPT.

Published as a conference paper at ICLR 2025
LLMOPT: LEARNING TO DEFINE AND SOLVE GEN- ERAL OPTIMIZATION PROBLEMS FROM SCRATCH
Caigao Jiang"; Xiang Shu Hong Qian Jun Zhou", Aimin Zhou², Yang Yu*
Xingyu Lu"
East China Normal University, China
Ant Group, China Nanjing University, China
shux@stu. ecnu edu en, {hqi an, amzhouf@cs ecnu. edu. {caigao. jeg, s sing. 1xy, jun. zhou (acouuuunaaaaan jun]eantgroup. com, yuyenju. edu.cn
ABSTRACT
Optimization problems are prevalent across various scenarios. Formulating and then solving optimization problems described by natural language often requires highly specialized human expertise, which could block the widespread applica- tion optimization- baaed decision making. To automate problem formulation and solving, leveraging large language models (LLMs) has emerged as a potential way. However, this kind of approach suffers from the issue of optimization gener- alization. Namely, the accuracy of most current LLM-based methods and the gen- crality optimization problem types that they can model are still limited. In this paper, we propose unified learning-based framework called LLMOPT boost optimization generalization. Starting from the natural language descriptions of op- timization problems and pre-trained LLM, LLMOPT constructs the introduced five-element formulation as a universal model for learning to define diverse op- timization problem types. Then, LLMOPT employs the multi-instruction tuning to enhance both problem formalization and solver code generation accuracy and generality. After that, to prevent hallucinations in LLMs, such as sacrificing solv- ing accuracy to avoid execution errors, the model alignment and self-correction mechanism are adopted in I.MOPT. We evaluate the optimization generalization ability of LLMOPT and compared methods across six real-world datasets cov- ering roughly 20 fields such as health, environment, energy and manufacturing. etc. Extensive experiment results show that LLMOPT is able to model various optimization problem types such as linear/nonlinear programming, mixed integer programming, and combinatorial optimization, and achieves notable 11 11.08% av- erage solving accuracy improvement compared with the state-of-the-art methods. The code is available at uthttps://githtt www.ttttubccooccte wwtom//aagaa/// ang/ L.MOPT
INTRODUCTION
Optimization problems are widespread in a variety of scenarios, such as job scheduling (Brandi-
marte 1993), path planning (Hong et al. 2016
Li al. 2023), matching problem (Wang et al.
2020 reinforcement leaning (Qian & Yu 2021 revenue management (Trimborn et al. [018]] 8) and
2012 Qian et al. (025), etc. Although powerful solvers are available, formally defining domain-specific optimization problems from natural language descriptions and then developing the solver code are highly complex tasks. It requires specialized domain knowl- edge, expert participation, and significant time investment. For example, in finance, the portfolio problems (Chen al., (02T) often manage thousands of variables and complex constraints, such as full investment constraints, cardinality constraints and pre-assignment constraints, where expert intervention is crucial for achieving accurate (i.e., low simple regret) solutions due the complexity and specialization of optimization problems.
security games (Brown
With the rapid development of large language models (LLMs) like ChatGPT (Brown et al. 2020) and GPT-4 (OpenAl 2023), using LLMs to automatically formulate and solve optimization prob-
Both authors contribute equally to this research.
Corresponding authors.

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Published as a conference paper at ICLR 2025
Problem Description
Five-Element Formulation
An accounting firm employs part time workers and full time workers. Full time workers work hours per shift while part time workers work hours per shift. In addition, full time workers are paid $300 per shift while part time workers are paid S100 per shift, Currently, the accounting firm has project requiring 450 hours of labor. If the firm how many of each type of worker should be scheduled to minimize the total number of workers.
Sets
Sct of employee types: 5 (f.p)
Hours per shift: nty 8. hp
Wages per shift: Wy = 300, W, 100
Parameters
Total labor time required: T = 450
Budget: B = 15000
Varlables
Employees shifts: Xp
Objective
Minimize employee number: min xy xy Xp
budget $1000
Labor time onstraint: hyx/ + hp 2 T
Constraints
Budget constraint: wyzz Wpp S
Positive integer constraint: Xp E
Problem Description
Five-Element Formulation
Generate Solver Code
Generate Solver Code
PytRo, import ()
fran ewviron Inport
model Concrete/baa())
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medel.t. Constraint expr-3 Xof model Kip 590)
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modet.22 Constrasm (xpe-320 rodel.x_ 100 redel.x_p 15800)
model.c2 Construint (eqpee200) mxlil.... f. mxll..pp 15830)
Optinal Solution Found:
Incorrect
Optinal Solution Found:
Correct
Hanboe of Full Virm Shifta ( 37.5
Number of Full Tiim Shifta (x_E):
thamber Part-Tine Shifts (K.)): 37.5
Missnn
Number Part-7ine shifts (x_P):
Corect definiiion and
Potal
Eeployess:
constraint
Mamber Reployses:
Figure 1: An example of the proposed five-element formulation, which provides a structured defini- tion for general optimization problems. Using the five-element formulation as an intermediate step can lead to more accurate solver code and solution.
lems described by natural language is becoming increasingly attractive. Existing work on it can be roughly divided into prompt-based and learning-based methods. Pioneering studies like Chain-of- Expert (Xiao et al. (024) and OptiMUS (AhmadiTeshnizi et al.) (2024) explore prompt-based meth- ods, using LLM interfaces to extract key information and solve optimization problems. Although these methods show promise, recent advances in learning- based methods, such as LLaMoCo (Mal et al. 2024) and ORLM (Tang et al. 2024), could show even greater potential. ORLM (Tang et al., 2024 reveals that a fine-tuned 7B open-source model can outperform larger pre-trained models like GPT4, highlighting the value of learning-based methods for optimization tasks.
Although previous research has established the use of LLMs for formulating and solving optimiza- tion problems described by natural language, their optimization generalization remains limited, re- stricting broader applicability and deployment across diverse problems. Optimization generuliza- tion, which involves both accuracy and generality in assessing whether LLMs can effectively solve general optimization problems, remains a significant challenge. Specifically, accuracy refers to solv- ing problems correctly, and generality refers to the ability model and solve diverse optimization problem types across different task scenarios.
In this paper, we propose a unified learning-basse framework called LLMOPT to significantly boost the optimization generalization. Through focusing on improving both accuracy and generality to effectively define and solve a wide range of optimization problems, this work tries narrow the gap between methods and practical applications. Specifically, we propose the five-element formulation define optimization problems to improve the accuracy of solving, as illustrated in Figure [ To ensure the effecctivneess of learning, we design a data augmentation process and perform data label- ing, resulting in high-quality datasets. Leveraging these datasets, we implement multi-ing multi-instruction supervised fine-tuning to improve the LLM's accuracy in both defining and solving optimization problems, Additionally, we introduce model alignment to further enhance accuracy and reduce the risk of hallucinations. We also develop auto-testing process with self-correction mechanism, enabling the accurate and automated resolution of optimization problems. Finally, the optimiza- tion generalization ability of LLMOPT is extensively evaluated on six real-world optimization and operation problem datasets involving roughly 20 domains, including health, environment, energy and manufacturing, etc. The results show that LLMOPT effectively handles various optimization
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problems, such linear and nonlinear programming, mixed integer programming and combinato- rial optimization, etc. Ablation studies validate the contributions of problem definition and model alignment to improving accuracy. Notably, LLMOPT achieves average accuracy improvement of 11.08% over stale-of-the-at methods.
The consequent sections respectively introduce the related work, present the proposed LLMOPT, show and analysis the empirical results, discuss important issues, and finally conclude the paper.
2 RELATED WORK
LLMs for Formulating and Solving Optimization Problems. The NL4Opt competition (Rama-
monjison et al. 2021) encourages researchers to explore how LLMs can be leveraged to solve opti-
mization problems efficiently (Huang et al.) 2024 Tang et al. 2024 Xiao et al. 2024 AhmadiTesh nizi et al. 2224 Ma et al.) 2024). Related work can be divided into prompt-based and learning-basee methods. The prompt-based methods utilize existing LLM interfaces to automate the solving opti- mization problems. Chain-of-Fxpert (CoE) (Xiao et al. 2024) introduces an agent-based workflow, where specific tasks are assigned to L.M agents at each stage of reasoning, while OptiMUS (Ah-) madiTeshnizi et al. 2024) employs tailored prompts designed specifically for solving linear pro- gramming and mixed-integer linear programming problems. The learning-based based method remains relatively underexplored. LLaMoCo (Ma al. 2024) introduces an instruction tuning framework in code-to-code manner and demonstrates its efficacy through model fine-tuning. ORLM (Tang) jet al. 2024) proposed a semi-automated approach to generating synthetic training data during the instruction tuning phase, addressing the data-hungry nature of optimization modeling. Besides, Mamo (Huang et al. 2024) offers a benchmark for mathematical modeling with two optimization datasets, and introduces standardized process to generate solver code using LLMs.
LLMS as Components of Optimizers. Yang et al. (202bb) propose to use LLMs as optimizers, inspired by Bayesian optimization (Garnett 2023). One LLM iteratively optimizes prompts, which
are then used to guide another LLM in performing specific optimization tasks. LLMs can also
serve components within optimizer (Guo et al. 2024 Yang & Li 2023 Liu et al. 2024a
2023 2024b). Guo et al.] (024) uses LLMS to as the crossover and mutation operators in genetic algorithms, thereby finding high-quality prompts. When solving the traveling salesman problem. Liu iet al. (2024a) uses LLMs to generate new trajectories based on cxisting ones, iteratively optimizing
them to produce higher-quality solutions. [iu al. (2023) and Yang & Li (2023) apply LLMs to
specific multi-objective optimization tasks. Liu et al. (2024b) employs LLMs to simulate both the acquisition function and the surrogate model in the Bayesian optimization process.
METHODOLOGY
3.1 AN OVERVIEW OF THE PROPOSED LLMOPT FRAMEWORK
This work aims to achieve better optimization generalization of defining and solving the optimiza- tion problems through a learning-baae based approach. To this end, this paper proposes the LLMOPT framework, as illustrated in Figure 2 which comprises three key components: data, learning, and auto-testing. The following sections provide a detailed introduction: First, in Section 3.2 we in- troduce the five-element formulation to define general optimization problems well, and discuss the augmentation and labeling of training data. Next, in Section 3.3 we detail the multiiinstruction supervised fine-tuning (SFT) process and model alignment to enhance solving accuracy. Finally, in Section 3.4
we present the auto-testing process with self-correction.
3.2
DATA: DEFINING AND LABELING GENERAL OPTIMIZATION PROBLEMS
3.2.1
UNIVERSAT. FORMULATION TO DEFINE GENERAL. OPTIMIZATION PROBLEMS
To enable LLMs formulate more general optimization problems described in natural language, we propose the five-element, universal formulation to define optimization problems. Firstly, in mathematics, an optimization problem can be formally represented as the following expression:
min R f(z), s.t. G(x)= S c.
(1)
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(a) Data
Natural Language Tonnonoenoainn
(b) Learning
Pre-trained LLM
Data for Formulation
Data for Solving
Optimization
(p.s)
Expert
GPT-4
Expert
(p.
SFT
(f+)
Label S Element
by Expert
Generate 5-Element by GPT-4 & Label by Expert
Labelad Expert
LLM SFT
Wodere/kaaeane Lubuled Capat
5-Element
Desiibility cubrac
Froblem
Proteem Disinablly by Labrrdd by Dipert
KTO
Pret with &Dvahnbalily By Lebelee Expent
Generate Code by GPT-4 by Expert
Algnment
Expert
Formulation:
LLM FINAL
Solving:
Solver Code
Solver Code dimeure Desiatillty cuaderlly fopits Salver Deciability
Learning to Define
Learning to Generate Code
Probiem
(c) Auto-Testing
Self-correction
Errors
BO
Problem
LINAL
3
5-Element
LLM
tug
Code
Solver
Logs
LLM Solutionn
Results
(Defining)
(Solving)
Figure 2: The framework LLMOPT. Sub-figure (a) shows the data labeling process, where experts and GPT-4 work together to label both the five-element formulation and solver code. Sub-figure (b) shows the learning process, in which inlti-instruuction supervised fine-tuning and model alignment employed learn-to-define and generate code. Sub-figure (c) shows the auto-testing process with self-correction mechanism, which can define and solve optimization problems automatically.
where x = (I1, (2,... TD) T is the D-dimensional decision variable, X' C RPP is the feasible domain of æ, and the goal is minimize the objective function f X IR. The constraints represented by the vector-valued function G(x)): RD + IR" where G(xx) < c denotes a series of inequality constraints, and c (c1, c,.... Cm. is the vector of upper bounds for these constraints.
In practice, a vast majority of optimization problems can be formulated as shown in Formula T Based on it, the five-element formulation provides structured and intuitive way define optimiza- tion problems, retaining essential descriptions and making them more accessible and understand- able, even for L.Ms. As the name suggests, the five-element consist of five key components of an optimization problem: Sets, Parameters, Variables, Objective and Constraints. As shown in Figure [ the element Variables, Objective, and Constraints describe the decision variables æ, the objective function f(x), and the constraints G(æ) in Formula [ along with concise descriptions for each. Meanwhile, Sets and Parameters provide detailed information indices and descriptions, well the specific numerical values of the parameters involved in the objective function and constraints. Importantly, the five-element provides a more accurate formulation of the problem by mining and preserving key descriptions of the problem. For example, in Figure the positive integer constraint is implicit in the problem but is effectively captured by the five-element formulation.
As universal formulation, five-element can define various types of optimization problems, thereby enhancing the optimization generalization ability. For instance, integer programming can be mod- eled by adjusting the feasible region X, while linear and non-linear programming constraints are captured by different forms of G(z). More complex problems, such as multi-objective optimiza- tion, extend the objective function to a vector-valued form F(x). Additional examples illustrating the formulation of various optimization problems are provided in Appendix
3.2.2 DATA AUGMENTATION AND LABELING
The five-element formulation enables the mapping of optimization problems described in natural language to a universal structure, which can then be used to generate solver code. However, the ef- fectiveness of learning is significantly influenced by the quality, quantity, and distribution of training data. Current available datasets are limited these aspects and lack proper labels of formulations and solver code. Therefore, data augmentation and labeling employed to address these gaps.
Firstly, we collect almost all existing optimization problem datasets, including NL40pt (Ramamon
jison al. 2021), Mamo (FasyL.P and Complexl.P (Huang et al. 2024), IndustryOR (Tang al.
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2024). NLP4LP (AhmadiTeshnize al. 2024) and ComplexOR (Xiao et al. 2024) whose detailed information is introduced in the Appendix [ Subsequently, 100 samples are randomly selected from each dataset as the reserved test dataset. For datasets with fewer than 100 samples, all data are used for testing. The remaining samples are used for data augmentation, ensuring a clear separation between training and testing data. In the data augmentation process, LLMS effectively generate data through prompt engincering (Tang et al., 2024, Luo et al. 2023). To build a high-quality dataset, seven distinct instructions are applied to 1,763 seed problems. These instructions comprehensively extend the original problems from various perspectives, such as modifying constraints, changing scenarios, altering optimization types, and branching out from the original problem, among other approaches. Finally, experts review the generated problems, removing those with unclear descrip- tions or infeasible solutions to ensure data diversity and quality. Prompts are detailed in Appendix
Then, the labeling of optimization problems using five-element and solver code is performed by experts. GPT-4 (OpenAl 2023) is also used to generate labels with uncertain correctness, which can be used in the model alignment. As shown in Figure 2 a). for each optimization problem p. experts label the five-element formulation ff and solver code 5, and GPT-4 generates the formulation f' and code s', respectively. Since GPT-4 may produce errors, experts validate it and assign desirability label d' to indicate whether GPT-4's s label is accurate (True) or not (False). This process produces types of labeled data: fully accurate formulations and solver codes, and those with potential errors but validated by experts. The two kinds of data are well-suited for the processes of multi-instruction supervised fine-tuning and model alignment, discussed in the next section. The detailed process of data labeling and augmentation is introduced in Appendix K
3.3
LEARNING: MULTI- SFT AND MODEL ALIGNMENT
3.3.1
MULTT-INSTRUCTION SET
Direct utilization of LLM to address optimization problems described in natural language often results in inaccuracies, primarily due to their inability to comprchensively capture implicit informa- tion. To address this issue, we enhance the capability to both define and solve the problems through multi-instructiin supervised fine-tuning (SFT).
The multi-instruction dataset DSFT D' U DDSTT labeled by experts, is designed to improve both the problem formulation and solving code generation abilities of LLM.. The formulation dataset D {u, ) focuses enhancing the LLM's ability to accurately define general optimiza- tion problems by providing data pairs (u, B.), where u, represents a problem after being applied to template, and v; is its corresponding five-element formulation f. Mcanwhile, the solving dataset D [u1, ux) ), focuses
improving the LLM's ability generate solver code. It contains two types of data: in one, u; is the problem p and V/ is the solver code s; in the other, u; is the formulation ʃ and v; is the solver code 8. Both are aligned with the appropriate templates and all the labels in DSFT are correctly labeled by experts. The instruction templates used in the learning process are detailed in Appendix H which provides comprehensive explanation, including more in-depth description of the five-element to facilitate better understanding by the LLM.
Then, given the instruction u and its label v, the objective of SFT to maximize the conditional probability p(u u). Assuming that p(u u) = [ p(v; U0s-1, u), this leads to minimizing the negative likeliiohoo:::
(0) -E Dan
Σ log π (v; 10:0-1. u)
where DSFT denotes the multi instruction SFT dataset, which combines instructions from D'ET and DEFT to train the LLM in defining and solving optimization problems. TT-) represents the predicted probability distribution of the I.M,, and 0 denotes its parameters.
3.3.2 MODEL ALIGNMENT
Despite SFT training the model to learn how to write solving code, LLMs may still exhibit hal- lucination when faced with novel problems. This hallucination manifests as outputs that appear plausible but are, in fact, fabricated or inaccurate. To mitigate hallucinations, we incorporate model
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alignment, which is not utilized by other learning-basse methods. We use Kahneman- Op- timization (tthayaaa et al.; 2024) as our alignment method, as it avoids reward model bias and mitigates the data demands of ranking-based approaches like PPO(Ouyang et al. 2022 and
DPO [Rafallov et al. 2023).
The KTO algorithm aligns the model using (instruction, completion, desirability) data, where desir- ability reflects the correctness of the completion using a binary label: True or False Specifically, the dataset DKTO {uus, Vi, d;)); NATO used by KTO, contains GPT-4 completions with expert-assigned desirability labels. The instructions v. in DETO match those in DSFT, covering formulation and code generation tasks. However. the completions v, generated by GPT-4, do not always fulfill the instructions u, correctly, and experts assign desirability labels de achieve the model alignment.
Building upon the previous work by Rafailov et al. (2023). it is evident that the LLM can be inter- preted as reward function. Consequently, we can express the optimal reward for KTO as follows:
rk,o v) = Bog x² (v u) Tea(2)) u)
(3)
In this equation, r-) denotes the optimal reward function of the completion in response to the instruction u, and 7T' represents the optimal model (i.c., the KTO model), (er(.)) indicates the reference model (i.e., the SFT model), and serves the scaling factor. This ratio measures the relative confidence of the optimal model in generating v compared to the reference model.
Inspired by classic prospect theory (Tvrrsky& & Kahneman [1992), the value function employs lo- gistic transformation on the adjusted ratio of log probabilities, thereby assessing the value of each generation in terms of its desirability and the divergence of the policy from the reference. By sub- stituting the exponential function with a sigmoid function, we derive the value function as follows:
ΦктO (u, () ( - (K, u)) d = False u, U,
(rkTo v) Zief) if d True
u, V
(4)
where 4te is defined as the reference point, formulated as: 2 = 3KL((* (v' u) || Tee u)).
Another essential component of KTO loss is the weight function, which can be expressed as follows:
Ap if d True u, v,
w( (v) =
Xu if d False u,
The weight function w(v) assigns weights to the loss depending on the desirability of the outcome v, with ÅD and Ny representing the weights for desirable and undesirable outcomes, respectively. Finally, the KTO loss function is formulated as:
LKIO(TT, meet) E ((u. (w()) - éxro v; 3)]]
The KTO loss function encourages the optimal model x² to produce completions that align more closely with expert-labeled data, improving the overall optimization generalization of both problem formulation and solver generation.
3.4 AUTO-TESTING: FORMULATION, SOLVING AND CELF-CORRECTION
In the auto-testing process, first, the optimization problem is formulated using the five-element framework based on its natural language description; second, the solver code is generated for the five-element formulation and executed; finally, the solver's running logs, including output results and errors, are analyzed determine whether self-correction is necessary. The auto-testing pro- cess automates the entire workflow of problem definition and solver code generation, while also integrating the self-correction mechanism for continuous improvement.
Inspired by Chen et al. (2024). to enhance optimization generalization, we implement self-correction to automatically analyze the output results and identify errors arising during the execution of the solver code. Specifically, the instruction is organized around the problem, the five-element formula- tion, the solver code, and the execution output results, following a template outlined in Appendix H The self-correction LLM then determines whether further resolution is necessary optimal solution has been achieved. If resolution is needed, the model generates an analysis with suggestions and decides to return to whether the problem formulation step the code generation step. The self-correction mechanism ensures a more robust and adaptive optimization process, im- proving optimization generalization especially in the accuracy of formulations and final solutions.
whether the

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4 EXPERIMENT
To analyze the performance of LLMOPT, conduct SFT and model alignment based on the open- source LLM Qwen 1.n.1.5111BB (Bai et al. 2023) and compare with various learning- based and prompt- based methods extensive datasets. The experiments aim to answer four key questions below.
(Q1)
Learning-based vs. Prompting-based Methods: What advantages do learning-basede meth- ods, such as ILMOPT, have over L.Ms that rely solely on prompt engineering?
(Q2) Optimization Generalization (Accuracy and Generality): To what extent can LLMOPT improve optimization generalization ability compared with existing methods?
(Q3) Importance of Problem Definition in LLMOPT: How does the proposed five-element for- mulation as an intermediate step contribute to boost accuracy solving optimization tasks?
(Q4) Effectiveness of Model Alignment in LLMOPT: How effective is model alignment in en- hancing the solving accuracy LLMs for optimization tasks?
The four questions are answered in order in the next sections which include a detailed introduction to the experiments, followed by the analysis of results.
4.1 EXPERIMENTAL SETUP
The experiments are conducted on six real-world optimization and operation task datasets, namely NL.Opt (Ramamonjison et al. 202T), Mamo (EasyLP and ComplexLP) (Huang al., [2224, In- dustryOR (Tang et al. 2024), NLP4LP (AhmadiTeshnizi et al.) 2024). ComplexOR (Xiao et al. 2024). These datasets encompass about 20 scenario and 7 types optimization problems. Detailed descriptions of these datasets are provided in Appendix A Training and testing data are strictly sep- arated. For the NL.Opt, Mamo (EasyL. and ComplexLP) datasets, we shuffle the original datasets and randomly extract 100 data from each dataset the test datasets, and the remaining data are used as seed data for data augmentation. The IndustryOR dataset retains its original partitioning. Due to the limited data in NLP4LP and ComplexOR, all data from these datasets used for testing and excluded from the training process.
In the experiment, we three performance metrics to comprehensively evaluate the optimization generalization of the algorithm, namely, Execution Rate (ER), Solving Accuracy (SA), and Average Solving Times (AST). Specifically, ER refers to the proportion of solutions whose code without any errors and has running results output. SA refers to the proportion of solutions that correctly solve the optimization problem, i.e., find the optimal solution. AST refers the average number of times the self-correction process is performed during the test. In our experiment, the maximum number of self-correction re-solves set to 12 times.
4.2 ANALYSIS OF OPTIMIZATION GENERALIZATION
In this section, we compare ILMOPT with prompt-based methods (Reflexion Shiin et al. 202.3)
Chain-of-Expert (Xiao et al. 2024), OptiMUS (AhnadiTeshnizi al. 20244), learning- based
methods (ORLM (Tang et al. 2024) built Mistral-7B, Deepseck-Math-17 -iBaath-BBBBa Base, LLaMa3-8B),
and GPT-4 (OpenAl 2023). demonstrating the optimization generalization capability of LLMOPT.
Table 1: Comparison of the SA metric between LLMOPT and learning-basee methods. The results for ORLM are cited from Tang al. (2024). Underlined results indicate the previous SOTA, while bold results indicate the current SOTA.
Datasets
NL40pt
Mamo Easy
Mamo Complex
IndustryOR
GPT-4 Directly
47.3%
66.5%
14.6%
28.0%
Mistral-7B
84.4%.
81.4
32.0%
27.0%
ORLM
Deetscek-Math-7B.. -ate-BB-BB
86.5%
82.2%
37.9%
33.0%
LLMMM--8B
85.7%
82.3%
37.4%
38.0%
LLMOPT (Ours)
Qwen 1.5-1BB
93.0%
97.0%
68.0%
46.0%
Improvement Rate
+6.5%
+14.7%
+30.1 %
+8.0%

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100%
Solving, Accuracy (SA)
GPT-46
100%
LEMOPT wio Beeee
80%
Qvard 5 with Ouly SFT
LEMOPT
Solving Accuracy
80%
60%
60%
55.%
40%
24
40%
20%
20%
0%
NL.4pp
C. IndusiryOR NLPALE ComplexCR
NL40pt
IndustyOR (b) Ablation of five-element.
ComplexCR
(a) Comparison between GP"T-4o and Qwenl.5 14B with only SFT.
Salving, Accuracy (SA)
100%
LLSOPIT wio KTO
Solving Accuracy (SA)
100%
80%
FEAN
LLMOFT
80%
15.%
72.74
60%
60%
SALTT
40%
40%
20%
20%
L2MOPT wio WIRKTO SElman
12.MPFT
0%
0%
N0, 40
InavrrOR NLP4IP ComplexCR
NL40pt
IndueryOR NLMIP Complex
(c) Ablation KTO.
(d) Ablation of both five-element and KTO.
Figure 3: The results of the SA metric. Sub-figure (a) compares the SA performance of GPT-4o and the @owen1551444 model with only SFT, showing the potential of learning- reanning-based methods. Sub- figures (b), (c) and (d) perform ablation on five-element and KTO. The Mamo Easy and Mamo Complex datasets are abbreviated as Mamo E. and Mamo C., due to space constraints.
Table 2: Comparison of the SA metric between LLMOPT and prompt- methods. The results for Reflexion, Chain-of of-experts and OptiMUS cited from Xiao al. (2024). Underlined results indicate the previous SOTA, while bold results indicate the current SOTA.
Datasets
NI40pt
NLPALP
ComplexOR
GPT-4 Directly
47.3%
35.8%
9.5%
Reflexion
53.0%
46.3%
19.1%
Exxpersf-Experts
64.2%
53.1%
38.1%
OptiMUS
78.8%
72.0%
66.7%
LLMOPT (Ours)
93.0%
83.8%
72.7%
Improvement Rate
+14.2 %
+11.8
+6.0% %
Learning-based vs. Prompting-based Methods (Answer to Q1). To demonstrate the potential of the learning- based method, we perform SFT on 1iven1.5-14BB Bai al. (2023) and compare it with GPT-4o using the same prompt (Appendix H). As shown in Figure 3a), [), the results indicate that LLM with only SFT can achieve comparable performance to GPT-4o. For the complete LLMOPT (including model alignment and self-correction), shown in Figure 4 LLMOPT outperforms both GPT-4o and GPT-4-Turbo, achieving higher solving accuracy with fewer solving times across all six datasets. All these results demonstrate the potential of learning- barning-based methods.
Accuracy (Answer to Q2-1). We compare the solving accuracy (SA) of LLMOPT with prompt- based and learning-basee methods shown in Table T and 2 respectively. To ensure consistency in reproduction, we only cite the results from the original paper. LLMOPT achieves state-of-the-an (SOTA) performance SA across six datasets, outperforming both prompt-based and learning- based methods. Compared to learning-based methods, defining with the five-clement formulation and self-correction process improves the accuracy, with average increase of 14.83% on four datasets. For prompt-based methods. SFT and model alignment enhance the LLM's ability to solve optimization problems. yielding a 10.67% average improvement across three datasets. Overall, ILMOPT improves SOTA performance by an average of 11.08% across the six datasets.
Generality (Answer Q2-2). To analyze the generality of LLMOPT on general optimization problems, we conduct testing on six datasets, each covering diverse optimization types, such as linear programming, nonlinear programming, integer programming, mixed-integer programming. multi-objective optimization, and combinatorial optimization, detailed in Appendix G These datasets span over 20 different fields, including agriculture, energy, and healthcare, as detailed in Appendix F LLMOPT achieves state-of-te -ftte-rrt-e--tttt results across all datasets including variety of optimization problems, demonstrating its generality. Appendix [ shows the specific experimental results by question types. Moreover, despite the absence of NI NL.4I.P and ComplexOR data in the
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Easy
Complex
IndustryOR
NLPEE
ComplexOR
100%
100%
Solving Accuracy (SA)
Salving Accuracy (SA)
80%
72%
60%
58%
LLMOPT
40%
LLMOPT (Ous)
wio Element
GPT-4o
wio KTO
GPT-4-Turee
wio S-Element KTO
20%
30%
10
3.0
5.0
7.0
9.0
11.0
1.0
3.0
5.0
7.0
9.0
11.0
Average Solving Times (AST)
Average Solving Times (AST)
(a) Comparison with GP and GPr-o-Turb.
(b) Comparison with ablation versions.
Figure Comparison of SA and AST between LLMOPT, GPT-4, and ablated versions.
Table 3: The ablation results of LLMOPT without (w/)) five-element formulation, KTO alignment and self-correction, respectively. In the experiments without self ceff-correction, the model is called only once, resulting an AST of 1.00. Bold indicates the best performance for each metric.
Metries
SA
AST
ER
AST
SA
AST
Dataset
NL.0pt
Mamo Easy
Mamo Complex
ILMOPT
99.0%
93.0%
3.07
100.0%
97.0%
1.70
97.0%
68.0%
6.30
w/o five-element
99.0%
91.0%
2.95
100.0%
96.0%
2.36
98.0%
64.0%
6.38
w/o KTO
99.0%
90.0%
3,47
100.0%
97.0%
2.80
95.0%
65.0%
6.79
w/o selffcoorretion
79.0%
57.0%
(1.00)
85.0%
71.0%
(1.00)
52.0%
32.0%
(1.00)
Dataset
IndustryOR
NT,P4IPP
ComplexOR
LLMOPT
92.0%
46.0%
8.35
100.0%
83.8 %
7.00
94.7%
72.7
10.27
w/o five-element
92.0%
41.0%
8.50
96.9%
73.0%
7.43
94.7%
72.7%
10.05
w/o KTO
92.0%
43.0%
8.88
89 2%
64.9%
8.32
100.0%
54.6%
10.84
w/o self-correction
55.0%
31.0%
(1.00)
47.7%
35.2%
(1.00)
47.4%
18.2%
(1.00)
training dataset, LLMOPT still achieves SOTA performance on them, as shown in Table 1 fur- ther proving its generality. Furthermore, seven examples of the five-element formulation applied to various optimization problems are provided in Appendix to illustrate the generality of this defi- nition. The high accuracy and generality demonstrate the optimization generalization capability of ILMOPT.
4.3 ABLATION STUDY
In this section, conduct comprehensive ablation experiments, including LLMOPT without five- element formulation, without KTO alignment, and without self-correction. The results of these experiments are shown in Figure 3 Figure 4 and Table We also design detailed ablation experi- ments on the self-correction mechanism, which are introduced in Appendix M
Importance of Problem Definition (Answer to Q3). In order to explore the importance of problem definition, we evaluate three metrics on the ablation experiments: Execution Rate (ER), Solving Accuracy (SA), and Average Solving Time (AST). As detailed in Table 3 and Figure (])), these results show that using the five-element formulation as the problem definition improves SA across all six datasets. However, this definition can sometimes lower the ER, as the L.LM may oversimplify the problem without it, producing error-free but inaccurate code. In contrast, using the five-clement approach ensures correct code generation, improving SA at the cost of slightly reduced ER.
Effectiveness of Model Alignment (Answer to Q4). Alignment typically enhances the efficiency and effectiveness of LLMs on specific tasks. As shown Figure 4b) and B(c), the ablation results shows that KTO alignment not only significantly boosts SA but also reduces AST across all six datasets. Furthermore, as shown in Figure bb. d), and Table the combination of five-element and KTO alignment improves performance in terms of SA and AST complex datasets.

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5 DISCUSSION
Attempt on Larger Models. To explore the potential for further performance improvement of LL- MOPT on larger models, we deploy it on Qwen2-72B (Yang al. 2024a). The detailed results are provided in Appendix [ We conduct experiments on two complex task datasets, Mamo Com- plex and IndustryOR, and the results show that LLMOPT based on Qwen2-72B shows significant improvements in both SA and ER compared with its performance on Qwenl.5. 14B. Given the trade- off between performance and the costs of training and deployment, i.e. green computing issue, we opt not to use larger models, since Qwen .5-14B already achieves state-of-the-art performance.
Compared with OpenAI o1 Model. OpenAl ol model (OpenAl 2024) has recently garnered attention for its strong reasoning ability. However, due to the limited availability of its API and restrictions on the number of weekly trials, our evaluation of the ol model's 's performance is limited. We conduct experiments on 10 easy and 10 complex problems from the Mamo Complex dataset. With a single call, the ol model successfully generated code to solve 7 easy problems and 5 complex problems. These results suggest that the ol model is more accurate in solving optimization problems compared with GPT-4 series models. However, due to the lack of open access, the absence of detailed technical specifications and training data description of o1, as well as the limitations usage and high costs, fine-tuning open-source large models with LLMOPT presents a more cost- effective solution for achieving better optimization generalization in real industrial scenarios.
The Seesaw Issue of LLMs. The seesaw issue in LLMs refers to the trade-off between improving the performance on specialized tasks and its generalization across diverse tasks, where gains in one area often result in losses in another. To assess whether enhancing the LLM's ability to define and solve optimization problems affects its performance on other tasks, we compared the model's per- formance before and after fine-tuning across 10 general tasks, including math, code, classification, information extraction, open QA, closed QA, text generation, brainstorming. rewriting and summa- rization. The results indicate that the fine-tuned model showed performance improvements in 6 tasks and declines 4 tasks, with an average performance increase of 0.3% across all tasks. Importantly, no significant trade-off or seesaw effect is observed. Detailed results are provided in Appendix
About High-Quality Training Data. High-quality data are crucial for fine-tuning LLMs Villalo- bos et al.) 2024). Accurate and diverse training data allow the model to improve its task-specific performance. However, optimization problem data described in natural language are relatively scarce. Although we have collected many optimization problem datasets as possible, which detailed in Appendix A these datasets often lack high-quality labels. For instance, in the IndustryOR dataset (Tang et al.) (2024), some data have incorrectly labeled optimal solutions, and the NL4Opt dataset (Ramamonjison et al. 202I) does not provide optimal solution annotations but only entity labels. This highlights the scarcity of optimization problems training data. Although prompt engi- neering for data augmentation is employed in this work, expert labeling remains a time-consuminn and labor-intensive process. Efficiently gathering, synthesizing and generating more diverse and well-labeled high-quality data remains an issue that cannot be ignored in this research direction in future. Moreover, for large-scale problems, data are typically stored in specific databases or files rather than extracted from natural language descriptions. The understanding of these data structures remains a new topic that requires further exploration.
6 CONCLUSION
This paper focuses on the challenge of optimization generalization in LLMs, including the accuracy in solving optimization problems and the generality of the problem types that LLMs can handle. We propose a learning-basse to define and solve general optimization problems through multi-instruction supervised fine-tuning and model alignment. I.LMOPT introduces the five-element formulation as a universal definition for various types of optimization problems to enhance solving accuracy. LLMOPT is extensively eval- uated on a wide range of optimization tasks. It achieves state-of-the- optimization generalization ability across all of them, i.c., a notable 11.08% average solving accuracy improvement.
framework called LLMOPT, which significantly improves LLMs' ability
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ACKNOWLEDGMENTS
We would like to thank the anonymous reviewers for their constructive and valuable suggestions. This work supported by the National Natural Science Foundation of China (No. 62476091), Ant Group and the Ant Group Research Intern Program.
ETHICS AND REPRODUCIBILITY STATEMENT
Ethics. This work does not involve any human subjects, personal data, or sensitive information. All the test datasets used publicly available, and no proprietary confidential information is used. We follow recommendations to use the Azure OpenAl service when using GPT models.
Reproducibility. Experimental settings are described in Section 4 and Appendix B and datasets
included in Appendix A The code is available at https: / http//gttttute thub. com/ www.ccaaaacaoo cann
LLMOPT
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APPENDIX
A DATASETS
A. THE INTRODUCTION OF DATASETS
We conduct experiments on the following real-world optimization task datasets.
Table 4: The statistics of the optimization problem datasets.
Dateset Name
# of Data
Notes
NL4 Ramamomjison et al. 2021
1101
All data are unlabeled optimal solution.
Mamo Complex (Huang ct al. 2024
211
Mamo Easy (Huang et al. 2024
652
IndustrvOR (Tag et al. 2024
100
Another 3000 data without optimal solution.
NLP4LP (Ahmadi Leshnizi et 2024
65
All data are unlabeled optimal solution.
ComplexOK (X1a0 al.
2024
19
All data unlabeled optimal solution.
NL40pt Ramaamonjsonn et al. 2021). The NL4Opt Competition curates dataset comprising 1101 annotated LPWPs across 6 diverse domains. Each set contained LPWPs from source domains like sales, advertising, and investment, ensuring representation across all splits. However, LPWPs from target domains such as production, transportation, and sciences were exclusively reserved for the development and test sets.
Mamo (Huang et al. 2024). The optimization dataset of Mamo benchmark consists of two parts, Easy LP and Complex LP. Easy LP contains 652 high school-level MILP problems for basic learn- ing of linear and mixed integer linear programming. Complex LP provides 211 undergraduate- level challenges, integrating LP and MILP, suitable for advanced learners and researchers. These problems more complex, covering different applications and theoretical challenges, suitable for advanced courses and research projects, and provide comprehensive development of optimization skills from basic to complex.
IndustryOR (Tang et al. 2024). IndustryOR is the first industrial dataset designed specifically for optimization modeling. incorporates data from different industries and covers a variety of real- world scenarios. IndustryOR consists of real operations research (OR) problems from eight different industries. It covers five types of optimization problems: linear programming, integer programming. mixed integer programming. nonlinear programming, and other special problem types. These prob- lems are also divided into three difficulty levels. The training dataset of IndustryOR contains 3000 instances without labeling optimal solution and the test dataset contains 100 instances with optimal solution.
NLP4LP (AhmadiTeshnizi et al. 2024). NLP4LP (Natural Language Processing for Linear Pro- gramming) is dataset that includes 65 samples we identified from its repository. These problems sourced from optimization textbooks and lecture notes covering areas such as facility location, network flow, scheduling, and portfolio management. Each instance in NLP4LP includes a de- scription, sample parameter data file, and optimal value derived from textbook solutions manual solving, offering range of complex optimization challenges of varying lengths.
Co (Xiao et 2024). ComplexOR dataset is developed with the collaboration of three specialists in operations research. We identify 19 samples from its repository, sourced from diverse references such academic papers, textbooks, and real-world industrial scenarios. These problems encompass broad spectrum of topics, including supply chain optimization, scheduling problems, and warehousing logistics.
A.2 TRAINING DATASETS FOR MULTI-INSTRUCTION SFT AND MODEL ALIGNMENT
We introduce the training datasets used for multi- iunlti-instruction SFT and model alignment, as illustrated in Table 5 After data augmentation, the SFT dataset comprises 9,828 instances, while the KTO alignment dataset contains 19,563 instances. To keep the generalization capability of the language model, incorporated additional data into the datasets, which is open source and available
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Table 5: The number of samples in the dataset for SFT and model alignment.
# of Optimization Data
of Other Data
Total
SFT
9828
20000
29828
True Label
9827 9735
26384
36212
KTO
False Label
19563
3616
30000
13351
49563
and httpps://huginggacce co/ wiiarriiiiiadddd-eeeeeeeeeeeeeee wwterepeeeeeeeeeeeeeeeeeeeee
It is important note that we do not utilize the entire datasets, but instead randomly selected 20,000 and 30,000 instances for the learning process.
B DETAIL SETTING OF MULTI-INTTUCTION SFT AND MODEL ALIGNMENT
We implement all model training using the PyTorch framework and utilize Qwen 1.5 with 14 billion parameters (Bai al.) 2023) the base model. We utilize NVIDIA 8* 8*A100 Tensor Core GPUs with 80 GB each for model training and employ 1*A100 GPU for model inference. The hyper- parameters of the training are shown Table 6 (Un-)DesirableWeighh in the table represents the hyperparameters λ and Xn of KTO in the paper.
Table 6: The detail training settings for SFT and KTO alignments.
LoRA Dropout
LoRA
LoRA Alpha
LearningRate
WarmU Ratio
ButchSize
MoxLength
Epochs
(Un-IDedrrbhlWWeight
Beta
SET KTO
0.05
64
16
3,00E: 04
0,1
24
2048
20
0.05
16
5.06-077
0.1
2048
20
1.0
0,1
For SFT, we adopt the code from the public Github repository at httts:://itthu: com/ TwenIM/Qwen) and for KTO training, we adopt the code from the public Github repository at And our project code is available at httpp::
http://hhhu...o////uggg///////// //githab. //githab.com/caatt http://coocccaaaaaae wtt.c//aaaaaaccaaannioooooo LLMOPT
C DETAILED RESULTS
In this section, we present detailed results of comparing LLMOPT with the GPT-4 series models, including the Solving Accuracy (SA) shown in Table 7 the Execution Rate (ER) shown in Table and the Average Solving Times (AST) shown in Table 9 All the results LLMOPT are based on Qwenl.5-14P
all the tables, the values in bold represent the best performance achieved by LL- МОPT, while the underlined values indicate the best performance among the GPT-4 series models. Across all six datasets, LLMOPT outperforms GPT-4 in both Solving Accuracy (SA) and Execution Rate (ER) metrics. Additionally, LLMOPT often achieves better performance than GPT-4 in terms of Average Solving Times (AST).
D COMPARISON BETWEEN QWEN1.5 AND QWEN2
In this section, we compare the performance of LLMOPT based on Qwen1.5-144 and Qwen2- 72B (Yang al. 2024a), focusing on challenging tasks: Mamo Complex and IndustryOR, As illustrated in Figure LLMOPT based on Qwen2-72B demonstrates superior performance in both tasks. Specifically. LLMOPT on Qwen2-72B outperforms Qwenl.5- 155BBBB across both the Solving Accuracy (SA) and Execution Rate (ER) metrics, highlighting the enhanced capabilities of the larger model handling complex optimization problems.
E GENERALIZATION ON OTHER TASKS
In this section, we compare the model's performance before and after fine-tuning across tasks: Math, Code, Classification, Extraction, Open QA, Closed QA, Text Generation, Brainstorming.
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Table 7: Detailed results of solving accuracy (SA). Bold indicates LLMOPT's best performance, while underlined indicates the best results from GPT-4 models.
NL4Opt
Easy
Complex
IndustryOR
ComplexOR
Dircetly
13.0%
GPT 4 urbo
48.0%
23.0%
Directly
51.0%
67.0%
19.056
31,0%5
18.9
9.19
GPT-4o
S-clen
67.0%
18.2%
wro debug
Eie-elleeent KTO
56.0%
65.0%
70.8%
23.0%
16.25
9.1%
KTO
52.0%
63.0%
33,0%
23.05
ILMOPT
W/o live element
60.%
7200%
31.05
27.0%
16.2%
9.19
M.PP dehugg?
$7.05
71.0%
32.8%
31.0%
35.29)
18.2%
debug eleth debug
75.0%
51.0%
25.0%5
30.015
32.4%
18.2%
76.0%
85.0%
27.016
29.0%
37.8%
27.3%
+ debug S-elem + deltg
$1.0%
01.8%
34.0%
$4.0%
12.4%
27 316
$3.08
90.05
38.0%
$4.0%
$5.05
36.458
with
w/o live elemett KTO
$8.0%
89.858
45.0%%
33,095
54.1%
45.5%
wil KTO
90.0%%
97.098
65.015
43.0%5
64.9%
54.0%
LLMOPT
wis Eve olement L.MOPT (Full)
91.0%
95.0%
64.0%
72.7%
93.0%
07.0%
68.0%
46.0%
72.7%
Table 8: Detailed results of execution rate (ER). Bold indicates LLMOPT's best performance, while underlined indicates the best results from GPT-4 models.
NL40pt
Mamo Easy
Mamo Complex
IndustryOR
NLPALP
ComplexOR
Directly +S-elem
64.0%
71.0%
24.0%
34,0%
21.5%
15.8%
GPT-4-curho
23.0%
Directly S-elem
21.0%
71.0%
42.0%
40.0%
35.4%
42.1%
69.055
72.0%
38.0%
36.055
33.9%
36.8%
w/o debug
wio KTO
73.0%
72.0%
39.0%
38.0%
33.9%
26.3%
LLMOPT
weo five element
82.0%
86.0%
53.0%
63.0%
38.5%
42.1%
debug)
79.0%
85.0%
47.7%
47.4%
debug
95.0%
99.0%
95.0%
82.0%
81.55
79. 0.
S-elem debug
93.0%
98.0%
91.0%
83.0%
78.5%
79.0%
debug
GPT-4a
Svelem debug
96.0%
100.0%
96.0%
$2.0%
84.6%
84,2%)
debug
w/o five-clement & KTO 100.0%
100.0%
96.0%
90.0%%
73.9%
73, 73.7%
KTO
LLMOPT
w/o five- element
99.0%
100.0%
98.0%
92.0%
96.9%
94.7%
LLMOPT (Full)
99.0%
100.0%
97.0%
92.0%
100.0%
94,7%
Table 9; Detailed results of average solving times (AST). Bold indicates LLMOPT's best perfor- mance, while underlined indicates the best results from GPT-4 models.
NL40pt
Mamo Easy
Mamo Complex
IndustryOR
NLP4LP
ComplexOR
debug
3.87
9.76
9.13
9,97
11.32
GPT-4-tubbo
5-elem dabug
4.75
3.59
9.81
9.47
10.08
11.32
debug debug
3.89
3.26
9.15
8.96
9.89
11.13
GPT-4o
9.06
10.95
w/o element & KTO
3.58
3.45
8.11
9.25
8.68
10.79
w/o KTO
3.47
2,80
6.79
8.88
8.32
10.84
ILMOPT
w/o five-element
2.95
2.36
6.38
8.50
7.43
10.05
LLMOPT (Full)
3.07
1.70
6.30
8.35
7.00
10.27
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LLMOPT bused on Qwenl.S-j4B Qwend-728
LLMOPT lusel an Quail.5-44B
80%
Qwae2-728
100.0%
Solving Accuracy (SA)
97.0%
60.0%
Execution Rate (ER)
92.0%
80%
40%
20%
60%
0%
Complex
IndustryOR
Mamo Complex
IndustryOR
(a) Solving Accuracy Comparison.
(b) Execution Rale Comparison.
Figure 5: Comparison in (a) solving accuracy and (b) execution rate between Qwen1.5 and Qwen2.
Rewriting, and Summarization, as shown in Figure 6 The results indicate that LLMOPT does not significantly degrade performance on a broad range of tasks.
LIMOPT heeei on Qwanl
+1.1
$0.49
-0.5
SC.1 1E.15--54
Accuracy
$2.7%L
60%
Open
Besiscurming
Summeization
Avange
Figure Results of performance evaluation on other task types.
SCENARIOS OF DATASETS
The scenarios of the datasets presented in Figure
covering 20 scenarios approximately.
NL40pt
Mame Easy
Mamo Comples
IndustryOR
NLMIP
CompleOOR
Sum.
Agriculnure Energy
30
68
33
50
Health
Retall
34
47
37
132
Environment
40
Education
II
32
47
Services
Transportation
36
73
[8
217
Public Uttliies
Manufacturing
44
71
207
Software
Construction
56
62
Legal
2
Custonne Service
2
4
Entertinment
44
54
Oberrs
100
112
Data
245
652
211
100
65
1292
Figure 7: Scenarios of the datasets.
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G OPTIMIZATION TYPES OF DATASETS
The types of optimization problems are presented in Figure categorized into 7 different classes.
NL4Opt
Easy
Complex
IndustryOR
ComplexOR
Linear Programming
104
2
59
20
15
207
Inseger Programming
Mixed Integer Programming
35
412
48
44
35
582
Nonlinear Pragramming
Combinatvial Optimization
0
0
65
9
3
82
Multiobjecttve Programming
Others
25
34
Figure 8: Optimization types of the datasets.
H TEMPLATES FOR INSTRUCTIONS ON LEARNING AND AUTO-TESTING
In this section, all instruction templates are introduced. Each template contains instruction con- structed by fully filling the curly braces "(.)" specify the required content. The following templates are used both for constructing learning instructions and for prompts during auto-testing.
mathematics, optimi zat problem can modeled
the following ng
(x}), expression s\min_[\bol dsymbol (x) Vin \mathcal (x}} f(Abaldsymbol (x)])
(Arm s.t.) G(Aboldsymbol (x}) Aleq \boldsymbol (c)s, where $ s\boldsymbol/x) (x_,, 2, \ldots, x D "tops is the
SD$ 9DS-dimensional decision variable, $lmathcal(X) \subset
\mathbb/R) is the feasible domain, $f: \mathcal \rightarrow \mathbb(R) is the objective function and the goal is to find the minima of $f$, 3d(1boldsymbol Aleg \boldsymbol (c]$ are the constraints of shboldsymbol(x/) (x)8.
The above definition can be mapped to a five-element consisting of Wariiabls,, Objective, Constraints, Sets, Parameters', Variables indicates what $ $1boldsymbel is, Objective describes the form of the objective function $1 (1blooddymmbll indicates the constraints $G (c(ooddsymbolix and (X)) These three can abstract the optimization problem. Sets and Parameters are their specific explanations: Sets describes and explains the subscripts of the vectors matrices in them, and Parameters supplement their specific values.
and Constraints
You need to write the correspending five-element model based on the problem description and information provided.
The problem deseription is as follews: .**
(PROBLEM DESCRIPTION)
12 Please write the corresponding five-element model. Please use LaTeX and plain text environment to complete the following template to model the above optinization problem into five-element:
...
15 " Sets:
16 [You need to fill in]
## Parameters:
19 [You need fill in]
20
21 # Variables:
[You need to fill in]
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# Objective:
25 [You need fill in]
26
27 ## Constraints:
[You need to fill in]
...
Listing I: Instruction template of define the five-element formulation from the problem description.
The eive-element model is the abstraction of optimization problem, which transforms speci fic problem scenarios into formal mathemat ical problems. You need to write the corresponding Pyomo code based on the five-olement model provided.
The following the five-clement medel an optimization problem: 1.
[FIVE-Element)
Please write the corresponding Pyeno code. Please add 'irom. syomo.enviree import at the beginning of your code (You can add other 'import' as well). Please print the optimal solution and the value of the objective function. Please do not output the running
log. You need to write in the form a class and add a main function:
''yyhhon
IWrite your code here]
12 ...
Listing 2: Instruction template of generate the solver code from the five-element formulation.
The five-element ement model is the abstraction an optimization problem, which transforms specifie fic problem scenarios into formal mathematical problems. You need to write the corresponding Pyomo code based on the five-element model. provided.
The problem description is as follows:
4. ...
(PROBLEM DESCRIPTION)
...
Please write the corresponding Pyeme code. Please add 'from syomo.environ import x' at the beginning of your code (You can add other 'import' as well). Please print the optimal solution and the value of the objective function. Please do not output the running log. You need to write in the form a class and a main function:
'pythhon
[Writ your code here]
12
...
Listing Instruction template of generate the solver code from the problem description.
For the following iatiiiaioo problem, modeling is performed, and pyomo code is generated and executed based on the modeling. Please judge whether the modeling and code are correct. The problem is follows. *..
4. (PROBLEM DESCRIPTION)
The five-el formulation follows.
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1.. (FIVE-Element)
10
11
The code is as Eollows.
1.
(SOLVER CODE}
15
Run the code and get the following running information.
19
(OUTPUT INFORMAT LONS)
20
[ERROR INFORMATIONSS
21
23 Please judge whether the above five-element and code are correct, and give your analysis according to the template bolow.
24
The five-element is [Fill True/False here].
27
28 The code is [Fi1l in True/False here].
29
Analysis:
31 [Fill in your analysis here]
32 1*y
Listing 4: Instruction template of self-correction.
I TEMPLATES FOR DATA AUGMENTATION
In this section, general template for data augmentation is introduced. The same as previous section, the curly braces "(-3" should fully filled. The "ONE OF THE RULES BELOW" represents one of the rules of generating new questions, selected randomly from the ones introduced below.
Please generate an optimization problem according to the following requirements and the given format.
(ONE OF THE RULES BELOW)
The original optimization problem em is as fol lows:
6 ..*
(ORIGINAL OPTIMIZATION PROBLEM DESCRIPTION)
10 Please construct a new optimization problem according to the above requirements and the provided questions and in the following format:
[Write your new problem here]
13 1..
Listing 5: General template for data augmentation.
1. The following is an optimization problem. Please construct a new optimization problem based on the context of this problem.
2 2. The following is an optimization problem. Please find similar problems in other fields and construct new optimization problem scenario.
with
3. There are two optimization problems. Please construct new optimization problem based on the scenario of problem A and the optimization problem type of problem B.
4. The following is an aotimization problem. Please modi the constraints this probles and construct new optimization problem.
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5. The following is an optimization problem. Please modify the constraints and object of this problem and construct a new optimization problem.
6. The follcwing is an optimizaticn problem, Please medify the variables and parameters of this problem reasonably and construct a new optimization problem.
7. 7. The following is an optimization problem. Please medify the description of some statements and construct a new optimization problem without changing the meaning of the original problem.
Listing 6: Rules for data augmentation.
EXAMPLES OF FIVE-ELEMENT FORMULATION FOR DIFFERENT OPTIMIZATION PROBLEMS
This section presents examples of five-element formulations for different optimization problems, including seven examples spanning linear programming, integer programming, mixed-integer pro- gramming. nonlinear programming, and combinatorial optimization. By illustrating the five-element formulation across these diverse problem types, we aim to show the broad applicability of this mod- eling approach for general optimization problems.
Linear Programming
Problem Statement:
A person plans to invest $100,000 in stocks and bonds over the next year to maximize the return their investment portfolio. The expected annual return rate for stocks 8%, while the annual return rate for bonds is 4%. At the same time, the risk coefficient for stocks 0.08, and the risk coefficient for bonds is 0.02. The investor wants least 60% of the funds to be invested in bonds, and the total risk of the entire portfolio cannot exceed 0.05. The investor needs determine how allocate this $100,000 under the above conditions to achieve the maximum possible expected return.
Five-Element Formulation:
Sets
Constraints
Set of investment methods: S = {a b}
Total investment constraint:
Parameters
Is 26
Annual return rate for stocks: R.
Minimum investment in bonds constraint:
Annual return rate for bonds: Rs = 0.04
Risk coefficient for stocks: C. = 0.08
36 aT
Risk coefficient for bonds: C. 0.02
Total investment amount:
100000
Total portfolio risk constraint:
Minimum proportion of bonds: a 0.60
C. Tx Co Cmax
Maximum total risk allowed: Cmaz 0.05
Variables
Non-negativity constraint:
Amount of money be invested in stocks: at Amount of money to be invested in bonds: 27
0, 2 0
Objective
Maximize the total expected return:
max #9.2%
Rg + RgJS
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Integer Programming
Problem Statement:
An accounting firm employs part time workers and full time workers. Full time workers work hours per shift while part time workers work hours per shift. In addition, full time workers paid $300 per shift while part time workers the accounting firm has a project requiring 450 hours of labor. If the firm has a budget of $15000, how many of each type of worker should be scheduled to minimize the total number of workers.
paid $100 per shift. Currently,
Five-Element Formulation:
Sets
Objective Minimize employee number: minzy. 2 tp
Set of employee types: S = {f.p}
Constraints
Parameters
Labor time constraint:
Hours per shift: h = 8, hp = 4
Wages per shift: w/ 300, wpp 100
hy + hy 2 T
Total labor time required: T 450
Budget: 15000
Budget constraint:
Variables Employces shifts: Tp
Mydy + Up S
Positive integer constraint:
Ip E 2
Integer Programming
Problem Statement:
A company intends to construct residential buildings new property and needs to com- plete the design and construction drawings. Influenced by various factors such as the market, the time required for design and construction exhibits cyclical peaks and troughs throughout the 12 months of the year. Specifically, the time required to complete the design drawings for each month is follows: 3, 3, 4, 5, 6, 7, 6, 5, 4, 4, 3, months; the time required for construction as follows: 3, 4, 4, 5, 6, 7, 8, 9, 7, 5, months. What is the shortest
time required for the company to go from the start of the design phase the completion of construction?
Five-Element Formulation:
Sets
Objective
Month Set:
Minimize the total time span:
M 1,,
min V g+ta((y 1) mod 1) x,
Parameters
Constraints
Design time (per month) is ta(x):
Order constraint:
3,,3, 4, 5, 6, 7,6, 4,
2 x ta(x)
Construction time (per month) tz mod 12):
Value constraint: J (1, 2,.,, 12}
fc 3. 4,5,, 7,
Variables
Value constraint:
The month design The month construction starts:
# E Zz
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Mixed Integer Programming
Problem Statement:
MarketFlow Inc. must decide how to allocate resources for efficiently supplying six retail stores. They have identified four potential distribution centers, each with different opening costs and capabilities. The challenge is to choose the right mix of centers and optimal transportation routes meet retail demands the lowest total cost, which includes both opening expenses and transportation costs.
Specifically, the supply capacity has a capacity of 1,631 units, Center has 1,954 units, Center has 1, 1,446 units, and Center 4 has 820 units. The demand for each retail store, expressed in units, is as follows: Store I has demand of 910 units, Store has 875 units, Store has 589 units, Store 4 has 962 units, Store has 966 units, and Store has 643 units. The opening costs each distribution center are as follows: Center costs $151,000, Center 2 costs $192,000, Center costs $1 $114,000, and Center 4 costs $171,000. The transportation cost per unit from each distribution center to retail stores is as follows:
each distribution center, measured units, is as follows: Center
From Center $5 to Store 1, SS Store 2, Store 3, to Store $3 to Store 5, S3 to Store
From Center 2: $5 to Store I, S4 to Store 2. $3 to Store 3, $5 to Store 4. $2 to Store S, S4 to Store 6
From Center $2 to Store I, S4 Store 2. Store to Store $4 to Store 5, S2 to Store
From Center 4: $5 to Store 1, S4 Store 2, $1 to Store 3, $1 to Store $3 to Store 5, S3 to Store 6
six retail stores while minimizing costs related to opening distribution centers and transporting goods. This involves strategically allocating resources by selecting which centers to open and how much to transport to each store, all within supply con- straints. What the optimal total cost for MarketFlow Inc. open the necessary centers and transport goods, ensuring demands are met at lowest overall expense?
MarketFlow Inc. aims to efficiently meet demand
Five-Element Formulation:
Sets
Set of potential distribution centers: 1 = (1,2,3,4 Set of retail stores: J = 21, 3, 4, 6]
Parameters
Demand of each store: = (910, 875, 589, 962, 966, 643)
Supply capacity of each center: (1631, 1954, 1446, 820)
Opening cost of each center: = (151000, 192000, 114000, 171000)
Transportation cost per unit t.j from center to store
5
5
2
3
37
4
3
2
4
Tuxx
4
5
4
2
4
3
3
Variables
Amount of goods transported from distribution center i retail store j: zig Binary variable indicating whether distribution center is open: yi
Objective
min
Call +
+Σ>
tiy
JC.
Constraints
Demand fulfillment constraint:
Σ Fij dj. Vj JJ
134
Supply capacity constraint:
Σ Øij S siyi 13M JeJ
von-tiegtyvvty constraint:
Ty 20 J
v. E (0,1), TeT
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Nonlinear Programming
Problem Statement:
The Chinese University of Hong Kong, Shenzhen decides to build circular fountain on the campus. The school wants the fountain to be round and as large as possible restricted in a polygonal construction field, which is given by the following points: (0, 1), (0, 6), (4, 10), (8, 10), (11, 7), (11, (7, 0), and (1, 0), the unit is m. Give linear optimization formulation and find the maximal area. Keep your answer in four significant digit number.
must be
Five-Element Formulation:
Sets
Set of polygon vertices: / ((1,, U:) l .8
Parameters
Polygon vertices:
Pi = (0,
P3 = (0)6)
P3 = (4,10)
Pi = (8, 10)
P3 (11,7)
Ps (11,4)
P7 (7.0)
Ps = (1,0)
Variables
Center of the circular fountain: (dtes ye.) Radius of the circular fountain:
Objective
max mr> Feiller
Constraints
Center inside polygon constraint: each edge detined by vertices (I.,, Vs) and (I1441 1:41). ensure:
(U++1 M.).. (Z++1 2. )e + 24+184 1e41
for i 1....,8
V )² + (2:41 2,)2
Distance edges constraint:
The distance from the center (Ie. Ve) to each edge must be at least the radius r:
(1++1 vi (#++1 Ii 14+1
1it1
for 1,...,8
(iu+11 yz) + (7++1 x2
Non-negativity constraint:
0
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Combinatorial Optimization (0-1 Knapsack Problem)
Problem Statement:
There are 4 items with weights of 4, 3, 1, and 1, and their values are 300, 200, 150, and 200 respectively, with only one of each item available. If we select items such that the total weight does not exceed 5, what is the maximum value that can be obtained?
Five-Element Formulation:
Sets
Objective
Set of items: / = (1,
Maximize the total value:
max Σ
Parameters
0:302
Weight items: w = (4,3, 3, 1, 1)
Value of items: (300, 200, 150, 200)
Maximum allowable weight: W
Constraints
Weight constraint:
Variables
Binary variable indicator: signifies whether item i is selected 1) (æ; 0), for
Σ
wiz; S
IE
i I
Binary constraint:
2; {0, for all E
Combinatorial Optimization (Traveling Salesman Problem)
Problem Statement:
In a network consisting of four cities, namely A, B, C, and D, the distances between the cities are as follows: the distance from city A city B is 10 units, the distance from city A to city C is 15 units, the distance from city A to city D is 20 units, the distance from city B city is 35 units, the distance from city
city 25 units, and the distance from city C to city D is 30 units. All distances are symmetrical, meaning the distance from city to city j equal to distance from city j to city i. travel route is defined as: starting from certain city, visiting all cities exactly once, and ultimately returning to the starting city. Please find a travel route that meets the requirements and minimizes the total travel distance.
Five-Element Formulation:
Sets of cities: (A,
Objective Minimize travel distance:
min Σ
Σ
Parameters
Distance between city and city
VET SEL
10 if [i.j] {A, B)
Constraints
15 if [i,j] [A,C)
Visit and leave constraints:
20
if [i.j] = ()
Σ =y=1 ViEI jEf
d,
35 (t.g] (B.C)
25
if (i.s] (B,
30 (i.j) D)
Σ my=1 VjEI /ET, Subtour elimination constraint:
ifi=j
Variables
U: u +1|| ボッ - |
V. (
Binary route indicator: mgj Vi.j I Visit order city u, Vj E
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K DETAILED PROCESS OF DATA LABELING AND AUGMENTATION
In this section, we introduce the process of data labeling and augmentation, which is divided into four stages: preliminary review, expert labeling, expert review, and data aggregation.
1. Preliminary review. Initially review the data and remove unfeasible problems (unfeasible not means difficult). With the help of GPT-4o, we divide optimization problems into two categories according to their difficulty. Problems that meet one of the following conditions will be classified as difficult problems: at least out of 5 solutions using GPT-4o are inconsistent, the code generated by GPT-4o has enrors, and experts have found complex constraints, reasoning, or large amounts of data.
2. Expert labeling. For simple questions, experts independently annotate. And experts independently label for complex questions. For each question, five-element and solver code need to labeled, and the code must be run without errors. this stage, experts may use GPT-4o to generate text that meets the expert's intentions to reduce typing time and generate more formatted code. In order to improve data quality, experts may modify questions appropriately to make them more suitable for the problem scenario, or delete inappropriate questions (such as unfeasible ones).
3. Expert review. For each question, new expert checks the labels of other experts in the previous step. which is based on the correctness of the problem modeling and the consis- tency of the labels of different experts (the same question may have different but correct labels from different experts). Highly controversial questions or those with incorrect labels are included in an independent challenging dataset.
4. Data aggregation. Five experts discuss and analyze each of the data in the independent difficult dataset to determine whether the problem has a solution or can be adjusted to feasible problem, and decide whether abandon the problem based on this. If not, the experts will discuss and complete the labeling of these data. The correctly labeled questions that have passed the expert review will be summarized, And the other feasible questions that have been incorrectly labeled during the entire labeling process will be summarized.
The above steps are completed by 12 experts. The "preliminary review" and "expert labeling" are finished by 9 undergraduate students with bachelor's degrees or above (computer science mathe- matics, all of whom have taken optimization courses), including 4 master's 's students in related fields (I doctoral student). The "expert review" is finished by university professor whose research is optimization in machine learning and 2 algorithm engineers working on operations research opti- mization. "Data aggregation" is finished by the experts except undergraduates. During the data labeling process, we ensure that each expert completed the labeling independently. In the first three stages, question would not be assigned to the same expert twice.
The review pass rate for the seven experts assigned to simple question labeling exceeds 90%, while the pass rate for the four experts labeling complex question labeling is above 80% (2 experts la- beled both simple and complex questions). Considering the above statistics, along with the fact that the labeling results will undergo review by senior experts in the third stage, the reliability and effectiveness of the expert labeling process are well supported.
L MORE EXPERIMENTS FOR GENERALIZATION PERFORMANCE
In order to demonstrate the generality of LLMOPT, we re-analyzed all experimental results and clas- sified the types of optimization problems involved in each dataset, including Linear Programming (LP), Integer Programming (IP). Mixed Integer Programming (MIP), Nonlinear Programming (NP), Combinatorial Optimization (CO), Multi-objective Programming (MOP) and Others. The follow- ing table shows the SA performance of LLMOPT in solving various types of problems on different datasets. The data are the solving accuracy (SA), and the values in brackets represent "number of problems solved correctly/number of problems in the dataset".. Detailed statistics are shown in Ta- ble 10 The results show that LLMOPT is universal in various types of optimization problems and can solve almost all kinds of optimization problems. However, due to the different distribution of problem formulating difficulties, the accuracy of LLMOPT on these datasets is also different, which will be one of the focuses of future work.
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Table 10: SA of LLMOPT (based on (wen-1155-44)) organized by the type of optimization prob- lems. The values in brackets represent "number of problems solved correctly/number of problems in the dataset".
NL40pt
MamoEasy
MamoComplex
IndustryOR
NLP4LP
ComplexOR
LP
90.5% (38/42)
76.0% (19/25)
60.0% (12/20)
77.8% (7/9)
60.0% (3/5)
IP
(46/48) 95.8%
100.0% (36/36)
100.0% (5/5)
45.5% (5/1 I)
0.0% (0/1)
MIP
90.0% (9/10)
95.3% (61/64)
73.9% (17/23)
47.7% (21/44)
95.2% (20/21)
80.0% (4/5)
NP
100.0% (1/1)
100.0% (3/3)
CO
60.0% (21/35)
33.3% (3/9)
33.3% (1/3)
100.0% (1/1)
MOP Others
37.5% (3/8)
50.0% (5/10)
25.0% (2/8)
Total
93.0% (93/100) 97.0% (97/100)
68.0% (68/100)
46.0% (46/100)
83.8% (31/37)
72.7% (8/11)
Table 11: Performance of Different Inference Models and Correction Mechanisms
Inference Model
Correction Mechanism
IndustryOR Dataset
NL40pt Dataset
LLMOPT (Qwen-1.3 1.55
Self-correction Correction by GPT-4o
46.0%
93.0%
LLMOPT 1Qw)
41.0%
89.0%
LLMOPT (Qwen-la) (.5en-1.5)
Best of 12 repeats
42.0%
89.0%
GPT-4o OpenAl (2033
Correction by GPT-4o
34.0%
84.0%
GPT-4o OpenAl 2023
Best of 12 repeats
32.0%
84.0%
ORLM Tang et al.
2024
Best of 12 repeats
39.0%
88.0%
M MORE ABLATION EXPERIMENTS FOR CORRECTION MECHANISM
To further explore the superiority of self-correction and LLMOPT, we deploy experiments on the NL40pt and IndustryOR datasets (NL4Opt has relatively simple problems, while IndustryOR has relatively complex problems). We change two correction mechanisms, one is correction by GPT- 4o with the same prompt of self-correction, and the other to repeat the inference 12 times and manually judge the optimal solution (which means that only one optimal solution needs to be found in 12 repeated experiments). The reason we chose 12 is that self-correction is limited to a maximum of 12 repeated checks, so this is fair. We also conduct experiments on GPT-4o and ORLM Tang al. 2024). We reproduce the open source model of ORLM Tang et al. (2024), but found that this model seems have lost other abilities except writing coptpy code for optimization problems. We find that ORLM has serious seesaw problem, which is performance as without generalization ability, and cannot answer other questions. Therefore, only the "Best of 12 repeats" correction mechanism is experimented. The results are shown in Table T1. The results show that, when LLMOPT (based on Qwen-1.5) is used as the inference model, the correction performance of GPT-4 is lower than the self-correction solving accuracy of LLMOPT. This indicates that the Qwen- 1.5 model learned by LLMOPT shows stronger overall capabilities in both solving optimization problems and correction compared to other methods. Although manually selecting the best result from 12 repetitions shows performance improvement (considering once solving correct one out of 12 repetitions is accurate), it still falls short of the effectiveness compared with the self-correction mechanism. This highlights that identifying and correcting errors is more critical than simply repeating executions, emphasizing the necessity of implementing correction mechanism.
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Reviews:
[{"review_id": "wKlQN4dkW1", "reviewer": "Reviewer_izRU", "summary": "The authors of this paper introduced **LLMOPT**, a novel learning-based framework designed to enhance large language models' (LLMs) capability to define and solve general optimization problems described in natural language. The key innovation is the **five-element formulization**, which structures optimization problems into sets, parameters, variables, objectives, and constraints, enabling more accurate problem representation and solver code generation. LLMOPT also incorporates **multi-instruction supervised fine-tuning (SFT)** and **model alignment** using the Kahneman-Tversky Optimization (KTO) method to improve both the accuracy and generalization of solutions.\n\nAnother significant contribution is the **self-correction mechanism** in the auto-testing process, which automates error analysis and solution refinement during execution, ensuring robust performance without manual intervention in test-time. The framework was evaluated on six real-world datasets, covering diverse optimization types and fields, achieving an **average accuracy improvement of 11.08%** over state-of-the-art methods. This demonstrates LLMOPT's effectiveness in boosting both the accuracy and generalizability of LLMs in solving complex optimization problems.", "questions": "**Questions and Suggestions for the Authors:**\n\n1. **Clarification on Expert Involvement in Data Labeling**:\n   - **Question**: Could the authors provide more details on the qualifications, expertise, or at least agreement rates of the human experts involved in the data labeling process? \n   - **Suggestion**: Including a quantitative or qualitative analysis of expert reliability and consistency would strengthen the validity of the data labeling claims. Evidence such as statistics of certifications or relevant experience, and agreement rates among experts would be valuable to better assess the credibility of the labeled data.\n\n2. **Theoretical Justification for the Five-element Formulation**:\n   - **Question**: What is the theoretical basis for claiming that the five-element formulation is a universal method for defining optimization problems? \n   - **Suggestion**: Providing additional references or an in-depth theoretical analysis that supports the universality and applicability of the five-element formulation across various optimization scenarios would enhance the argument. Empirical comparisons to alternative problem formulations could also be beneficial.\n\n3. **Comparative Performance Analysis of Self-correction Mechanism**:\n   - **Question**: How does the self-correction mechanism in LLMOPT compare with similar correction methods, such as manual debugging or integration with other LLMs or methods like GPT-4 in a correction loop?\n   - **Suggestion**: Presenting a detailed comparison of LLMOPT’s self-correction mechanism with existing correction strategies or showing performance metrics of LLMOPT combined with baseline models could provide insights into its relative effectiveness. This would help demonstrate whether the self-correction offers unique advantages or is comparable to simpler, established methods.\n\n4. **Scalability Concerns**:\n   - **Question**: How do the authors envision scaling the current framework for large-scale practical deployments given the heavy reliance on expert validation?\n   - **Suggestion**: Suggestions for automating or streamlining the expert review process, possibly by incorporating semi-supervised or active learning techniques, could address concerns about scalability and long-term sustainability of the labeling process.\n\n5. **Detailed Explanation of Generalization Capabilities**:\n   - **Question**: Can the authors provide more specific examples or case studies where the five-element formulation showed clear advantages in generalization compared to alternative models?\n   - **Suggestion**: Adding empirical evidence or a breakdown of performance across different types of optimization problems, particularly ones not included in the training set, would strengthen the claim of its broad applicability and effectiveness.\n\n6. **Addressing Potential Limitations in Self-correction**:\n   - **Question**: How does the self-correction mechanism handle potential limitations or biases in repeated corrections, such as overfitting to a specific type of error?\n   - **Suggestion**: Discussing safeguards or mechanisms in place to ensure diverse error handling and avoiding biased correction loops would provide more confidence in the robustness of the self-correction feature.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper introduces a *novel five-element formulization* to define optimization problems, which significantly enhances the ability of large language models (LLMs) to accurately interpret and transform natural language descriptions into solvable optimization problems. This structured approach captures the essential elements of optimization scenarios, ensuring clearer problem representation and facilitating better code generation. The inclusion of elements such as sets, parameters, variables, objectives, and constraints helps the model produce more precise solutions and reduces the risk of omitting implicit problem aspects.\n\n- Another notable feature is the *self-correction in test-time*, an automated process called *Auto-testing* integrated within LLMOPT to identify and rectify errors in generated solver code during execution. This mechanism analyzes output logs and determines if adjustments are necessary, enhancing the robustness and accuracy of problem-solving without manual intervention. By automatically looping back to problem reformulation or code generation when needed, LLMOPT can iteratively improve the accuracy of its solutions and adapt effectively to complex challenges.\n\n- The paper boasts *strong evaluation results*, demonstrated by extensive testing across six real-world datasets encompassing approximately 20 different fields and various optimization problem types, such as linear and nonlinear programming and combinatorial optimization. LLMOPT shows superior performance, achieving a notable 11.08% average improvement in solving accuracy compared to existing state-of-the-art methods. This result underscores the framework’s generalization capabilities and effectiveness in diverse optimization scenarios, validated by comprehensive comparisons and ablation studies.", "weaknesses": "1. **Complexity of Data Labeling**: The proposed five-element formulation and the use of expert labeling for optimization problem formulations and solver code rely significantly on manual validation. While the authors mention the use of GPT-4 to assist in data generation, human experts are still required to verify the correctness of the outputs. This introduces potential scalability issues as extensive human oversight could limit the practicality of the approach, especially in large-scale deployments. Additionally, the authors do not provide a qualitative or quantitative analysis of the labeling quality or reliability performed by the human experts. Evidence demonstrating the expertise and qualifications of these experts should be presented to support the validity of the labeling process.\n\n2. **Insufficient Theoretical Justification for the Five-element Formulation**: The authors claim that the five-element formulation is a universal method for defining optimization problems, but they do not provide sufficient references or theoretical analysis to support this claim of 'universal'. Similarly, in the experimental results, a detailed and decomposed explanation of why the five-element formulation generalizes well and outperforms other models across different types of problems is lacking. Providing theoretical or empirical grounding beyond performance metrics would strengthen the argument for the effectiveness and universality of this approach.\n\n3. **More comparative Analysis of the Self-correction Mechanism**: While the authors compare the performance of LLMOPT with and without the self-correction mechanism, it would be beneficial to include an analysis of the performance gains when self-correction is combined with baseline methods (e.g., GPT-4 directly). Additionally, a longitudinal comparison between the current self-correction design and other correction methods, such as manual debugging, could further elaborate on the advantages and limitations of the self-correction approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "apYLZhIe4z", "reviewer": "Reviewer_JwJz", "summary": "This paper proposes LLMOPT, which finetunes an LLM to formulate optimization problems (from problem description -> mathematical model/solver model). The finetuning is performed in two stages: SFT stage (where the LLM is finetuned using MLE on ground truth outputs), and KTO-based alignment (where the LLM is finetuned based on desirability labels on LLM generated outputs).", "questions": "I also have some minor concerns:\n\n* **Dataset construction**: In L219, the authors mentioned they had 1763 'seed' problems. Did they label formulations for all 1763 problems? If so, this is a pretty big contribution if the authors choose to open-source their dataset.\n* **Augmented data**: My understanding is that additional examples were introduced by mutating the original seed problems. Were these also formulated and labeled by human experts? How many examples in total (original + mutated) were used in training? \n* Can the authors describe the procedure of collecting the labelled formulations (if human experts were indeed used). This seems like a huge undertaking especially if they are required to write out the mathematical models and code.\n* **Eq(4) and (5)**: The notation $\\nu \\sim \\cdot$ was slightly confusing, since it is not a random variable, which is where this notation is mainly used.\n* **Eq (5)**: Are $\\lambda_D$ and $\\lambda_U$ learnt parameters or hyperparameters?\n* **Solver**: What solver did LLMOPT use? Is it `Pyomo` as shown in the method figure? Are the baselines evaluated with the same solvers?\n* ps. this is just a suggestion, but the use of the word `alignment` is slightly misleading, the authors are not aligning the LLM to principles, but doing a different stage of finetuning with a different dataset.\n* pps. another minor suggestion, but the authors should make more prominent that KTO is an existing method, I saw the citation, but it is slightly buried, and might convey the impression that KTO is an original contribution of this work.\n\nI am going to start with a conservative rating, but open to revising if the authors address my concerns.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper addresses an interesting and timely problem. Optimization modeling is a huge topic, and introducing techniques to automate (partially) this task will have significant impact.\n\nTo my knowledge, this is also one of the first works that builds/finetunes LLMs specifically for writing optimization models.\n\nThe empirical analysis is comprehensive (spanning many benchmarks) and the resulting performance gains are impressive.", "weaknesses": "1. **Novelty**: The authors mentioned ORLM, which similarly trains an LLM to do optimization modeling, but did not provide a direct comparison. I also read ORLM (not in the most detail), but it appears to do some data augmentation to train an LLM for model formulation. It seems the main difference is the alignment step (using KTO) and the self-reflection step, can the authors explain the novelty of their method compared to ORLM?\n\n2. **KTO alignment**: There are a few comments on this:\n* **Writing/clarity**: The writing in S3.3.2 is quite hard to follow, I had to read the original paper to understand what this part is doing. Importantly, Equation (3) is not correct, in the original DPO paper, the optimal reward function has an additional log partition function term. I did not check if this affected the rest of the formulation.\n* **KTO dataset**: Based on my understanding: (1) the SFT step does not use the KTO dataset (which contains GPT4 generated responses, and desirability labels), and (2) the alignment step does not use the original dataset (which contains ground truth formulations). Is my understanding correct? If so, what is the motivation for not using the KTO dataset during SFT and original dataset in KTO alignment (where the ground truth are all labeled as 'desirable')? \n* **Purpose of KTO**: The authors state that `KTO loss function encourages the optimal model $\\pi^*$ to produce completions that align more closely with expert-labeled data`. **Minor**: this can't be the optimal model, but the learned policy $\\pi$? **Major**: This is exactly the same purpose as SFT, so we are back to the above point of why the KTO dataset is not used directly through SFT, it would likely be more stable and have lower variance compared to KTO updates.\n* **Ablation study**: Can the authors more comprehensively ablate the importance of the KTO step? I carefully examined the results in Table 3 (ablation study)---which seems to indicate that `w/o KTO` on many benchmarks does not significantly improve SA. I would be interested to also see how LLMOPT performs when the KTO dataset is used directly during SFT.\n\n3. **Questions about results**: \n* **AST**: Can the authors help me understand the big difference in average solution time (AST) plotted in Fig 4? I had a look at some examples in `NLP4LP`, these are very straightforward LP problems. As such, I am very surprised that LLMOPT and GPT4 have almost 2x difference in solution time. Given the simplicity of the problems in this dataset, (1) this difference is unlikely to be explained by clever reformulations that improve solution time, (2) and unlikely to be noise. \n* **Solution accuracy**: Can the authors elaborate on exactly how solution accuracy is calculated? Is this based on some test cases? My intuition is that it is extremely difficult to check the formulation directly, so how is SA computed and is this a robust evaluation method?\n* **Performance by problem type**: In L422-L431, the authors claim their method is more general, which I understood as achieving better performance on more types of problems (e.g. LP, MILPs, QPs). To confirm this claim, I would like to see the performance by problem type (similar to the Figure 8 in App G).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "w6LOlrcbCj", "reviewer": "Reviewer_iJY5", "summary": "This paper proposes to finetune LLMs to improve MILP modeling from natural languages. Specifically, from natural language description of a MILP problem, this paper proposes to formulate the MILP problem as a five element formulation, and then generate solver code from the  formulation. It uses data augmentation to expand the data set, and ask domain experts to label the data set to finetune the LLMs. Through extensive empirical study, it shows performance improvement from a variety of competitive baseline methods.", "questions": "- Five-element formulation: to my knowledge, OPTIMUS [1] also identifies components such as variables, constraints, parameters etc in the optimization description before they translate the optimization problem into code. Can the authors comment on the difference between the two modeling approaches?\n- Line 222: “experts review the generated problems, removing those with unclear descriptions or infeasible solutions to ensure data diversity and quality.” Can the authors comment on what are the criteria for the experts to determine the descriptions are “unclear”? How long is the expert labeling process? Can experts make mistakes? Can the authors comment on whether there is anyway to consider the expert mistakes to further improve the learning performance?\n- Line 276 equation (3): I find the description of KTO confusing. For example, what is the reference model pi_ref used by the authors? Also, what is the optimal model (is it the same thing as the learning model)?\n- Table 3: Can the authors comment on what is the setup w/o KTO? Is there still a training / fine-tuning component? what is the alternative loss?\n\nAdditional Feedback:\n\n- line 53: The authors provide Wrong citation for ORLM on page 1.\n- line 249: “are correct labeled by experts” → “are correctly labeled by experts”\n\n[1] AhmadiTeshnizi, Ali, Wenzhi Gao, and Madeleine Udell. \"OptiMUS: Scalable Optimization Modeling with (MI) LP Solvers and Large Language Models.\" ICML (2024).", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The authors did a good job benchmarking their method on a variety of benchmarks with many competitive baseline methods. The empirical evaluation seems to be comprehensive.\n- To my knowledge, there has not been many works that aim to fine-tune the LLM for MILP modeling tasks, so the task is relatively novel.\n- I appreciate the detailed discussion provided by the authors, which I think is valuable and can help guide the community thinking about future steps.", "weaknesses": "- The fine-tuning data collection requires expert manual labels, which limit the scalability and applicability of applying this method.\n- Despite strength 2 mentioned above, the related work ORLM cited by the paper already took an initial step in fine-tuning LLMs for MILP modeling, and RLHF/DPO/KTO has been commonly used in LLM literature to finetune LLMs, so I’m a bit concerned whether the novelty of this work is sufficient, especially the work requires expert manual labors for constructing the fine-tuning data.\n- Table 3: it seems like a majority of performance improvement is from self-correction instead of the five element components and KTO. If I understand correctly, the self-correction is prompting the LLM to correct any error it makes and has been used in previous papers such as [1], and it’s not related to the KTO fine-tuning pipeline. Given this, I’m a bit concerned about the contribution of the two main components (five elements and KTO) in this work.\n- I find certain parts of the paper missing details and somewhat confusing (see my questions below).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "98sg4s8lb8", "reviewer": "Reviewer_feJ7", "summary": "This paper proposes a learning-based LLM for solving Optimization Problems. The experimental results demonstrate the effectiveness of the proposed LLMOPT compared with prompt-based methods.", "questions": "Please refer to the weakness section.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The introduction of this manuscript is clear.\n\n2. Some improvement over baseline results.", "weaknesses": "1. It is questionable whether LLMs are capable of solving large-scale optimization problems. If their applicability is limited to small-scale problems, such as the Traveling Salesman Problem with just 10 nodes, the significance of this method is unclear. A problem of this size can even be solved by the Hopfield network proposed in the 1980s. In fact, both traditional heuristics and more recent neural combinatorial optimization methods are capable of handling problems with thousands of nodes [1, 2].\n\n2. In Abstract, the authors mention, \"to prevent hallucinations in LLMs, such as sacrificing solving accuracy to avoid execution errors.\" However, I do not believe this phenomenon should be described as hallucination. Hallucination typically refers to instances where the output of an LLM appears reasonable but is, in fact, fabricated. In this case, reducing solving accuracy to prevent execution errors does not align with the conventional meaning of hallucination. Please provide a more precise explanation or reconsider this terminology.\n\n3. As shown in Table 3, the self-correction mechanism plays a decisive role in the performance of LLMOPT. However, this mechanism is not originally proposed in this manuscript. As the authors describe on Page 6, \"Inspired by Chen et al. (2024), to enhance optimization generalization, we implement self-correction to automatically analyze the output results and identify errors arising during the execution of the solver code.\" Furthermore, the prior study [3] has also employed this mechanism to improve the performance of LLMs in solving vehicle routing problems.\n\n4. In Section 4, the authors do not report the training time of LLMOPT, making it unfair to directly compare it with prompt-based methods.\n\n5. Please provide the code generated by the LLM to solve the optimization problem. I am doubtful that current LLMs are capable of generating code sophisticated enough to solve medium- or large-scale problems.\n\n[1] Fu Luo, et al. Neural combinatorial optimization with heavy decoder: Toward large scale generalization. In Proceedings of the Advances in Neural Information Processing Systems, 2023.\n\n[2] Huigen Ye, et al. GNN&GBDT-guided fast optimizing framework for large-scale integer programming. In Proceedings of International Conference on Machine Learning, pp. 39864–39878. 2023.\n\n[3] Zhehui Huang, et al. Can Large Language Models Solve Robot Routing?. arXiv, 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)