以策略思考，以工具行动：面向组合问题的可追溯智能体优化

# Think in Strategies, Act with Tools: Traceable Agentic Optimization for Combinatorial Problems
Source: [https://arxiv.org/html/2610.14736](https://arxiv.org/html/2610.14736)
Shi XuPeng YingWeikang GongZhixiao WangXinzheng Xu††thanks:Corresponding author:xxzheng@cumt\.edu\.cnAffiliation:School of Computer Science and Technology / School of Artificial IntelligenceAffiliation:China University of Mining and TechnologyAffiliation:Xuzhou, ChinaEmail:[\{zhongnianli,xushi,pengying,wkgong,zhxwang,xxzheng\}@cumt\.edu\.cn](mailto:)

###### Abstract

Optimization by PROmpting \(OPRO\) directly leverages Large Language Models \(LLMs\) to generate candidate solutions through natural\-language prompting\. However, OPRO’s optimization process is difficult to track, which prevents LLMs from capturing useful information accumulated over previous iterations, thereby limiting solution quality\. To address this limitation, we propose a Tool\-Augmented Strategic Agent \(TASA\) which summarizes and exploits trajectory to improve solution quality\. Specifically, TASA plans the strategy for each step based on the optimization trajectory, which decomposes the optimization process into traceable strategy selections\. Moreover, we leverage LLM to construct problem\-specific tools and introduce a strategy\-guided adaptive tool planner, which efficiently utilizes the information of trajectory to update solutions\. Weestablishlocal convergence underpersistent exploration and evaluate the complete framework under a shared candidate\-evaluation budget\.Theresultssupport itssolution qualityand the traceability of strategy\-guided search\.

## 1Introduction

Recent advances in Large Language Models \(LLMs\)[Team et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib1);[Grattafiori et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib2);[Guo et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib3);[Yang et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib4)substantially improve their capabilities in reasoning[Wei et al\. \(2022\)](https://arxiv.org/html/2610.14736#bib.bib5);[Gao et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib6), planning[Yao et al\. \(2023a\)](https://arxiv.org/html/2610.14736#bib.bib7);[Zhou et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib8), and decision\-making[Yao et al\. \(2023b\)](https://arxiv.org/html/2610.14736#bib.bib9), providing possibilities for using LLMs to solve optimization problems\. Optimization by PROmpting \(OPRO\)[Yang et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib23)leverages an LLM as the optimizer to generate candidate solutions by iterative prompting process\. Specifically, OPRO prompts the LLM to directly generate promising candidate solutions by constructing a prompt that contains a description of the optimization problem and previously evaluated solutions\. Due to its advantages, OPRO attracts considerable attention, inspiring broader research on LLM\-based optimization[Guo et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib10);[Liu et al\. \(2024b\)](https://arxiv.org/html/2610.14736#bib.bib11);[van Stein and Bäck \(2025\)](https://arxiv.org/html/2610.14736#bib.bib12);[Zheng et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib13);[Ye et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib14)\.

Unfortunately, the limited traceability of OPRO’s optimization process makes it difficult for LLMs to capture useful information from previous iterations, thereby limiting solution quality\. As illustrated in Figure[1](https://arxiv.org/html/2610.14736#S1.F1)\(a\), the LLM directly outputs a candidate solution without explicitly stating the intention or trajectory behind its generation\. Consequently, OPRO retains only the candidate solutions generated across iterations, without an explicit account of how each new solution is generated\. This lack of the information hinders LLM from accurately assessing the current search state, thereby limiting ability of OPRO to identify promising search directions\. For example, when the candidate set is dominated by similar solutions, the OPRO may struggle to escape the corresponding region of the solution space and discover a better solution\.

To address this limitation, we propose a Tool\-Augmented Strategic Agent \(TASA\) to plan the strategy for each step based on the optimization trajectory\. Specifically, TASA constructs an explicit optimization trajectory by continuously recording information about its optimization process, primarily the selected optimization strategies and tool invocations\. As illustrated in Figure[1](https://arxiv.org/html/2610.14736#S1.F1), after constructing the trajectory, the agent analyzes the recorded information and accordingly selects the polish strategy to further shorten the current tour through local adjustments\. TASA aims to enable agent to exploit the optimization trajectory for more informed optimization decisions by shifting the LLM’s role from directly generating candidate solutions to planning optimization strategies\.

In this paper, we leverage an LLM to construct a library of problem\-specific tools according to the characteristics of each optimization problem\. The resulting library organizes executable, problem\-adaptive tools by optimization strategy\. Building on this tool library, we introduce a strategy\-guided adaptive tool planner, which efficiently translates the optimization strategies selected by agent into concrete solution updates\. This mechanism balances the exploitation of previously well\-performing tools, the exploration of underexplored tools, and the avoidance of repeatedly selecting poorly performing tools\. Finally, the planner applies the selected tool to the candidate solution, enabling the efficient execution of the LLM\-planned strategy\. Moreover, we analyze the probabilistic local convergence of TASA under strategy\-conditioned tool neighborhoods\. Experimental results across diverse combinatorial optimization problems demonstrate that TASA not only makes the optimization process traceable but also improves solution quality\.

Our main contributions are summarizedbelow\.

- •We propose a tool\-augmented strategic agent for traceable combinatorial optimization and provide a theoretical analysis of its probabilistic local convergence under strategy\-conditioned tool neighborhoods\.
- •We construct problem\-specific tools and introduce a strategy\-guided adaptive tool planner that invokes these tools to efficiently update solutions\.
- •Experimental results across diverse combinatorial optimization problems demonstrate that TASA not only makes the optimization process traceable but also improves solution quality\.

![Refer to caption](https://arxiv.org/html/2610.14736v1/pic1.png)Figure 1:*Motivation and overview of our Tool\-Augmented Strategic Agent\.*TASA shifts the role of the LLM from directly generating solutions to planning optimization strategies from optimization trajectories, and executes these strategies through strategy\-guided adaptive tool planner\.
## 2Related Work

### 2\.1LLM\-Based Combinatorial Optimization

Large Language Models \(LLMs\)[Grattafiori et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib2);[Team et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib1);[Guo et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib3);[Yang et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib4)apply to combinatorial optimization through both direct solution generation and automatic heuristic design[Da Ros et al\. \(2026\)](https://arxiv.org/html/2610.14736#bib.bib15)\. Optimization by PROmpting \(OPRO\)[Yang et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib23)treats the LLM as an optimizer and generates new candidates from previously evaluated solutions\. FunSearch[Romera\-Paredes et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib26)combines a LLM with an evaluator to evolve program components in function space, while Evolution of Heuristics \(EoH\)[Liu et al\. \(2024a\)](https://arxiv.org/html/2610.14736#bib.bib27)jointly evolves natural\-language heuristic ideas and their executable implementations\. These methods demonstrate the potential of LLMs to optimize over both candidate solutions and heuristic programs\.TASA complements these approaches with explicit links between trajectory\-conditioned strategies,executable tool updates, andobserved search outcomes\.

### 2\.2Tool\-Augmented Agents

Tool\-augmented agents[Karpas et al\. \(2022\)](https://arxiv.org/html/2610.14736#bib.bib29);[Paranjape et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib30);[Shen et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib31);[Song et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib32);[Hao et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib33);[Patil et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib34);[Qin et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib35);[Tang et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib36);[Lu et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib16);[Xu et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib17);[Wu et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib18);[Kong et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib19);[Gao et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib20)extend LLMs by enabling them to interact with external tools and environments[Cai et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib37);[Qian et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib38);[Yuan et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib39)\. ReAct[Yao et al\. \(2023b\)](https://arxiv.org/html/2610.14736#bib.bib9)integrates reasoning with action generation, allowing LLMs to update their plans based on observations from the environment, while Toolformer[Schick et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib40)trains language models to determine which APIs to invoke, when to invoke them, and how to incorporate their outputs\. In optimization[Li et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib41);[Ahmaditeshnizi et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib42), OptiGuide translates natural\-language queries into code that modifies and invokes existing solvers, supporting solution interpretation and what\-if analysis\.

However, these methods mainly target general task interaction or solver\-assisted analysis, rather than treating tools as structured operators for iterative solution improvement\. In contrast, TASA constructs problem\-specific tools and employs a strategy\-guided adaptive tool planner that selects them according to optimization trajectory\.

## 3Method

### 3\.1Problem Formalization

In this paper, we consider a combinatorial optimization problemPPdefined over a finite discrete candidate spaceΩP\\Omega\_\{P\}\. Since candidate solutions generated during optimization may violate problem constraints, we utilizevP​\(x\)v\_\{P\}\(x\)to represent the constraint\-violation status of a candidatexx\. Each component ofvP​\(x\)v\_\{P\}\(x\)corresponds to a specific constraint, taking the value 0 if the constraint is satisfied and 1 if it is violated\. The vector is divided into a hard\-constraint componentvPhard​\(x\)v\_\{P\}^\{\\mathrm\{hard\}\}\(x\)and a soft\-constraint componentvPsoft​\(x\)v\_\{P\}^\{\\mathrm\{soft\}\}\(x\)\(when the problem contain soft\-constraint\)\. Hard constraints determine the feasibility of a candidate, whereas soft constraints express desirable but nonmandatory requirements\. A candidate is considered feasible if none of its hard constraints are violated\. Accordingly, the feasible solution set is defined as

𝒳P=\{x∈ΩP∣vPhard​\(x\)=0\}\\mathcal\{X\}\_\{P\}=\\left\\\{x\\in\\Omega\_\{P\}\\mid v\_\{P\}^\{\\mathrm\{hard\}\}\(x\)=0\\right\\\}\(1\)
To unify the optimization directions of different combinatorial optimization problems, we transform original objective functiongP​\(x\)g\_\{P\}\(x\)of each problem into a higher\-is\-better score by a problem\-specific transformation functionϕP​\(⋅\)\\phi\_\{P\}\(\\cdot\)\. The unified internal scoring functionfP​\(x\)f\_\{P\}\(x\)defined as

fP​\(x\)=ϕP​\(gP​\(x\)\)−𝒫P​\(vP​\(x\)\)f\_\{P\}\(x\)=\\phi\_\{P\}\\\!\\left\(g\_\{P\}\(x\)\\right\)\-\\mathcal\{P\}\_\{P\}\\\!\\left\(v\_\{P\}\(x\)\\right\)\(2\)where𝒫P​\(⋅\)\\mathcal\{P\}\_\{P\}\(\\cdot\)is a problem\-specific penalty function used to penalize constraint violations and guide the search toward the feasible region\.

Under this formulation, our goal is to identify the best feasible solution found within a budget ofMMcandidate evaluationsas

x^PM∈arg⁡maxx∈𝒳P∩ℰM​fP​\(x\)\\hat\{x\}\_\{P\}^\{M\}\\in\\arg\\max\_\{x\\in\\mathcal\{X\}\_\{P\}\\cap\\mathcal\{E\}\_\{M\}\}f\_\{P\}\(x\)\(3\)whereℰM=x1,x2,…,xM\\mathcal\{E\}\_\{M\}=\{x^\{1\},x^\{2\},\\ldots,x^\{M\}\}denotes the set of candidates evaluated during the search\.

### 3\.2Problem\-Specific Tools Construction

To maintain adaptability of TASA across tasks, we utilize an LLM to construct a problem\-specific tools library based on the characteristics of each optimization problem\. TASA constructs tool library through the following two coupled stages\.

#### Tool Archetype Generation\.

Given a structured context containing the problem description, constraints, candidate\-solution schema, and strategy definitions, the LLM generates tool archetypes for each of the four strategies\. Each archetype describes the intended functionality and behavior rather than providing directly executable code\.

#### Problem\-Specific Tool Archetype Instantiation\.

Once generated, the tool archetypes are validated for structural correctness, safety, and compatibility with the available primitive operators\. Valid archetypes are then bound to registered operators and instantiated as callable tools adapted to the current problem, while invalid archetypes are corrected or rejected\.

The LLM specifies each tool’s strategy,operator sequence, and parameters\. Registered operators execute theseupdatesand may perform multiple updates per candidate\.Tool\-construction prompts appearin the supplementary material\.

### 3\.3Optimization Strategy Planning

TASA uses an LLM to select optimization strategies fromtheproblem description,available tools, and optimization trajectory\. This allowstheagent to relate thecurrent search statetothepurpose of the next solution update\.Weletz∈𝒵:=\{rebuild,repair,polish,escape\}z\\in\\mathcal\{Z\}:=\\\{\\texttt\{rebuild\},\\texttt\{repair\},\\texttt\{polish\},\\texttt\{escape\}\\\}denote an optimization strategy, with thefour strategiesdefined as follows\.

- •rebuildreconstructs a solution from a different structural configuration, enabling the exploration of a new region of the search space\.
- •repairfocuses on reducing constraint violations and restoring the feasibility of the current solution\.
- •polishperforms local refinement around a promising feasible solution to further improve score\.
- •escapeapplies stronger structural perturbations to move away from a stagnant search region\.

The allocation amongthese intentionsadaptsto thesearch state\. At each replanning iterationkk,sks\_\{k\}summarizes solution quality, feasibility,stagnation, remaining budget, andtool feedback from the recorded trajectory\. This representation connects the observed search progress to the next optimization intention\. Based on this representation, the strategy planner produceszk=ΦLLM​\(sk\)z\_\{k\}=\\Phi\_\{\\mathrm\{LLM\}\}\(s\_\{k\}\), whereΦLLM\\Phi\_\{\\mathrm\{LLM\}\}denotes an inference\-time decision process induced by a prompt, andzkz\_\{k\}denotes the optimization strategy selected at iterationkk\.

### 3\.4Strategy\-Guided Adaptive Tool Planner

To translate the strategy selected by the agent into concrete solution updates, we introduce a strategy\-guided adaptive tool planner that balances tool exploration and optimization gain\. The strategyzkz\_\{k\}represents the optimization intention at iterationkk, but does not directly specify which tool should be executed\.

#### Candidate Tool Library Construction\.

The planner first constructs a candidate tool library that is compatible with both the problem structure and the current strategy, given by𝒞k=𝒯⁡\(P\)∩𝒯⁡\(zk\)\\mathcal\{C\}\_\{k\}=\\mathcal\{T\}\(P\)\\cap\\mathcal\{T\}\(z\_\{k\}\), where𝒯⁡\(P\)\\mathcal\{T\}\(P\)denotes the set of tools applicable to problemPP, and𝒯⁡\(zk\)\\mathcal\{T\}\(z\_\{k\}\)denotes the set of tools consistent with strategyzkz\_\{k\}\. Their intersection maps the abstract optimization strategy to a library of executable candidate tools\.

#### Balancing Tool Exploration and Optimization Gain\.

After constructing the candidate tool library, the planner balances exploration and exploitation by evaluating less\-tested tools while prioritizing those with proven effectiveness\. Let𝒞k0=\{t∈𝒞k∣nt=0\}\\mathcal\{C\}\_\{k\}^\{0\}=\\left\\\{t\\in\\mathcal\{C\}\_\{k\}\\mid n\_\{t\}=0\\right\\\}denote the set of untried candidate tools, and define𝒞kmin=\{t∈𝒞k\|nt=mins∈𝒞k⁡ns\}\\mathcal\{C\}\_\{k\}^\{\\min\}=\\left\\\{t\\in\\mathcal\{C\}\_\{k\}\\;\\middle\|\\;n\_\{t\}=\\min\_\{s\\in\\mathcal\{C\}\_\{k\}\}n\_\{s\}\\right\\\}as the set of candidate tools with the smallest invocation count\. Here,ntn\_\{t\}denotes the number of times tooltthas been invoked\. We further define the gain\-maximizing tool astk⋆=arg⁡maxt∈𝒞k​pk​\(t\)t\_\{k\}^\{\\star\}=\\arg\\max\_\{t\\in\\mathcal\{C\}\_\{k\}\}p\_\{k\}\(t\), wherepk​\(t\)p\_\{k\}\(t\)denotes the priority score assigned to toolttat iterationkk\.

Conditioned on the optimization trajectorysks\_\{k\}, the tooltkt\_\{k\}is selected according to the following probability distribution

Pr⁡\(tk=t∣sk\)=\{𝕀\[t∈𝒞k0\]\|𝒞k0\|,𝒞k0≠∅,ϵ​𝕀\[t∈𝒞kmin\]\|𝒞kmin\|\+\(1−ϵ\)𝕀\[t=tk⋆\]𝒞k0=∅\.\\Pr\\\!\\left\(t\_\{k\}=t\\mid s\_\{k\}\\right\)=\\begin\{cases\}\\displaystyle\\frac\{\\mathbb\{I\}\\\!\\left\[t\\in\\mathcal\{C\}\_\{k\}^\{0\}\\right\]\}\{\\left\|\\mathcal\{C\}\_\{k\}^\{0\}\\right\|\},&\\mathcal\{C\}\_\{k\}^\{0\}\\neq\\varnothing,\\\\\[10\.0pt\] \\begin\{aligned\} &\\displaystyle\\epsilon\\frac\{\\mathbb\{I\}\\\!\\left\[t\\in\\mathcal\{C\}\_\{k\}^\{\\min\}\\right\]\}\{\\left\|\\mathcal\{C\}\_\{k\}^\{\\min\}\\right\|\}\\\\\[\-1\.0pt\] &\\displaystyle\\quad\+\(1\-\\epsilon\)\\mathbb\{I\}\\\!\\left\[t=t\_\{k\}^\{\\star\}\\right\]\\end\{aligned\}&\\mathcal\{C\}\_\{k\}^\{0\}=\\varnothing\.\\end\{cases\}\(4\)where𝕀⁡\[⋅\]\\mathbb\{I\}\[\\cdot\]is the indicator function andϵ∈\[0,1\]\\epsilon\\in\[0,1\]controls the exploration rate\. If untried tools remain, the planner selects one uniformly from𝒞k0\\mathcal\{C\}\_\{k\}^\{0\}, prioritizing the evaluation of previously unused candidates\. Once all current candidate tools have been invoked, the planner explores with probabilityϵ\\epsilonby selecting uniformly from the least\-invoked tools in𝒞kmin\\mathcal\{C\}\_\{k\}^\{\\min\}\. With probability1−ϵ1\-\\epsilon, the planner follows the current optimization trajectory by selecting highest\-priority tooltk⋆t\_\{k\}^\{\\star\}\. This mechanism maintains tool diversity and prevents premature concentration on a small subset of tools\.

#### Dynamic Tool Prioritypk​\(t\)p\_\{k\}\(t\)\.

When randomized exploration is not performed, the planner selects the tool with the highest dynamic priority, defined as

pk​\(t\)=α​r¯t\+\(1−α\)​rtmax−λ​ζt,p\_\{k\}\(t\)=\\alpha\\bar\{r\}\_\{t\}\+\(1\-\\alpha\)r\_\{t\}^\{\\max\}\-\\lambda\\zeta\_\{t\},\(5\)wherer¯t\\bar\{r\}\_\{t\}andrtmaxr\_\{t\}^\{\\max\}respectively denote the average and maximum historical gains of tooltt, andα∈\[0,1\]\\alpha\\in\[0,1\]controls their relative contributions\. Moreover,ζt\\zeta\_\{t\}is the number of consecutive invocations without improvement, andλ≥0\\lambda\\geq 0controls the stagnation penalty\. Once the tool produces an improvement,ζt\\zeta\_\{t\}is reset to zero\.

The planner thus maintains tool diversity through initial and randomized exploration while progressively allocating more resources to tools that produce larger and more reliable improvements\.

### 3\.5Probabilistic Local Convergence Analysis

At each replanning time, the agent maps the optimization trajectory to a strategyzz\. Rather than updating the candidate solutions directly, TASA activates the compatible tool library𝒞⁡\(z\):=𝒯⁡\(P\)∩𝒯⁡\(z\)\\mathcal\{C\}\(z\):=\\mathcal\{T\}\(P\)\\cap\\mathcal\{T\}\(z\), from which the tool planner selects an executable tool\. We first characterize the neighborhoods induced by the strategies and their compatible tools, and then establish the corresponding local convergence result\.

#### Definition 1 \(Strategy\-Conditioned Tool Neighborhood\)

For eacht∈𝒞⁡\(z\)t\\in\\mathcal\{C\}\(z\), letqt​\(x′∣x\)q\_\{t\}\(x^\{\\prime\}\\mid x\)denote the probability that toolttgenerates a feasible candidatex′x^\{\\prime\}using the reference solutionxx\. The neighborhood induced by strategyzzis

𝒩𝒯​\(x∣z\):=⋃t∈𝒞⁡\(z\)\{x′∈𝒳P:qt​\(x′∣x\)\>0\}\.\\mathcal\{N\}\_\{\\mathcal\{T\}\}\(x\\mid z\):=\\bigcup\_\{t\\in\\mathcal\{C\}\(z\)\}\\\{x^\{\\prime\}\\in\\mathcal\{X\}\_\{P\}:q\_\{t\}\(x^\{\\prime\}\\mid x\)\>0\\\}\.\(6\)
Since TASA may activate different strategies, we aggregate their neighborhoods to define the overall local search region\.

#### Definition 2 \(TASA\-Induced Local Optimum\)

Define the overall reachable neighborhood𝒩TASA​\(x\)\\mathcal\{N\}\_\{\\mathrm\{TASA\}\}\(x\)and the improving setℐTASA​\(x\)\\mathcal\{I\}\_\{\\mathrm\{TASA\}\}\(x\)of TASA, respectively, as

𝒩TASA​\(x\)\\displaystyle\\mathcal\{N\}\_\{\\mathrm\{TASA\}\}\(x\):=⋃z∈𝒵𝒩𝒯​\(x∣z\),\\displaystyle:=\\bigcup\_\{z\\in\\mathcal\{Z\}\}\\mathcal\{N\}\_\{\\mathcal\{T\}\}\(x\\mid z\),\(7\)ℐTASA​\(x\)\\displaystyle\\mathcal\{I\}\_\{\\mathrm\{TASA\}\}\(x\):=\{x′∈𝒩TASA​\(x\):fP​\(x′\)\>fP​\(x\)\}\.\\displaystyle:=\\\{x^\{\\prime\}\\in\\mathcal\{N\}\_\{\\mathrm\{TASA\}\}\(x\):f\_\{P\}\(x^\{\\prime\}\)\>f\_\{P\}\(x\)\\\}\.\(8\)A feasible solution is TASA\-locally optimal ifℐTASA​\(x\)=∅\\mathcal\{I\}\_\{\\mathrm\{TASA\}\}\(x\)=\\varnothing\. The set of all TASA\-locally optimal solutions is denoted byℒTASA\\mathcal\{L\}\_\{\\mathrm\{TASA\}\}\.

At iterationkk,x^Pk\\hat\{x\}\_\{P\}^\{k\}denotes the best feasible solution found so far\.ℬk\\mathcal\{B\}\_\{k\}denotes the generated batch\.xkx\_\{k\}denote the reference solution\.xk′x^\{\\prime\}\_\{k\}denote the candidate solution\.

#### Assumption 3 \(Strict Improvements\)

The feasible set𝒳P\\mathcal\{X\}\_\{P\}is finite\.Assume an initial feasible solution is available\.TASA updates

x^Pk\+1=arg⁡maxx∈\{x^Pk\}∪\(ℬk∩𝒳P\)​fP​\(x\),\\hat\{x\}\_\{P\}^\{k\+1\}=\\arg\\max\_\{x\\in\\\{\\hat\{x\}\_\{P\}^\{k\}\\\}\\cup\(\\mathcal\{B\}\_\{k\}\\cap\\mathcal\{X\}\_\{P\}\)\}f\_\{P\}\(x\),\(9\)breaking ties in favor ofx^Pk\\hat\{x\}\_\{P\}^\{k\}\. Thus, the best feasible solution changes only after a strict improvement\. Since strict improvements can occur only finitely many times, persistent exploration is needed to ensure that TASA does not remain at a solution that is not locally optimal\.

#### Assumption 4 \(Persistent Local Exploration\)

Letℱk\\mathcal\{F\}\_\{k\}contain the full search history before batchℬk\\mathcal\{B\}\_\{k\}\.Foreveryx∉ℒTASAx\\notin\\mathcal\{L\}\_\{\\mathrm\{TASA\}\}, there existspx\>0p\_\{x\}\>0, uniform over iterations and histories withx^Pk=x\\hat\{x\}\_\{P\}^\{k\}=x,such thatalmost surely

Pr⁡\(ℬk∩ℐTASA​\(x\)≠∅∣ℱk\)≥px\.\\Pr\\\!\\left\(\\mathcal\{B\}\_\{k\}\\cap\\mathcal\{I\}\_\{\\mathrm\{TASA\}\}\(x\)\\neq\\varnothing\\mid\\mathcal\{F\}\_\{k\}\\right\)\\geq p\_\{x\}\.\(10\)
Assumptions 3 and 4 ensure finite improvement steps and exclude non\-locally optimal solutions as limiting states\. To formalize these implications, we establish two auxiliary lemmas, whose statements and proofs are deferred to supplementary material\.

#### Theorem5 \(TASA\-Local Convergence\)

Under Assumptions 3 and 4, the probability that the best feasible solution reaches a TASA\-induced local optimum in finite time and remains unchanged thereafter is given by

Pr⁡\(CLOSE\\displaystyle\\Pr\\\!\\bigl\(∃K<∞,∃x∞∈ℒTASA,\\displaystyle\\exists\\,K<\\infty,\\;\\exists\\,x\_\{\\infty\}\\in\\mathcal\{L\}\_\{\\mathrm\{TASA\}\},\(11\)OPENsuch that​x^Pk=x∞​for all​k≥K\)=1\.\\displaystyle\\text\{such that \}\\hat\{x\}\_\{P\}^\{k\}=x\_\{\\infty\}\\text\{ for all \}k\\geq K\\bigr\)=1\.Here,KKis a finite random time andx∞x\_\{\\infty\}is the random limiting best feasible solution\. The proofappearsin the supplementary material\.Assumption 4 requirespersistent explorationthrough strategy planningandtool execution, whereas equation[4](https://arxiv.org/html/2610.14736#S3.E4)describes tool selection for the current strategy\. Together withstrict improvement, this assumption yields localconvergence\.

Table 1:Performance comparison on eight benchmark tasks\. Results are reported as meanzz\-score±\\pmstandard deviation over seeds 1–3\. Higher values indicate better performance\. The best result for each task is shown inbold, and the second\-best result isunderlined\. — indicates that no feasible solution was obtained for at least one seed\.

## 4Experiments

### 4\.1Experimental Setup

#### Datasets\.

To comprehensively evaluate our method, we consider eight representative combinatorial optimization problems in two groups\. The first group includes discrete structure optimization problems, namelythe 0–1 Knapsack Problem \(KP\), Maximum Cut Problem \(MaxCut\), Balanced Graph Partition Problem \(BGP\), and Traveling Salesman Problem \(TSP\)\. These problems assess the adaptability of our method to diverse solution representations and combinatorial search spaces\. The second group covers routing and scheduling problems, namelythe Capacitated Vehicle Routing Problem \(VRP\), Pickup\-Delivery Routing Problem \(PDR\), Job\-Shop Scheduling Problem \(JSSP\), and Unrelated Parallel\-Machine Scheduling Problem \(MS\)\. They evaluate the method’s ability to handle complex constraints, including capacity limits, precedence relations, resource competition, and load balancing\. Detailed instance sizes, parameter settings, and generation procedures are provided in the supplementary material\.

#### Baselines\.

We compare our method with twelve baselines, including one approximate global\-optimum baseline, four traditional optimization baseline, four direct LLM baseline, and three LLM\-generated heuristic baseline\.

Approximate Global\-Optimum Baseline\.RefSolver\(*Reference Solver*\)[Perron et al\. \(2023\)](https://arxiv.org/html/2610.14736#bib.bib21)applies an exact algorithm when tractable and otherwise returns a solution produced by a deterministic strong heuristic\.Traditional optimization baselines\.RandSearch\(*Random Search*\)[Bergstra and Bengio \(2012\)](https://arxiv.org/html/2610.14736#bib.bib22)independently samples candidate solutions without exploiting evaluation history\.BO\(*Bayesian Optimization*\) uses a lightweight kernel\-regression model to estimate objective values and selects candidates based on an upper confidence bound\.GA\(*Genetic Algorithm*\) maintains a population of candidate solutions and generates new candidates through selection, crossover, and mutation\.CB\(*Combinatorial Bandit*\) treats solution components as bandit arms and constructs candidates according to their estimated rewards and exploration bonuses\.

Direct LLM solver baselines\.OPRO\(*Optimization by PROmpting*\)[Yang et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib23)prompts the LLM with the problem description and previously evaluated solutions to directly generate complete candidate solutions\.E2E Solver\(*End\-to\-End LLM Solver*\)[Jiang et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib24)uses a fixed prompt to directly generate complete candidate solutions for problem instance, without using previously evaluated solutions or optimization feedback\.Block E2E Solver\(*Block\-wise End\-to\-End LLM Solver*\)[Zheng et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib25)decomposes a high\-dimensional solution into blocks and asks the LLM to update one block at a time\.Block\-LS\(*Block\-wise LLM Solver with Local Search*\)[Jiang et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib24);[Zheng et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib25)alternates block\-wise LLM candidate generation with local refinement\.

LLM\-generated heuristic baselines\.FunSearch[Romera\-Paredes et al\. \(2024\)](https://arxiv.org/html/2610.14736#bib.bib26)is a lightweight archive\-guided search proxy that periodically injects randomly generated candidates and otherwise mutates high\-scoring historical solutions\.EoH\(*Evolution of Heuristics*\)[Liu et al\. \(2024a\)](https://arxiv.org/html/2610.14736#bib.bib27)periodically introduces random candidates and otherwise recombines and mutates high\-scoring historical solutions\.CoT\(*CoT Direct Solver*\)[Sprague et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib28)uses chain\-of\-thought prompting to guide the LLM through step\-by\-step problem analysis and directly generate candidate solutions\. Detailed implementation settings are provided in the supplementary material\.The supplementary material details these implementations, including LLM\-call limits and subsequent local\-search operations\.

#### Implementation Details\.

Qwen\-Plus[Yang et al\. \(2025\)](https://arxiv.org/html/2610.14736#bib.bib4)is accessed via API and used as the language model\. Each experiment is conducted using random seeds 1, 2, and 3\. Under each seed, all methods are evaluated on the same problem instance, with randomness in instance generation and initialization controlled by the corresponding seed\. Each run is limited to 60 candidate solution evaluations with a batch size of 2\. The budget counts generated candidates\. Tool construction, LLM calls, and computations within tools depend on the method, so results compare solution quality under this budget and the reported implementations\.

#### Metrics\.

For fair comparison across problems with different objective scales and optimization directions, all native objectives are converted into higher\-is\-better scores\. For problempp, seedss, and methodmm, the best feasible score is standardized aszp,s,m=yp,s,m−μp,sσp,sz\_\{p,s,m\}=\\frac\{y\_\{p,s,m\}\-\\mu\_\{p,s\}\}\{\\sigma\_\{p,s\}\}\. Here,yp,s,my\_\{p,s,m\}denotes the best feasible value offP​\(x\)f\_\{P\}\(x\)achieved by methodmm\. Details of the computation offP​\(x\)f\_\{P\}\(x\)are provided in the supplementary material\. For each problemppand seedss,μp,s\\mu\_\{p,s\}is the mean of these values across all methods that produce a feasible solution, andσp,s\\sigma\_\{p,s\}is the corresponding population standard deviation\. Accordingly,zp,s,m\>0z\_\{p,s,m\}\>0indicates above\-average performance, whereaszp,s,m<0z\_\{p,s,m\}<0indicates below\-average performance for the corresponding problem and seed\.

### 4\.2Comparative Performance on Benchmark Tasks

From the results presented in Table[1](https://arxiv.org/html/2610.14736#S3.T1), it can be observed that TASA achieves the strictly best mean score on five tasks, ties RefSolver for the best result on KP, and ranks second on BGP and PDR\. On BGP, it nearly matches RefSolver, with a gaps of only 0\.009\. Its averagezz\-score across all tasks is 1\.806, almost identical to RefSolver’s 1\.809\. These results show that, under the same evaluation budget, TASA is competitive with specialized solvers while retaining a unified framework across problems with diverse structures and constraints\.

TASA also consistently outperforms direct LLM\-based methods\. It exceeds OPRO on every task by 2\.376zz\-score points on average and surpasses the strongest direct LLM baseline by an average of 1\.762 points\. The advantage is particularly clear on BGP, VRP, JSSP, and MS, where direct LLM solvers obtain negative mean scores while TASA remains strongly positive\. These negative values arise fromzz\-score normalization, as a method receives a negative score when its best feasible score falls below the cross\-method mean for the corresponding problem and seed\. These resultssupportthecombined useof optimization trajectories, strategy planning,and problem\-specific toolsin TASA under the reported settings\.

Finally, TASA finds feasible solutions for every seed on all eight tasks and has a standard deviation below 0\.2 on six tasks, demonstrating reliable performance across repeated runs\. Because some baselines do not reliably produce feasible solutions across random seeds, we use “—” in Table[1](https://arxiv.org/html/2610.14736#S3.T1)to indicate that no feasible solution was obtained for at least one seed\. On four of the five tasks where it ranks first, its standard deviation is no larger than that of RefSolver\. Its main weakness is PDR, where it trails RefSolver by 1\.145 points, suggesting that more specialized routing and repair tools are needed\. Overall, the results demonstrate the effectiveness, generality, and robustness of our method\.

![Refer to caption](https://arxiv.org/html/2610.14736v1/pic2.png)
Figure 2:Traceability of TASA over 60 evaluations on a representative run of VRP \(Seed 3\)\. The curve shows the best\-so\-far scores, while the markers and colored timelines indicate the optimization strategies selected throughout the search\.
![Refer to caption](https://arxiv.org/html/2610.14736v1/pic3.png)
Figure 3:Effect of the tool\-planning mechanism on best\-so\-far performance over 60 evaluations for TSP and VRP\. The curves show the mean best\-so\-far scores over seeds 1\-3, and the shaded regions indicate one standard deviation around the mean\. Ours denotes the strategy\-guided adaptive tool planner, while Random and UCB1 are alternative tool\-planning policies\.

![Refer to caption](https://arxiv.org/html/2610.14736v1/pic4.png)
Figure 4:Effect of tool\-library size on best\-so\-far performance over 60 evaluations for JSSP and VRP\. The curves show the mean scores over seeds 1–3, and the shaded regions indicate one standard deviation around the mean\. All Tools uses the full library, while Half Tools and Quarter Tools retain one\-half and one\-quarter of the tools, respectively\.
![Refer to caption](https://arxiv.org/html/2610.14736v1/pic5.png)
Figure 5:Effect of feedback granularity on best\-so\-far performance for BGP and MaxCut\. The bars report the mean best\-so\-far scores over seeds 1–3 at evaluations 1, 20, 40, and 60\. Full Feedback provides the strategy planner with therecordedoptimization trajectory, whereas Solution\-Level Feedback retains only solution\-level information\.

#### Traceability of TASA\.

Figure 2 makesstrategyselections and subsequent score improvements traceable ina representative VRP run\.Rebuild precedes improvements near evaluations 8 and 28, andescapenear16and49\.

### 4\.3Effect of the Tool Planner\.

To evaluate the effectiveness of the tool\-planning mechanism, we compare our strategy\-guided adaptive tool planner with random selection and UCB1\. As shown in Figure[3](https://arxiv.org/html/2610.14736#S4.F3), our planner achieves the highest final scores on both TSP and VRP\. On TSP, it rapidly reaches 14\.92, whereas Random and UCB1 converge to only 13\.88 and 13\.27\. These resultssupport using historical tool gains and consecutive invocations without improvement to selecteffective toolsforthe intendedoptimizationstrategy\.

### 4\.4Effect of Tool Library Size\.

To evaluate the effect of tool\-library size, we compare TASA using all tools with variants retaining one\-half or one\-quarter of the library\. As shown in Figure[4](https://arxiv.org/html/2610.14736#S4.F4), the full library achieves the best final scores on both JSSP and VRP, reaching 7\.01 and 5\.63, respectively\. The advantage is especially pronounced on VRP, where using half and one\-quarter of the tools yields only 4\.42 and 2\.95\.Thelargertool library provides more problem\-specifictools forexecuting the selected strategies,supporting the contributionof tool diversity to solution quality\.

### 4\.5Effect of Feedback Granularity\.

To evaluate the effect of feedback granularity, we compare Full Feedback with Solution\-Level Feedback\. As shown in Figure[5](https://arxiv.org/html/2610.14736#S4.F5), Full Feedback achieves higher scores on both BGP and MaxCut\. On BGP, Solution\-Level Feedback plateaus by 20th evaluation, whereas Full Feedback continues to improve in subsequent evaluations\. On MaxCut, both variants improve until around 40th evaluation and then change little, but Full Feedback consistently maintains higher scores\. These resultssupport using theoptimizationtrajectory toprovidethe LLM strategy planner withinformationaboutsearch progress and stagnation\.

## 5Conclusion

In this paper, we propose TASA, which shifts the role of the LLM from directly generating solutions to planning optimization strategies based on optimization trajectories and executing them through a strategy\-guided adaptive tool planner\. Experimental results on eight benchmark problems demonstrate that TASA provides an effective and traceable framework for LLM\-based combinatorial optimization\.

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