Large Language Models as Evolutionary Optimizers
作为进化优化器的大型语言模型

IEEE Publication Technology
电气和电子工程师学会出版技术Manuscript received xxx.
收到手稿 xxx。 Shengcai Liu, Caishun Chen, Xinghua Qu, Ke Tang, and Yew-Soon Ong
Shengcai Liu、Caishun Chen、Xinghua Qu、Ke Tang 和 Yew-Soon OngShengcai Liu and Caishun Chen are with the Centre for Frontier AI Research (CFAR), Agency for Science, Technology and Research (A*STAR), Singapore (email: liu_shengcai@cfar.a-star.edu.sg; chen_caishun@cfar.a-star.edu.sg). Xinghua Qu is with Shanda Group (email: quxinghua17@gmail.com). Ke Tang is with the Guangdong Provincial Key Laboratory of Brain-Inspired Intelligent Computation, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China (tangk3@sustech.edu.cn). Yew-Soon Ong is with the Centre for Frontier AI Research (CFAR), Agency for Science, Technology and Research (A*STAR), Singapore, and is also with the School of Computer Science and Engineering, Data Science and Artificial Intelligence Research Centre, Nanyang Technological University (NTU), Singapore (email: asysong@ntu.edu.sg).
Shengcai Liu和Caishun Chen就职于新加坡科技研究局（A*STAR）人工智能前沿研究中心（CFAR）（电子邮件：liu_shengcai@cfar.a-star.edu.sg; chen_caishun@cfar.a-star.edu.sg）。瞿兴华就职于盛大集团（电子邮件：quxinghua17@gmail.com）。唐珂，南方科技大学计算机科学与工程系脑启发智能计算广东省重点实验室，深圳，中国 (tangk3@sustech.edu.cn)。Yew-Soon Ong是新加坡科技研究局（A*STAR）人工智能前沿研究中心（CFAR）的成员，同时也是新加坡南洋理工大学（NTU）计算机科学与工程学院、数据科学与人工智能研究中心的成员（电子邮件：asysong@ntu.edu.sg）。

Abstract 摘要

Evolutionary algorithms (EAs) have achieved remarkable success in tackling complex combinatorial optimization problems. However, EAs often demand carefully-designed operators with the aid of domain expertise to achieve satisfactory performance. In this work, we present the first study on large language models (LLMs) as evolutionary combinatorial optimizers. The main advantage is that it requires minimal domain knowledge and human efforts, as well as no additional training of the model. This approach is referred to as LLM-driven EA (LMEA). Specifically, in each generation of the evolutionary search, LMEA instructs the LLM to select parent solutions from current population, and perform crossover and mutation to generate offspring solutions. Then, LMEA evaluates these new solutions and include them into the population for the next generation. LMEA is equipped with a self-adaptation mechanism that controls the temperature of the LLM. This enables it to balance between exploration and exploitation and prevents the search from getting stuck in local optima. We investigate the power of LMEA on the classical traveling salesman problems (TSPs) widely used in combinatorial optimization research. Notably, the results show that LMEA performs competitively to traditional heuristics in finding high-quality solutions on TSP instances with up to 20 nodes. Additionally, we also study the effectiveness of LLM-driven crossover/mutation and the self-adaptation mechanism in evolutionary search. In summary, our results reveal the great potentials of LLMs as evolutionary optimizers for solving combinatorial problems. We hope our research shall inspire future explorations on LLM-driven EAs for complex optimization challenges.
进化算法（EA）在解决复杂的组合优化问题方面取得了巨大成功。然而，进化算法通常需要借助领域专业知识精心设计算子，才能获得令人满意的性能。在这项工作中，我们首次将大型语言模型（LLMs）作为进化组合优化器进行研究。这种方法的主要优点是只需最少的领域知识和人力，也无需对模型进行额外的训练。这种方法被称为LLM驱动的 EA（LMEA）。具体来说，在每一代进化搜索中，LMEA 都会指示LLM从当前种群中选择父解，并执行交叉和变异以生成子解。然后，LMEA 对这些新方案进行评估，并将其纳入下一代种群。LMEA 配备了自适应机制，可以控制LLM的温度。这使它能够在探索和利用之间取得平衡，防止搜索陷入局部最优状态。我们研究了 LMEA 在组合优化研究中广泛使用的经典旅行推销员问题（TSP）上的威力。值得注意的是，研究结果表明，在多达 20 个节点的 TSP 实例中，LMEA 在寻找高质量解决方案方面的表现优于传统启发式方法。此外，我们还研究了进化搜索中LLM驱动的交叉/突变和自适应机制的有效性。总之，我们的研究结果揭示了LLMs作为进化优化器解决组合问题的巨大潜力。我们希望我们的研究能为未来探索LLM驱动的进化优化器应对复杂优化挑战提供启发。

Index Terms:

Evolutionary algorithms, large language model, combinatorial optimization, pre-trained model
索引术语：进化算法、大型语言模型、组合优化、预训练模型

I Introduction I 引言

Evolutionary algorithms (EAs) are a class of algorithms that draw inspiration from natural evolution [1]. By simulating the principles of natural selection and genetic variation, EAs have been widely utilized in tackling complex combinatorial optimization problems arising in various domains, including logistics [2, 3, 4, 5], cloud computing [6, 7], manufacturing [8, 9, 10], and robotics [11]. Despite the huge success achieved thus far, EAs, in general, still require carefully handcrafted operators to achieve high performance. Typically, the design process of an EA involves algorithm experts analyzing the problem structure, tailoring specialized genetic operators (e.g., crossover and mutation) that exploit this structure most effectively, and continually refining these operators [12]. This process heavily relies on domain expertise and human efforts, and can become even more burdensome when dealing with new problems.
进化算法（EAs）是一类从自然进化中汲取灵感的算法[1]。通过模拟自然选择和遗传变异的原理，进化算法已被广泛用于解决物流[2, 3, 4, 5]、云计算[6, 7]、制造[8, 9, 10]和机器人[11]等各个领域出现的复杂组合优化问题。尽管迄今为止取得了巨大成功，但一般来说，EA 仍然需要精心设计的人工操作才能实现高性能。通常情况下，EA 的设计过程包括算法专家分析问题结构，定制最有效地利用这一结构的专门遗传算子（如交叉和突变），并不断完善这些算子[12]。这一过程严重依赖领域专业知识和人力，在处理新问题时可能会变得更加繁重。

Motivated by the above challenge, recently there has been a surge of research interest in automating (part of) the design process of EAs, primarily through a meta-optimization paradigm. Specifically, these approaches construct a meta-level optimization problem where the decision variables are design choices of the algorithm, and the optimization objective is the algorithm’s performance on a pre-collected set of problem instances (called the training set). Then, by solving this meta-optimization problem, the design choices of the algorithm are automatically determined. In the literature, there are various ways to represent the algorithm design choices, ranging from algorithm parameters [13, 14], search heuristics [15, 16, 17, 18], to end-to-end deep neural networks [19, 20]. However, the meta-optimization approaches still pose non-trivial challenges, including how to select a representative training set that can sufficiently reflect the target cases where the algorithm will be applied [21, 22], how to build a compact and effective algorithm design space [23], and how to solve the meta-optimization problem [24]. Moreover, these approaches often need to consume significant computational resources to obtain a high-performing algorithm.
在上述挑战的推动下，最近出现了一股研究热潮，主要是通过元优化范式，将 EA 的（部分）设计过程自动化。具体来说，这些方法构建了一个元优化问题，其中的决策变量是算法的设计选择，优化目标是算法在预先收集的问题实例集（称为训练集）上的性能。然后，通过解决这个元优化问题，就能自动确定算法的设计选择。文献中有多种表示算法设计选择的方法，从算法参数[13, 14]、搜索启发式[15, 16, 17, 18]到端到端深度神经网络[19, 20]。然而，元优化方法仍然面临着非同小可的挑战，包括如何选择一个有代表性的训练集来充分反映算法应用的目标案例[ 21, 22]，如何构建一个紧凑有效的算法设计空间[ 23]，以及如何解决元优化问题[ 24]。此外，这些方法通常需要消耗大量计算资源才能获得高性能算法。

Large language models (LLMs) have recently yielded impressive results in a wide range of domains [25, 26, 27, 28, 29, 30, 31, 32, 33]. These models extract human knowledge by learning from vast amounts of text data and have demonstrated remarkable reasoning and decision-making capabilities [34, 35, 36, 37, 38, 39, 40, 41]. From this perspective, it is plausible that the knowledge embedded in LLMs also encompasses human experiences and intuitions in designing optimization algorithms. This naturally raises an interesting question: can LLMs be used to help EAs solve complex optimization problems?
大型语言模型（LLMs）最近在众多领域取得了令人瞩目的成果[25, 26, 27, 28, 29, 30, 31, 32, 33]。这些模型通过从海量文本数据中学习来提取人类知识，并展示了卓越的推理和决策能力[ 34, 35, 36, 37, 38, 39, 40, 41]。从这个角度看，LLMs中蕴含的知识也包含了人类在设计优化算法时的经验和直觉，这一点是可信的。这自然引出了一个有趣的问题：LLMs能否用于帮助EA解决复杂的优化问题？

This work provides an affirmative answer to the above question. Specifically, we propose an innovative approach, named LLM-driven EA (LMEA), for solving combinatorial optimization problems. In each generation of the evolutionary search, LMEA constructs a prompt to instruct the LLM to select parent solutions from the current population, and perform crossover and mutation to generate offspring solutions. Then, these new solutions are evaluated and added to the population for the next generation. Additionally, a simple self-adaptation mechanism is integrated into LMEA to control the temperature of the LLM, thus balancing its exploration and exploitation.
本研究为上述问题提供了肯定的答案。具体来说，我们提出了一种创新方法，名为LLM驱动进化搜索（LMEA），用于解决组合优化问题。在每一代进化搜索中，LMEA 都会构建一个提示，指示LLM从当前种群中选择父解，并执行交叉和变异以生成子解。然后，对这些新方案进行评估，并将其添加到下一代种群中。此外，LMEA 还集成了一个简单的自适应机制，用于控制LLM的温度，从而平衡其探索和利用。

From the perspective of designing EAs, LMEA has two appealing features. First, due to the capabilities of LLMs, in LMEA we can describe the optimization problem and the desired solution properties in natural language to instruct the LLM. In consequence, optimization with LMEA enables quick adaptation to different optimization problems by changing the problem description and solution specifications in the prompt. Compared to the traditional practice of formally defining the problem and implementing operators through programming, LMEA follows an approach that is more direct and only demands minimal domain knowledge and human efforts. Second, LMEA leverages the LLM in a zero-shot manner. Here, the term “zero-shot” means that no additional training of the model is required, which is a significant advantage compared to meta-optimization approaches that require extensive computational resources to optimize the algorithm.
从设计 EA 的角度来看，LMEA 有两个吸引人的特点。首先，由于LLMs的功能，在LMEA中，我们可以用自然语言描述优化问题和所需的解决方案属性，以指示LLM。因此，使用 LMEA 进行优化时，只需更改提示中的问题描述和解决方案说明，即可快速适应不同的优化问题。与正式定义问题并通过编程实现运算符的传统做法相比，LMEA 采用的方法更为直接，只需极少的领域知识和人力。其次，LMEA 以 "0-shot "的方式利用LLM。这里的 "零次 "指的是不需要对模型进行额外的训练，与需要大量计算资源来优化算法的元优化方法相比，这是一个显著的优势。

We investigate the power of LMEA on the classical traveling salesman problems (TSPs) widely used in combinatorial optimization research. The experimental results demonstrate that, despite its minimal reliance on domain expertise, LMEA performs competitively to the traditional heuristics on TSP instances with up to 20 nodes. Surprisingly, it consistently obtains the optimal solutions on TSP instances with 10 nodes and 15 nodes. Furthermore, we conduct experiments to verify the effectiveness of the LLM-driven genetic operators and the self-adaptation mechanism.
我们研究了 LMEA 在组合优化研究中广泛使用的经典旅行推销员问题（TSP）上的威力。实验结果表明，尽管 LMEA 对领域专业知识的依赖性极低，但它在多达 20 个节点的 TSP 实例上的表现与传统启发式算法相比仍具有竞争力。令人惊讶的是，它还能在 10 节点和 15 节点的 TSP 实例中持续获得最优解。此外，我们还通过实验验证了LLM驱动遗传算子和自适应机制的有效性。

To the best of our knowledge, this is the first attempt of utilizing LLMs in evolutionary combinatorial optimization. Also, we would like to note that this work is not to show that LMEA can outperform those sophisticated specialized solvers for classical combinatorial optimization problems like TSPs. Instead, the goal is to introduce an approach with a significant departure from previous design paradigms of EAs. Importantly, this approach indeed demonstrates its capacity to solve non-trivial hard combinatorial optimization problems. We hope that our results will inspire further exploration of LLM-driven EAs for combinatorial optimization challenges.
据我们所知，这是首次尝试在进化组合优化中使用LLMs。此外，我们还想指出的是，这项工作并不是要证明 LMEA 在处理 TSP 等经典组合优化问题时可以超越那些复杂的专门求解器。相反，我们的目标是介绍一种与以往的 EA 设计范式大相径庭的方法。重要的是，这种方法确实证明了其解决非难组合优化问题的能力。我们希望，我们的研究结果将激励人们进一步探索LLM驱动的EA，以应对组合优化挑战。

The remainder of this paper is organized as follows. Section II briefly reviews the literature on EAs for combinatorial optimization, LLMs and prompts, as well as the intersection between EAs and LLMs. Section III presents the approach LMEA. Experiments on TSPs are presented in Section IV. Finally, Section V concludes the paper with discussions.
本文的其余部分安排如下。第二节简要回顾了有关组合优化、LLMs和提示的EA文献，以及EA与LLMs之间的交叉。第三节介绍了 LMEA 方法。第四节介绍了 TSP 实验。最后，第五部分对本文进行了总结和讨论。

II Related Works II 相关作品

II-A EAs for Combinatorial Optimization
II-A 用于组合优化的 EAs

Combinatorial optimization involves finding the best possible solution from a finite solution set. It has many applications in various domains [42]. From the perspective of computational complexity, many combinatorial optimization problems are NP-hard due to their discrete and nonconvex nature [43]. For these problems, exact methods such as the branch and bound algorithms [44] can find the optimal solutions, but generally suffer from exponential time complexity. In contrast, meta-heuristics seek to find good (but not necessarily optimal) solutions within reasonable computation time. EAs, as one of the mainstream meta-heuristics, have achieved significant progress in solving combinatorial optimization problems over the past decades and are still undergoing diverse and flourishing development. Currently, EAs have established themselves as state-of-the-art methods for various combinatorial optimization problems [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. These algorithms range from classic EAs like genetic algorithms [2, 3], hybrid approaches such as memetic algorithms [5, 4, 7, 11], to co-evolutionary algorithms [45].
组合优化是指从有限的解集中找出可能的最佳解决方案。它在各个领域都有很多应用 [ 42]。从计算复杂度的角度来看，由于离散和非凸的性质，许多组合优化问题都是 NP 难问题[43]。对于这些问题，精确方法（如分支与边界算法[44]）可以找到最优解，但一般具有指数级的时间复杂性。与此相反，元启发式试图在合理的计算时间内找到好的（但不一定是最优的）解。EA 作为主流元启发式方法之一，在过去几十年里在解决组合优化问题方面取得了重大进展，目前仍在经历着多样化的蓬勃发展。目前，EA 已成为解决各种组合优化问题的最先进方法[2, 3, 4, 5, 6, 7, 8, 9, 10, 11]。这些算法包括遗传算法[2, 3]等经典 EA、记忆算法[5, 4, 7, 11]等混合方法以及协同进化算法[45]。

In general, when applying EAs to solve combinatorial optimization problems, algorithm practitioners often need to design operators tailored to the problem to obtain satisfactory performance. These operators can be specific to the solution representations, such as the various crossover variants [46] (e.g., single-point, multi-point, shuffle, and matrix) for binary solution representations in set maximization problems, as well as the k-opt mutation [47] and edge assembly crossover [48] for the permutation-based solution representations in TSPs and vehicle routing problems (VRPs). In certain cases, one also needs to design repair operators [49] to ensure that the solutions found by EAs adhere to problem constraints, and regrouping operators [2] to maintain population diversity. Overall, the design of effective operators relies on a deep understanding of the problem of interest, which requires extensive domain expertise and human efforts [12].
一般来说，在应用 EA 解决组合优化问题时，算法实践者往往需要设计适合问题的算子，以获得令人满意的性能。这些算子可以是特定的解表示，例如集合最大化问题中二元解表示的各种交叉变体[46]（如单点、多点、洗牌和矩阵），以及 TSP 和车辆路由问题（VRP）中基于排列的解表示的 k-opt 突变[47]和边缘集合交叉[48]。在某些情况下，我们还需要设计修复算子[49]来确保 EA 找到的解符合问题约束条件，以及设计重组算子[2]来保持群体多样性。总之，有效算子的设计依赖于对相关问题的深入理解，这需要广泛的领域专业知识和人力[12]。

II-B Large Language Models (LLMs) and Prompts
II-B 大型语言模型（LLMs）和提示

LLMs are large deep neural networks (often with billions of parameters) which are trained on vast amounts of text data to perform next-token prediction, i.e., predict the proceeding token given a sequence of tokens seen before. In the last two years, scaling up LLMs has yielded groundbreaking performance across a broad spectrum of tasks [25, 26, 27, 28, 29, 30, 31, 32, 33]. Among them of particular interest are tasks involving reasoning and decision-making, such as planning [40, 41, 39] and solving mathematical problems [37, 36, 35]. Given the inherent inter-connections between optimization, reasoning, and decision-making, it is likely that LLMs could also demonstrate competence in optimization tasks, which is indeed one motivation behind this work.
LLMs是大型深度神经网络（通常有数十亿个参数），在海量文本数据的基础上进行训练，以执行下一个标记预测，即根据之前看到的标记序列预测下一个标记。在过去的两年中，LLMs的扩展在各种任务中取得了突破性的性能[25, 26, 27, 28, 29, 30, 31, 32, 33]。其中特别值得关注的是涉及推理和决策的任务，如规划[40, 41, 39]和解决数学问题[37, 36, 35]。鉴于优化、推理和决策之间的内在联系，LLMs也有可能在优化任务中表现出能力，而这正是这项工作的动机之一。

A prompt is the instruction that guides LLMs into generating desired output. Although a prompt can be in various forms such as a question, a sentence, or a keyword, there has been much evidence [34, 35, 36] showing that the prompt format can significantly influence the quality of the LLM’s output. In general, when faced with a specific task, one needs to carefully craft the prompt to align with the desired outcome. This involves considering the context, the level of detail required, and the clarity of the instruction. Previous research has identified several effective techniques for prompting. For example, one can include several task examples as demonstrations in the prompt, i.e., in-context learning [50], and can explicitly instruct the LLM to think step by step, i.e., chain of thoughts [34]. In this work, to make LLMs function effectively as the operators of EAs, we carefully design a well-structured prompt that consists of several parts including task descriptions, solution properties, population information, and operator instructions.
提示是引导LLMs生成所需结果的指令。虽然提示可以是问题、句子或关键词等多种形式，但许多证据[ 34, 35, 36]表明，提示格式会极大地影响LLM的输出质量。一般来说，面对特定任务时，我们需要精心设计提示语，使其与预期结果相一致。这就需要考虑背景、所需的详细程度以及指令的清晰度。以往的研究发现了几种有效的提示技巧。例如，可以在提示中加入几个任务示例作为示范，即情境学习[50]，还可以明确指示LLM一步一步地思考，即思维链[34]。在这项工作中，为了让LLMs有效地发挥 EA 操作员的作用，我们精心设计了一个结构良好的提示，其中包括任务描述、解决方案属性、种群信息和操作员指令等几个部分。

II-C Intersection between EAs and LLMs
II-C 《环境协定》与 LLMs 的交叉点

Currently, research at the intersection of LLMs and EAs is still in its early stage. Due to the remarkable capabilities of LLMs in code generation, several recent works have attempted to combine LLMs with EAs to generate neural network structures [51], programs that fulfill specific functionalities [52], and meta-heuristics [53]. Furthermore, the core ideas of EAs, including natural selection and genetic variations, have also been employed to evolve prompts to enhance the performance of LLMs [54]. Regarding directly applying LLMs to solve optimization problems, there is very limited research. One such work is [55], which presents preliminary results of using LLMs for solving linear regression problems and TSPs. Another very recent work [56] incorporates LLMs as the mutation operator in continuous multi-objective optimization algorithm. However, the utilization of LLMs for evolutionary combinatorial optimization still remains unexplored.
目前，LLMs与EA的交叉研究仍处于早期阶段。由于LLMs在代码生成方面的卓越能力，最近有几项研究尝试将LLMs与EA结合起来，生成神经网络结构[51]、实现特定功能的程序[52]以及元启发式算法[53]。此外，包括自然选择和遗传变异在内的EA核心思想也被用于进化提示，以提高LLMs的性能。[ 54].关于直接应用LLMs解决优化问题，目前的研究非常有限。其中一篇论文[ 55]介绍了使用LLMs解决线性回归问题和TSP的初步结果。另一项最新研究[ 56] 将LLMs作为突变算子纳入连续多目标优化算法。然而，LLMs在进化组合优化中的应用仍有待探索。

Refer to caption — Figure 1: An overview of LMEA. The right half of this diagram demonstrates an example of the constructed prompt when utilizing LMEA to solve TSPs. The contents within “{}” in the prompt will be replaced with the corresponding input.
图 1：LMEA 概述。图的右半部分展示了利用 LMEA 解决 TSP 时构建的提示示例。提示中"{}"内的内容将被替换为相应的输入内容。

Input: The optimization problem

T

, maximum number of generations

G

, population size

N

;
输入优化问题

T

，最大代数

G

，种群数量

N

；

Output: the best found solution

s^{*}

输出：找到的最佳解决方案

s^{*}

P\leftarrow

randomly initialize

N

solutions to

T

;
1

P\leftarrow

随机初始化

N

解为

T

；

g=1

;

3 while $g\leq G$ do
3 当

g\leq G

做

prompt\leftarrow

construct prompt based on

T

and

pop

;
4

prompt\leftarrow

在

T

和

pop

的基础上构建提示；

P^{\prime}\leftarrow

instruct LLM with

prompt

to generate

N

offspring solutions;
5 个

P^{\prime}\leftarrow

用

prompt

指导LLM，生成

N

个子代方案；

P\leftarrow

the top

N

solutions among

P\cup P^{\prime}

;
6

P\leftarrow

P\cup P^{\prime}

中的顶级

N

解；

7 Self-adapt the temperature of LLM if necessary;
7 必要时自行调整 LLM 的温度；

g\leftarrow g+1

;

10 end while 10 结束 while

s^{*}\leftarrow

the best solution in

P

;
11

s^{*}\leftarrow

是

P

中的最佳解决方案；

return $s^{*}$ 返回

s^{*}

Algorithm 1 LLM-driven EA (LMEA)
算法 1 LLM 驱动的 EA（LMEA）

III LMEA for Combinatorial Optimization
III 用于组合优化的 LMEA

In this section, we first present the framework of LMEA, then delve into the construction of prompts to instruct the LLM, and finally describe the self-adaptation mechanism of the LLM’s temperature. An overview of LMEA is also illustrated in Figure 1.
在本节中，我们将首先介绍 LMEA 的框架，然后深入探讨如何构建提示来指导LLM，最后介绍LLM的温度自适应机制。图1也展示了LMEA的概况。

III-A Algorithm Framework
III-A 算法框架

As presented in Algorithm 1, LMEA follows the conventional EA framework [1]. Given an optimization problem $T$ , LMEA first randomly generates $N$ solutions to form the initial population (line 1) and then proceeds with the evolutionary process (lines 3-9). In each generation, LMEA constructs a prompt (line 4) to guide LLM in selecting parent solutions from the current population and then conduct crossover and mutation based on them, generating $N$ offspring solutions (line 5). These $N$ solutions are then combined with the current population (line 6) and the top $N$ solutions are retained for the next generation (line 7). In addition, the LLM’s temperature would be adjusted if necessary (see Section III-C). Finally, LMEA would terminate when the number of generations reaches a predefined number and the best found solution is returned (lines 10-11).
如算法 1 所示，LMEA 遵循传统的 EA 框架[ 1]。给定一个优化问题 $T$ ，LMEA 首先随机生成 $N$ 解，形成初始种群（第 1 行）。LMEA首先随机生成 $N$ 解，形成初始种群（第1行），然后继续演化过程（第3-9行）。在每一代中，LMEA 会构建一个提示（第 4 行），引导LLM从当前种群中选择父解，然后在此基础上进行交叉和变异，生成 $N$ 子解（第 5 行）。然后将这些 $N$ 解与当前种群相结合（第 6 行），并将最优秀的 $N$ 解保留到下一代（第 7 行）。此外，LLM的温度也会在必要时进行调整（见第 III-C 节）。最后，当世代数达到预定数目时，LMEA 将终止，并返回找到的最佳解决方案（第 10-11 行）。

III-B LLMs as Evolutionary Optimizers
III-B 作为进化优化器的LLMs

LMEA employs the LLM as evolutionary operators in a zero-shot manner. Specifically, the parent selection and genetic variations (crossover and mutation) are accomplished through the in-context learning process of the LLM [50], which is facilitated by carefully constructed prompts.
LMEA 采用LLM作为进化算子，以零点的方式进行进化。具体来说，父本选择和遗传变异（交叉和突变）是通过LLM的上下文学习过程完成的。[50]，而精心设计的提示则为这一过程提供了便利。

To be concrete, the prompt consists of three parts:
具体来说，提示包括三个部分：

•

Problem description and solution properties: this part includes the description of the optimization problem to be solved and specifications of the desired solution properties.

- 问题描述和解决方案属性：这一部分包括待解优化问题的描述和所需解决方案属性的说明。
•

In-context examples: some solutions to the optimization problem and their corresponding fitness are provided as demonstrations for the LLM. Typically, these solutions are derived from the current population.

- 上下文示例：为LLM提供一些优化问题的解决方案及其相应的适应度作为示范。通常情况下，这些解决方案来自当前种群。
•

Task Instructions: this part provides explicit instructions for the LLM to perform parent selection and carry out crossover and mutation, generating new solutions.

- 任务指令：这部分为LLM提供了明确的指令，用于执行父代选择、交叉和变异，生成新的解。

Figure 1 illustrates an example of the constructed prompt when using LMEA to solve TSPs. The problem description contains the coordinates of the points in the TSP instance, while the solution properties specify the constraints that TSP solutions must meet (traversing each point exactly once) and that shorter lengths are preferable. In-context examples consist of TSP solutions from the current population and their corresponding lengths. Task Instructions guide the LLM in generating new TSP solutions.
图 1 展示了使用 LMEA 解决 TSP 时所构建的提示示例。问题描述包含 TSP 实例中各点的坐标，而解决方案属性则规定了 TSP 解决方案必须满足的约束条件（每个点精确遍历一次）以及长度越短越好。上下文中的示例包括当前列表中的 TSP 解及其相应长度。任务指令可指导LLM生成新的 TSP 解决方案。

It is important to note that, unlike the traditional practice of implementing evolutionary operators step-by-step through programming, LMEA does not instruct the LLM on how to precisely perform parent selection, crossover, and mutation. Instead, LMEA instructs the LLM at a higher level using natural language. This approach only requires minimal reliance on domain expertise. Finally, the prompt also strictly defines the format of the LLM’s output to enable LMEA to interpret the output easily. For example, the results of parent selection are enclosed between $<$ selection $>$ and $<$ /selection $>$ , and the generated solutions are enclosed between $<$ res $>$ and $<$ /res $>$ .
值得注意的是，与通过编程逐步实现进化运算符的传统做法不同，LMEA 不会指导LLM 如何精确执行父代选择、交叉和变异。相反，LMEA 使用自然语言在更高层次上对LLM进行指导。这种方法对领域专业知识的依赖性极低。最后，提示还严格定义了LLM的输出格式，使 LMEA 能够轻松解释输出结果。例如，父级选择的结果括在 $<$ 选择 $>$ 和 $<$ 之间。/选择 $>$ 生成的解决方案括在 $<$ res $>$ 和 $<$ 之间。/res $>$ .

III-C Self-Adaptation of the LLM’s Temperature
III-C LLM的温度自适应

The temperature of LLMs, which is used in the sampling process when generating text, is a parameter that controls the randomness or entropy of the text [50]. A higher temperature value increases the randomness, while a lower value makes the model’s output more deterministic.
LLMs的温度用于生成文本时的采样过程，是一个控制文本随机性或熵的参数[50]。温度值越高，随机性越强，而温度值越低，模型输出的结果越具有确定性。

From the perspective of search-based optimization, the temperature of LLMs can be understood as a parameter that controls the exploration in the search process. A higher temperature equips the LLM with stronger exploratory ability. Based on this, we propose a simple rule for adaptively adjusting the temperature: If LMEA fails to find a solution better than the current best for $K$ consecutive generations, the temperature value will be increased by $\alpha$ . Typically, the default temperature of LLMs is 1.0 [50]; in this work, we always set $K=20$ and $\alpha=0.1$ .
从基于搜索的优化角度来看，LLMs的温度可以理解为一个控制搜索过程中探索的参数。温度越高，LLM的探索能力就越强。在此基础上，我们提出了一个自适应调整温度的简单规则：如果 LMEA 连续 $K$ 代都未能找到优于当前最优解的方案，则温度值将增加 $\alpha$ 。.通常情况下，LLMs的默认温度为 1.0 [ 50]；在本文中，我们始终设置 $K=20$ 和 $\alpha=0.1$ 。.

IV Experiments IV 实验

We investigate the power of LMEA on TSPs. Specifically, the experiments aim to address the two questions below.
我们研究了 LMEA 在 TSP 上的威力。具体来说，实验旨在解决以下两个问题。

•

How good is LMEA’s performance, especially compared to those traditional hand-designed heuristics?

- LMEA 的性能如何，尤其是与那些传统的手工设计的启发式算法相比？
•

Are LLM-driven genetic operators and the self-adaptation mechanism useful in improving the optimization performance?

- LLM驱动的遗传算子和自适应机制是否有助于提高优化性能？

IV-A Test Problem Instances
IV-A 测试问题实例

We considered EUC-2D TSPs, where the nodes are defined on a two-dimensional plane and the distances between two nodes are the same in both directions. Specifically, two different types of TSP instances were generated through the generators which has been used to create testbeds for the 8-th DIMACS Implementation Challenge [57].¹¹1Generators available at: http://dimacs.rutgers.edu/archive/Challenges/TSP
发电机见： http://dimacs.rutgers.edu/archive/Challenges/TSP
我们考虑的是 EUC-2D TSP，其中节点定义在二维平面上，两个节点之间的距离在两个方向上都相同。具体来说，我们通过生成器生成了两种不同类型的 TSP 实例，该生成器曾用于为第 8 届 DIMACS 实施挑战赛创建测试平台[ 57]。 ¹

•

The portgen generator generates a TSP instance (called a rue instance) by uniformly and randomly placing nodes on a two-dimensional plane.

- 波特根生成器通过在一个二维平面上均匀、随机地放置节点来生成一个 TSP 实例（称为 rue 实例）。
•

The portcgen generator generates a TSP instance (called a clu instance) by randomly placing nodes around different central nodes.

- portcgen 生成器通过在不同中心节点周围随机放置节点，生成一个 TSP 实例（称为 clu 实例）。

For both types (rue and clu), the instances were generated such that both $x$ and $y$ coordinates of the nodes lie within [0, 100]. For each instance type, we considered four different problem sizes (number of nodes, denoted as $n$ ), i.e., $n=10,15,20,25$ . In summary, there were eight different combinations of instance types and problem sizes. We denote them as rue/clu-10/15/20/25, respectively, and for each of them, we generated five TSP instances. For all the 40 generated instances, Concorde [58], an exact TSP solver, was used to obtain their optimal solutions.²²2Concorde available at https://www.math.uwaterloo.ca/tsp/concorde.html
协和式飞机可在 https://www.math.uwaterloo.ca/tsp/concorde.html
对于两种类型（芸香和簇香），生成的实例都要求节点的 $x$ 和 $y$ 坐标都在[0, 100]以内。对于每种实例类型，我们都考虑了四种不同的问题规模（节点数，用 $n$ 表示），即 $n=10,15,20,25$ ..总之，有八种不同的实例类型和问题大小组合。我们将它们分别记为 rue/clu-10/15/20/25，并为每种组合生成 5 个 TSP 实例。对于所有生成的 40 个实例，我们都使用 TSP 精确求解器 Concorde [ 58] 来获得其最优解。 ²

IV-B Baseline Algorithms
IV-B 基准算法

We considered the following four traditional heuristics for solving TSPs as baseline algorithms.³³3Implementations available at https://github.com/Valdecy/pyCombinatorial
可在 https://github.com/Valdecy/pyCombinatorial 上查阅实施情况
我们将以下四种解决 TSP 的传统启发式算法视为基准算法。 ³

•

Nearest neighbor (NN). This heuristic first randomly chooses a node which is the starting point of the tour. Then at each step, the next node is chosen as the one nearest to the last node of the tour and is appended to the tour as the new last node. This process finishes when the tour contains all the nodes.

- 最近邻居 (NN)。这种启发式方法首先随机选择一个节点，作为游程的起点。然后，在每一步中，选择下一个节点作为最靠近游程最后一个节点的节点，并将其作为新的最后一个节点添加到游程中。当游程包含所有节点时，该过程结束。
•

Farthest/nearest/random insertion (FI/NI/RI). The insertion heuristics minimize the cost of inserting a node into the tour. At each step, for each node $k$ , its position of insertion is determined such that the cost $c(k)=d_{i,k}+d_{k,j}-d_{i,j}$ is minimized, where $i$ and $j$ are adjacent nodes in the current tour and $d$ indicates the distance. The different variants of insertion heuristics differ in how the inserted node is selected. FI selects the node with the largest distance to any node of the tour. NI selects the node that is nearest to any node in the tour. RI selects a random node.

- 最远/最近/随机插入（FI/NI/RI）。插入启发式方法最大限度地降低了将节点插入巡回过程的成本。在每一步中，对于每个节点 $k$ 中， $i$ 和 $j$ 是当前游程中的相邻节点， $d$ 表示距离。不同的插入启发式方法在选择插入节点的方式上有所不同。FI 选择与游程中任意节点距离最大的节点。NI 选择与游程中任何节点距离最近的节点。RI 随机选择一个节点。

Additionally, we considered the very recent approach Optimization by PROmpting (OPRO) [55] as the baseline. OPRO is also driven by LLMs, but it differs from LMEA in that OPRO does not employ LLM to perform crossover and mutation. Instead, in each generation, OPRO directly instructs the LLM to generate $N$ new solutions based on the current population. Therefore, OPRO can be viewed as a variant of LMEA without LLM-driven genetic operators (crossover and mutation). The comparison between them can validate the effectiveness of LLM-driven genetic operators.
此外，我们还将最新的优化方法（Optimization by PROmpting，简称 OPRO）[ 55] 作为基准。OPRO 也由 LLMs 驱动，但与 LMEA 不同的是，OPRO 不使用 LLM 进行交叉和变异。相反，在每一代中，OPRO 直接指示LLM根据当前种群生成 $N$ 个新方案。因此，OPRO 可以看作是不使用LLM驱动遗传算子（交叉和变异）的 LMEA 的变种。两者之间的比较可以验证LLM驱动遗传算子的有效性。

IV-C Experimental Setup IV-C 实验装置

To make a fair comparison, for LMEA and OPRO, the population size $N$ and maximum generation number $G$ were set the same, i.e., $N=16$ and $G=250$ . For LMEA, we used the chat-turbo-0613 version of the GPT-3.5 API as the LLM.⁴⁴4Details of the API available at: https://platform.openai.com/
有关 API 的详细信息，请访问： https://platform.openai.com/
为了进行公平比较，LMEA 和 OPRO 的种群规模 $N$ 和最大世代数 $G$ 设置相同，即 $N=16$ 和 $G=250$ 。.在 LMEA 中，我们使用 chat-turbo-0613 版本的 GPT-3.5 API 作为 LLM 。 ⁴ On each test set, i.e., rue/clu-10/15/20/25, we executed each algorithm and reported its average optimality gap on the set. The optimality gap is defined as the difference between the length of the best solution found by the algorithm (denoted as $len(s^{*})$ ) and the length of the optimal solution (denoted as $opt$ ), calculated using $(len(s^{*})-opt)/opt$ .
在每个测试集（即 rue/clu-10/15/20/25）上，我们执行了每种算法，并报告了其在测试集上的平均最优性差距。最优性差距被定义为算法找到的最佳解决方案长度（用 $len(s^{*})$ 表示）与最优解决方案长度（用 $opt$ 表示）之差，计算方法为 $(len(s^{*})-opt)/opt$ 。.

TABLE I: Test results on eight test sets with different numbers of nodes (

n=10,15,20,25

) and TSP types (rue and clu). Each test set contains 5 TSP instances, and the average optimality gap (%)

\pm

standard deviation achieved on them is reported. On each test set, the best average optimality gap is indicated in bold. “# Generations” represents the mean

\pm

standard deviation of the generation numbers that LMEA and OPRO finds the optimal solution. “# Successes” counts the number of TSP instances that LMEA and OPRO finds the optimal solution. “N/A” means that no optimal solution is found for any TSP instance in the test set.
表一：不同节点数（

n=10,15,20,25

）和 TSP 类型（芸香和簇香）的八个测试集的测试结果。每个测试集包含 5 个 TSP 实例，并报告了在这些实例上取得的平均最优性差距（%）

\pm

标准偏差。在每个测试集上，最佳平均优化差距用粗体表示。"# 代数 "表示 LMEA 和 OPRO 找到最优解的平均

\pm

标准偏差。"# 成功次数 "表示 LMEA 和 OPRO 找到最优解的 TSP 实例数。"不适用 "表示在测试集中没有为任何 TSP 实例找到最优解。

Test set 测试装置	Optimality gap (%) 优化差距 (%)						# Generations (# Successes) # 世代（# 成功）
Test set 测试装置	NN	FI	NI	RI	LMEA	OPRO	LMEA	OPRO
rue-10 芸-10	11.22 $\pm$ 3.35	2.23 $\pm$ 1.26	0.58 $\pm$ 0.58	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	35.80 $\pm$ 7.17 (5)	60.60 $\pm$ 13.68 (5)
rue-15 芸-15	9.84 $\pm$ 3.34	1.08 $\pm$ 1.01	0.79 $\pm$ 0.79	2.45 $\pm$ 1.18	0.06 $\pm$ 0.06	5.23 $\pm$ 2.01	235.25 $\pm$ 6.12 (4)	189.00 $\pm$ 0.00 (1)
rue-20 芸-20	21.47 $\pm$ 2.01	1.99 $\pm$ 0.86	2.65 $\pm$ 1.43	2.15 $\pm$ 1.26	3.94 $\pm$ 1.54	26.30 $\pm$ 3.58	197.00 $\pm$ 0.00 (1)	N/A (0) 不适用 (0)
rue-25 芸-25	10.71 $\pm$ 3.36	2.33 $\pm$ 1.34	1.41 $\pm$ 0.72	2.41 $\pm$ 0.74	18.72 $\pm$ 3.31	53.59 $\pm$ 8.37	N/A (0) 不适用 (0)	N/A (0) 不适用 (0)
clu-10 组-10	16.48 $\pm$ 2.02	1.28 $\pm$ 0.79	0.99 $\pm$ 0.99	1.37 $\pm$ 0.84	0.00 $\pm$ 0.00	0.00 $\pm$ 0.00	19.00 $\pm$ 3.30 (5)	90.40 $\pm$ 16.55 (5)
clu-15 区块-15	23.39 $\pm$ 4.39	0.00 $\pm$ 0.00	0.48 $\pm$ 0.48	0.08 $\pm$ 0.08	0.11 $\pm$ 0.11	8.13 $\pm$ 4.83	152.25 $\pm$ 18.80 (4)	153.00 $\pm$ 0.00 (1)
clu-20	21.29 $\pm$ 2.77	0.78 $\pm$ 0.48	3.81 $\pm$ 1.45	1.97 $\pm$ 0.87	4.05 $\pm$ 0.69	19.83 $\pm$ 4.76	N/A (0) 不适用 (0)	N/A (0) 不适用 (0)
clu-25 组 25	22.36 $\pm$ 1.29	2.10 $\pm$ 0.55	3.58 $\pm$ 0.68	1.83 $\pm$ 0.62	10.06 $\pm$ 1.69	48.25 $\pm$ 5.86	N/A (0) 不适用 (0)	N/A (0) 不适用 (0)

IV-D Results and Analysis
IV-D 结果与分析

The test results are presented in Table I. In addition to the optimality gap, the numbers of generations needed by LMEA and ORPO to find the optimal solutions are also reported.
测试结果见表 I。除了最优性差距外，还报告了 LMEA 和 ORPO 找到最优解所需的代数。

First, it can be observed that on TSP instances with 10 nodes and 15 nodes, LMEA outperforms the heuristics on three out of four test sets, i.e., rue-10/15 and clu-10. Taking a closer look, on the rue/clu-10/15 test sets, it is interesting to find that LMEA consistently finds the optimal solution on 19 out of 20 instances. Considering the size of the solution space of a rue/clu-15 TSP instance is at least in the order of billions, LMEA can find the optimal solution to it within a total fitness evaluation number not exceeding $250\times 16=4000$ (note that $G=250$ and $N=16$ ). This clearly shows that, despite its minimal reliance on domain expertise, LMEA has the capability to optimize non-trivial NP-hard combinatorial optimization problems like TSPs.
首先，我们可以发现，在具有 10 个节点和 15 个节点的 TSP 实例上，LMEA 在四个测试集中的三个测试集（即 rue-10/15 和 clu-10）上的表现优于启发式算法。仔细观察，在 rue/clu-10/15 测试集上，有趣的是，在 20 个实例中，LMEA 始终能在 19 个实例上找到最优解。考虑到 rue/clu-15 TSP 实例的解空间大小至少在数十亿数量级，LMEA 可以在不超过 $250\times 16=4000$ 的总适配性评价数内找到其最优解（注意 $G=250$ 和 $G=250$ ）。(注意是 $G=250$ 和 $N=16$ ）。这清楚地表明，尽管 LMEA 对领域专业知识的依赖程度极低，但它仍有能力优化类似 TSP 这样的非难 NP 组合优化问题。

Second, on TSP instances with 20 nodes, LMEA performs slightly worse than the heuristics. As the node number increases further ( $n=25$ ), the optimality gap of LMEA increases rapidly. Addressing the scalability limitations of LMEA is a crucial area for future research.
其次，在有 20 个节点的 TSP 实例上，LMEA 的表现略逊于启发式算法。随着节点数的进一步增加（ $n=25$ ），LMEA 的最优性差距迅速扩大。解决 LMEA 的可扩展性限制是未来研究的一个重要领域。

Finally, on all the eight test sets, LMEA outperforms ORPO, and the performance gap becomes larger as the node number increases. Figure 2 also illustrates the convergence curves of LMEA and ORPO on all the test sets. It can be observed that in general LMEA can find better solutions more quickly than ORPO. Since OPRO can be viewed as a variant of LMEA without LLM-driven crossover and mutation, these results have confirmed the effectiveness of the LLM-driven genetic operators.
最后，在所有八个测试集上，LMEA 的性能都优于 ORPO，而且随着节点数的增加，性能差距也越来越大。图 2 还展示了 LMEA 和 ORPO 在所有测试集上的收敛曲线。可以看出，一般来说，LMEA 比 ORPO 能更快地找到更好的解决方案。由于 OPRO 可以看作是没有LLM驱动交叉和变异的 LMEA 的变种，这些结果证实了LLM驱动遗传算子的有效性。

TABLE II: Comparison of LMEA and its variant without self-adaptation (LMEA*) on the rue-20 test set.
表 2：LMEA 及其不带自适应功能的变体（LMEA*）在芸香-20 测试集上的比较。

Test set 测试装置	Optimality gap (%) 优化差距 (%)
Test set 测试装置	LMEA	LMEA*
rue-20 芸-20	3.94 $\pm$ 1.54	11.82 $\pm$ 2.21

IV-E Effectiveness of Self-Adaptation
IV-E 自我适应的有效性

To validate the effectiveness of the self-adaptation of the LLM’s temperature in LMEA, we tested a variant of LMEA without self-adaptation (LMEA*) on the rue-20 test set. The results are presented in Table II. The convergence curves of LMEA and LMEA* are also illustrated in Figure 3. It can be clearly observed that, LMEA can achieve significantly better optimality gaps than LMEA*. Moreover, based on Figure 3, one can observe that even when the quality of the random initial solutions are worse than that of LMEA*, LMEA is still capable of finding better solutions more quickly. These results demonstrate the effectiveness of self-adaptation.
为了验证 LMEA 中LLM温度自适应的有效性，我们在 rue-20 测试集上测试了不带自适应功能的 LMEA 变体（LMEA*）。结果见表 II。图 3 也展示了 LMEA 和 LMEA* 的收敛曲线。可以清楚地看到，LMEA 的最优性差距明显优于 LMEA*。此外，根据图 3 可以观察到，即使随机初始解的质量比 LMEA* 差，LMEA 仍然能够更快地找到更好的解。这些结果证明了自适应的有效性。

V Discussions and Conclusion
V 讨论和结论

In this work, we explored on employing LLMs as evolutionary combinatorial optimizers, where the LLM repeatedly generates offspring solutions based on the current population. Our investigation demonstrates that LMEA has the capacity of solving non-trivial NP-hard combinatorial optimization problems such as TSPs. Nonetheless, there are many open questions that remain to be explored in future works.
在这项工作中，我们探索了将LLMs作为进化组合优化器，其中LLM根据当前种群反复生成子代解。我们的研究表明，LMEA 有能力解决诸如 TSP 等非难 NP 组合优化问题。尽管如此，还有许多问题有待在今后的工作中探索。

•

Improving the scalability of LMEA. Currently LMEA still has limitations in handling relatively large problems. One possible approach to improve the scalability of LMEA is to instruct it to only focus on improving the local parts of the solutions, rather than the whole solution.

- 提高 LMEA 的可扩展性。目前，LMEA 在处理相对较大的问题时仍有局限性。要提高 LMEA 的可扩展性，一种可行的方法是指示它只专注于改进解决方案的局部，而不是整个解决方案。
•

Learning lessons from unsuccessful solutions. Instructing LLMs to learn lessons from incorrect answers has been proven effective in improving their performance [59]. Hence, it is interesting to investigate how such a strategy can boost the performance of LMEA for solving optimization problems.

- 从失败的答案中吸取教训。事实证明，指导LLMs从错误答案中吸取经验教训可以有效提高它们的性能[59]。因此，研究这种策略如何提高 LMEA 在解决优化问题时的性能是很有意义的。
•

Leveraging state-of-the-art prompt engineering. Techniques like chain of thoughts [34] and self-consistency [35] have the potential to enhance the performance of LMEA.

- 利用最先进的提示工程。思维链 [ 34] 和自洽性 [ 35] 等技术有可能提高 LMEA 的性能。
•

Applying LMEA to other problems. It is always interesting to investigate how LMEA would perform on different combinatorial optimization problems.

- 将 LMEA 应用于其他问题。研究 LMEA 在不同组合优化问题上的表现总是非常有趣的。

Acknowledgments 致谢

This work is partially supported by the Centre for Frontier AI Research (CFAR), Agency for Science, Technology and Research (A*STAR), and the School of Computer Science and Engineering at Nanyang Technological University. The research work is carried out in CFAR.
本研究工作得到了南洋理工大学科技研究机构（A*STAR）前沿人工智能研究中心（CFAR）和南洋理工大学计算机科学与工程学院的部分支持。研究工作在 CFAR 进行。

References 参考资料

[1] J. H. Holland, “Genetic algorithms,” Scientific american, vol. 267, no. 1, pp. 66–73, 1992.
J. H. Holland，"遗传算法"，《科学美国人》，第 267 卷，第 1 期，第 66-73 页，1992 年。
[2] T. Vidal, T. G. Crainic, M. Gendreau, N. Lahrichi, and W. Rei, “A hybrid genetic algorithm for multidepot and periodic vehicle routing problems,” Operations Research, vol. 60, no. 3, pp. 611–624, 2012.
T. Vidal, T. G. Crainic, M. Gendreau, N. Lahrichi, and W. Rei, "A hybrid genetic algorithm for multidepot and periodic vehicle routing problems," Operations Research, vol. 60, no.3, pp.
[3] Ç. Koç, T. Bektaş, O. Jabali, and G. Laporte, “A hybrid evolutionary algorithm for heterogeneous fleet vehicle routing problems with time windows,” Computers & Operations Research, vol. 64, pp. 11–27, 2015.
Ç.Koç, T. Bektaş, O. Jabali, and G. Laporte, "A hybrid evolutionorithm for heterogeneous fleet vehicle routing problems with time windows," Computers & Operations Research, vol. 64, pp.
[4] Y. Qi, Z. Hou, H. Li, J. Huang, and X. Li, “A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows,” Computers & Operations Research, vol. 62, pp. 61–77, 2015.
Y.Qi, Z. Hou, H. Li, J. Huang, and X. Li, "A decomposition based memetic algorithm for multi-objective vehicle routing problem with time windows," Computers & Operations Research, vol. 62, pp.
[5] L. Feng, Y.-S. Ong, M.-H. Lim, and I. W. Tsang, “Memetic search with interdomain learning: A realization between CVRP and CARP,” IEEE Transactions on Evolutionary Computation, vol. 19, no. 5, pp. 644–658, 2015.
L. Feng, Y.-S.Ong、M.-H.Lim, and I. W. Tsang, "Memetic search with interdomain learning：A realization between CVRP and CARP," IEEE Transactions on Evolutionary Computation, vol. 19, no.5, pp.
[6] Z.-H. Zhan, X.-F. Liu, Y.-J. Gong, J. Zhang, H. S.-H. Chung, and Y. Li, “Cloud computing resource scheduling and a survey of its evolutionary approaches,” ACM Computing Surveys, vol. 47, no. 4, pp. 1–33, 2015.
Z.-H. Zhan, X.-F.Zhan, X.-F. Liu, Y.-J. Gong, J. Zhang, H. S.-H.Liu、Y.-J. Gong、J. Zhang、H. S.-H.Chung, and Y. Li, "Cloud computing resource scheduling and a survey of its evolutionary approaches," ACM Computing Surveys, vol. 47, no.4, pp.
[7] C. Wang, H. Ma, G. Chen, and S. Hartmann, “Memetic eda-based approaches to qos-aware fully automated semantic web service composition,” IEEE Transactions on Evolutionary Computation, vol. 26, no. 3, pp. 570–584, 2022.
C. Wang, H. Ma, G. Chen, and S. Hartmann, "Memetic eda-based approaches to qos-aware fully automated semantic web service composition," IEEE Transactions on Evolutionary Computation, vol. 26, no.3, pp.
[8] Z. Pan, D. Lei, and L. Wang, “A knowledge-based two-population optimization algorithm for distributed energy-efficient parallel machines scheduling,” IEEE transactions on cybernetics, vol. 52, no. 6, pp. 5051–5063, 2022.
Z. Pan, D. Lei, and L. Wang, "A knowledge-based two-population optimization algorithm for distributed energy-efficient parallel machines scheduling," IEEE transactions on cybernetics, vol. 52, no. 6, pp.
[9] S. Zhou, L. Xing, X. Zheng, N. Du, L. Wang, and Q. Zhang, “A self-adaptive differential evolution algorithm for scheduling a single batch-processing machine with arbitrary job sizes and release times,” IEEE transactions on cybernetics, vol. 51, no. 3, pp. 1430–1442, 2019.
S. Zhou, L. Xing, X. Zheng, N. Du, L. Wang, and Q. Zhang, "A self-adaptive differential evolution algorithm for scheduling a single batch-processing machine with arbitrary job sizes and release times," IEEE transactions on cybernetics, vol. 51, no.3, pp.
[10] F. Zhang, Y. Mei, S. Nguyen, and M. Zhang, “Multitask multiobjective genetic programming for automated scheduling heuristic learning in dynamic flexible job-shop scheduling,” IEEE Transactions on Cybernetics, vol. 53, no. 7, pp. 4473–4486, 2023.
F. Zhang, Y. Mei, S. Nguyen, and M. Zhang, "Multitask multiobjective genetic programming for automated scheduling heuristic learning in dynamic flexible job-shop scheduling," IEEE Transactions on Cybernetics, vol. 53, no. 7, pp.
[11] S. Starke, N. Hendrich, and J. Zhang, “Memetic evolution for generic full-body inverse kinematics in robotics and animation,” IEEE Transactions on Evolutionary Computation, vol. 23, no. 3, pp. 406–420, 2019.
S. Starke, N. Hendrich, and J. Zhang, "Memetic evolution for generic full-body inverse kinematics in robotics and animation," IEEE Transactions on Evolutionary Computation, vol. 23, no.3, pp.
[12] S. Liu, Y. Zhang, K. Tang, and X. Yao, “How good is neural combinatorial optimization? a systematic evaluation on the traveling salesman problem,” IEEE Computational Intelligence Magazine, vol. 18, no. 3, pp. 14–28, 2023.
S. Liu, Y. Zhang, K. Tang, and X. Yao, "How good is neural combinatorial optimization? A systematic evaluation on the traveling salesman problem," IEEE Computational Intelligence Magazine, vol. 18, no.3，第 14-28 页，2023 年。
[13] L. C. Bezerra, M. López-Ibánez, and T. Stützle, “Automatic component-wise design of multiobjective evolutionary algorithms,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 3, pp. 403–417, 2015.
L. C. Bezerra, M. López-Ibánez, and T. Stützle, "Automatic component-wwise design of multiobjective evolutionary algorithms," IEEE Transactions on Evolutionary Computation, vol. 20, no.3, pp.
[14] C. L. Camacho-Villalón, M. Dorigo, and T. Stützle, “Pso-x: A component-based framework for the automatic design of particle swarm optimization algorithms,” IEEE Transactions on Evolutionary Computation, vol. 26, no. 3, pp. 402–416, 2021.
C. L. Camacho-Villalón, M. Dorigo, and T. Stützle, "Pso-x：Pso-x: A component-based framework for the automatic design of particle swarm optimization algorithms," IEEE Transactions on Evolutionary Computation, vol. 26, no.3, pp.
[15] L. Feng, Y. Huang, I. W. Tsang, A. Gupta, K. Tang, K. C. Tan, and Y.-S. Ong, “Towards faster vehicle routing by transferring knowledge from customer representation,” IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 2, pp. 952–965, 2022.
L. Feng, Y. Huang, I. W. Tsang, A. Gupta, K. Tang, K. C. Tan, and Y.-S. Ong, "Towards faster vehicle routing by transferring knowledge from customer representation," IEEE Transactions Intelligent Transportation Systems, vol. 23, no.Ong, "Towards faster vehicle routing by transferring knowledge from customer representation," IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 2, pp.
[16] J. Lin, Z.-J. Wang, and X. Li, “A backtracking search hyper-heuristic for the distributed assembly flow-shop scheduling problem,” Swarm and Evolutionary Computation, vol. 36, pp. 124–135, 2017.
J. Lin, Z.-J. Wang, and X. Li, "A backtracking search hyper-heuristic for the distributed assembly flow-shop scheduling problem," Swarm and Evolutionary Computation, vol. 36, pp.
[17] E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, and R. Qu, “Hyper-heuristics: A survey of the state of the art,” Journal of the Operational Research Society, vol. 64, pp. 1695–1724, 2013.
E. K. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. Özcan, and R. Qu, "Hyper-heuristics：A survey of the state of the art," Journal of the Operational Research Society, vol. 64, pp.
[18] J. H. Drake, A. Kheiri, E. Özcan, and E. K. Burke, “Recent advances in selection hyper-heuristics,” European Journal of Operational Research, vol. 285, no. 2, pp. 405–428, 2020.
J. H. Drake, A. Kheiri, E. Özcan, and E. K. Burke, "Recent advances in selection hyper-heuristics," European Journal of Operational Research, vol. 285, no. 2, pp.
[19] O. Vinyals, M. Fortunato, and N. Jaitly, “Pointer networks,” in Proceedings of Advances in Neural Information Processing Systems, NeurIPS’2015, 2015, pp. 2692–2700.
O. Vinyals, M. Fortunato, and N. Jaitly, "Pointer networks," in Proceedings of Advances in Neural Information Processing Systems, NeurIPS'2015, 2015, pp.
[20] Y. Bengio, A. Lodi, and A. Prouvost, “Machine learning for combinatorial optimization: a methodological tour d’horizon,” European Journal of Operational Research, vol. 290, no. 2, pp. 405–421, 2021.
Y. Bengio, A. Lodi, and A. Prouvost, "Machine learning for combinatorial optimization: a methodological tour d'horizon," European Journal of Operational Research, vol. 290, no. 2, pp.
[21] K. Smith-Miles and S. Bowly, “Generating new test instances by evolving in instance space,” Computers & Operations Research, vol. 63, pp. 102–113, 2015.
K. Smith-Miles and S. Bowly, "Generating new test instances by evolving in instance space," Computers & Operations Research, vol. 63, pp.
[22] K. Tang, S. Liu, P. Yang, and X. Yao, “Few-shots parallel algorithm portfolio construction via co-evolution,” IEEE Transactions on Evolutionary Computation, vol. 25, no. 3, pp. 595–607, 2021.
K. Tang, S. Liu, P. Yang, and X. Yao, "Few-shots parallel algorithm portfolio construction via co-evolution," IEEE Transactions on Evolutionary Computation, vol. 25, no.3, pp.
[23] A. Biedenkapp, M. Lindauer, K. Eggensperger, F. Hutter, C. Fawcett, and H. H. Hoos, “Efficient parameter importance analysis via ablation with surrogates,” in Proceedings of the 31st AAAI Conference on Artificial Intelligence, AAAI’2017, 2017, pp. 773–779.
A. Biedenkapp, M. Lindauer, K. Eggensperger, F. Hutter, C. Fawcett, and H. H. Hoos, "Efficient parameter importance analysis via ablation with surrogates," in Proceedings of the 31st AAAI Conference on Artificial Intelligence, AAAI'2017, 2017, pp.
[24] I. Bello, H. Pham, Q. V. Le, M. Norouzi, and S. Bengio, “Neural combinatorial optimization with reinforcement learning,” arXiv preprint arXiv:1611.09940, 2016.
I. Bello, H. Pham, Q. V. Le, M. Norouzi, and S. Bengio, "Neural combinatorial optimization with reinforcement learning," arXiv preprint arXiv:1611.09940, 2016.
[25] B. Min, H. Ross, E. Sulem, A. P. B. Veyseh, T. H. Nguyen, O. Sainz, E. Agirre, I. Heintz, and D. Roth, “Recent advances in natural language processing via large pre-trained language models: A survey,” ACM Computing Surveys, vol. 56, no. 2, pp. 1–40, 2023.
B. Min、H. Ross、E. Sulem、A. P. B. Veyseh、T. H. Nguyen、O. Sainz、E. Agirre、I. Heintz 和 D. Roth，"通过大型预训练语言模型进行自然语言处理的最新进展：A survey," ACM Computing Surveys, vol. 56, no. 2, pp.
[26] W. X. Zhao, K. Zhou, J. Li, T. Tang, X. Wang, Y. Hou, Y. Min, B. Zhang, J. Zhang, Z. Dong et al., “A survey of large language models,” arXiv preprint arXiv:2303.18223, 2023.
W. X. Zhao、K. Zhou、J. Li、T. Tang、X. Wang、Y. Hou、Y. Min、B. Zhang、J. Zhang、Z. Dong 等：《大型语言模型概览》，arXiv 预印本 arXiv:2303.18223, 2023。
[27] H. Tian, W. Lu, T. O. Li, X. Tang, S.-C. Cheung, J. Klein, and T. F. Bissyandé, “Is chatgpt the ultimate programming assistant–how far is it?” arXiv preprint arXiv:2304.11938, 2023.
H. Tian, W. Lu, T. O. Li, X. Tang, S.-C. Cheung, J. Klein, and T. F. Bissyandé, "Is chatgpt ultimate programming assistant-how far is it?Cheung, J. Klein, and T. F. Bissyandé, "Is chatgpt the ultimate programming assistant-how far is it?" arXiv preprint arXiv:2304.11938, 2023.
[28] P. Lee, S. Bubeck, and J. Petro, “Benefits, limits, and risks of gpt-4 as an ai chatbot for medicine,” New England Journal of Medicine, vol. 388, no. 13, pp. 1233–1239, 2023.
P. Lee, S. Bubeck, and J. Petro, "Benefits, limits, and risks of gpt-4 as an ai chatbot for medicine," New England Journal of Medicine, vol. 388, no. 13, pp.
[29] J. Blocklove, S. Garg, R. Karri, and H. Pearce, “Chip-chat: Challenges and opportunities in conversational hardware design,” arXiv preprint arXiv:2305.13243, 2023.
J. Blocklove、S. Garg、R. Karri 和 H. Pearce，"Chip-chat：对话式硬件设计的挑战与机遇》，arXiv 预印本 arXiv:2305.13243, 2023。
[30] M. Zheng, X. Su, S. You, F. Wang, C. Qian, C. Xu, and S. Albanie, “Can gpt-4 perform neural architecture search?” arXiv preprint arXiv:2304.10970, 2023.
M. Zheng, X. Su, S. You, F. Wang, C. Qian, C. Xu, and S. Albanie, "Can gpt-4 perform neural architecture search?" arXiv preprint arXiv:2304.10970, 2023.
[31] A. J. Thirunavukarasu, D. S. J. Ting, K. Elangovan, L. Gutierrez, T. F. Tan, and D. S. W. Ting, “Large language models in medicine,” Nature medicine, vol. 29, no. 8, pp. 1930–1940, 2023.
A. J. Thirunavukarasu, D. S. J. Ting, K. Elangovan, L. Gutierrez, T. F. Tan, and D. S. W. Ting, "Large language models in medicine," Nature medicine, vol. 29, no. 8, pp.
[32] E. Kasneci, K. Seßler, S. Küchemann, M. Bannert, D. Dementieva, F. Fischer, U. Gasser, G. Groh, S. Günnemann, E. Hüllermeier et al., “Chatgpt for good? on opportunities and challenges of large language models for education,” Learning and individual differences, vol. 103, p. 102274, 2023.
E. Kasneci, K. Seßler, S. Küchemann, M. Bannert, D. Dementieva, F. Fischer, U. Gasser, G. Groh, S. Günnemann, E. Hüllermeier et al., "Chatgpt for good? on opportunities and challenges of large language models for education," Learning and individual differences, vol. 103, p. 102274, 2023.
[33] Y. Liu, T. Han, S. Ma, J. Zhang, Y. Yang, J. Tian, H. He, A. Li, M. He, Z. Liu et al., “Summary of chatgpt-related research and perspective towards the future of large language models,” Meta-Radiology, p. 100017, 2023.
Y. Liu, T. Han, S. Ma, J. Zhang, Y. Yang, J. Tian, H. He, A. Li, M. He, Z. Liu et al.
[34] J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia, E. H. Chi, Q. V. Le, and D. Zhou, “Chain-of-thought prompting elicits reasoning in large language models,” in Proceedings of Advances in Neural Information Processing Systems, NeurIPS’2022, 2022, pp. 24 824–24 837.
J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia, E. H. Chi, Q. V. Le, and D. Zhou, "Chain-of-thought prompting elicits reasoning in large language models," in Proceedings of Advances in Neural Information Processing Systems, NeurIPS'2022, 2022, pp.
[35] X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou, “Self-consistency improves chain of thought reasoning in language models,” in Proceedings of the 11th International Conference on Learning Representations, ICLR’2023, 2023.
X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou, "Self-consistency improves chain of thought reasoning in language models," in Proceedings of the 11th International Conference on Learning Representations, ICLR'2023, 2023.
[36] D. Zhou, N. Schärli, L. Hou, J. Wei, N. Scales, X. Wang, D. Schuurmans, C. Cui, O. Bousquet, Q. V. Le, and E. H. Chi, “Least-to-most prompting enables complex reasoning in large language models,” in Proceedings of the 11th International Conference on Learning Representations, ICLR’2023, 2023.
D. Zhou, N. Schärli, L. Hou, J. Wei, N. Scales, X. Wang, D. Schuurmans, C. Cui, O. Bousquet, Q. V. Le, and E. H. Chi, "Least-to-most prompting enables complex reasoning in large language models," in Proceedings of the 11th International Conference on Learning Representations, ICLR'2023, 2023.
[37] A. Lewkowycz, A. Andreassen, D. Dohan, E. Dyer, H. Michalewski, V. V. Ramasesh, A. Slone, C. Anil, I. Schlag, T. Gutman-Solo, Y. Wu, B. Neyshabur, G. Gur-Ari, and V. Misra, “Solving quantitative reasoning problems with language models,” in Proceedings of Advances in Neural Information Processing Systems, NeurIPS’2022, 2022, pp. 3843–3857.
A. Lewkowycz, A. Andreassen, D. Dohan, E. Dyer, H. Michalewski, V. V. Ramasesh, A. Slone, C. Anil, I. Schlag, T. Gutman-Solo, Y. Wu, B. Neyshabur, G. Gur-Ari, and V. Misra, "Solving quantitative reasoning problems with language models," in Proceedings of Advances in Neural Information Processing Systems, NeurIPS'2022, 2022, pp.
[38] J. Huang and K. C. Chang, “Towards reasoning in large language models: A survey,” in Findings of the Association for Computational Linguistics: ACL Findings’2023, A. Rogers, J. L. Boyd-Graber, and N. Okazaki, Eds., 2023, pp. 1049–1065.
J. Huang and K. C. Chang, "Towards reasoning in large language models：A survey," in Findings of the Association for Computational Linguistics：A. Rogers, J. L. Boyd-Graber, and N. Okazaki, Eds., 2023, pp.
[39] S. Yao, J. Zhao, D. Yu, N. Du, I. Shafran, K. R. Narasimhan, and Y. Cao, “React: Synergizing reasoning and acting in language models,” in Proceedings of the 11th International Conference on Learning Representations, ICLR’2023, 2023.
S. Yao, J. Zhao, D. Yu, N. Du, I. Shafran, K. R. Narasimhan, and Y. Cao, "React：第 11 届学习表征国际会议论文集，ICLR'2023，2023 年。
[40] W. Huang, P. Abbeel, D. Pathak, and I. Mordatch, “Language models as zero-shot planners: Extracting actionable knowledge for embodied agents,” in Proceedings of the International Conference on Machine Learning, ICML’2022, 2022, pp. 9118–9147.
W. Huang, P. Abbeel, D. Pathak, and I. Mordatch, "Language models as zero-shot planners：Extracting actionable knowledge for embodied agents," in Proceedings of the International Conference on Machine Learning, ICML'2022, 2022, pp.
[41] Z. Xi, W. Chen, X. Guo, W. He, Y. Ding, B. Hong, M. Zhang, J. Wang, S. Jin, E. Zhou et al., “The rise and potential of large language model based agents: A survey,” arXiv preprint arXiv:2309.07864, 2023.
Z. Xi, W. Chen, X. Guo, W. He, Y. Ding, B. Hong, M. Zhang, J. Wang, S. Jin, E. Zhou et al：A survey," arXiv preprint arXiv:2309.07864, 2023.
[42] B. H. Korte, J. Vygen, B. Korte, and J. Vygen, Combinatorial optimization. Springer, 2011, vol. 1.
B.H. Korte, J. Vygen, B. H. Korte, and J. Vygen, Combinatorial optimisation.Korte, and J. Vygen, Combinatorial optimisation.Springer, 2011, vol. 1.
[43] D. S. Hochba, “Approximation algorithms for np-hard problems,” ACM Sigact News, vol. 28, no. 2, pp. 40–52, 1997.
D. S. Hochba, "Approximation algorithms for np-hard problems," ACM Sigact News, vol. 28, no. 2, pp.
[44] E. L. Lawler and D. E. Wood, “Branch-and-bound methods: A survey,” Operations research, vol. 14, no. 4, pp. 699–719, 1966.
E. L. Lawler 和 D. E. Wood, "Branch-and-bound methods：A survey," Operations research, vol. 14, no.4，第 699-719 页，1966 年。
[45] A. Chaabani, S. Bechikh, and L. B. Said, “A new co-evolutionary decomposition-based algorithm for bi-level combinatorial optimization,” Applied Intelligence, vol. 48, pp. 2847–2872, 2018.
A. Chaabani, S. Bechikh, and L. B. Said, "A new co-evolutionary decomposition-based algorithm for bi-level combinatorial optimization," Applied Intelligence, vol. 48, pp.
[46] A. J. Umbarkar and P. D. Sheth, “Crossover operators in genetic algorithms: a review.” ICTACT journal on soft computing, vol. 6, no. 1, 2015.
A. J. Umbarkar 和 P. D. Sheth，"遗传算法中的交叉算子：综述"。ICTACT 软计算期刊》，第 6 卷，第 1 期，2015 年。
[47] K. Helsgaun, “General k-opt submoves for the lin–kernighan tsp heuristic,” Mathematical Programming Computation, vol. 1, pp. 119–163, 2009.
K. Helsgaun, "General k-opt submoves for the lin-kernighan tsp heuristic," Mathematical Programming Computation, vol. 1, pp.
[48] Y. Nagata and S. Kobayashi, “A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem,” INFORMS Journal on Computing, vol. 25, no. 2, pp. 346–363, 2013.
Y. Nagata and S. Kobayashi, "A powerful genetic algorithm using edge assembly crossover for the traveling salesman problem," INFORMS Journal on Computing, vol. 25, no. 2, pp.
[49] S. Salcedo-Sanz, “A survey of repair methods used as constraint handling techniques in evolutionary algorithms,” Computer science review, vol. 3, no. 3, pp. 175–192, 2009.
S. Salcedo-Sanz, "A survey of repair methods used as constraint handling techniques in evolutionary algorithms," Computer science review, vol. 3, no.3, pp.
[50] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. M. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei, “Language models are few-shot learners,” in Proceedings of the Advances in Neural Information Processing Systems, NeurIPS’2020, 2020, pp. 1877–1901.
T. B. Brown、B. Mann、N. Ryder、M. Subbiah、J. Kaplan、P. Dhariwal、A. Neelakantan、P. Shyam、G. Sastry、A. Askell、S. Agarwal、A. Herbert-Voss、G. Krueger、T. Henighan、R. Child、A. Ramesh、D. M. Ziegler、J. Wu、C. Winter、C. Hesse、M. Chen、E. Sigler、M. Litwin、S. Gray、B. Chess、J. Clark、C. Berner、S. McCandlish、A. Radford、I. Sutskever 和 D. Amodei。Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei, "Language models are few-shot learners," in Proceedings of the Advances in Neural Information Processing Systems, NeurIPS'2020, 2020, pp.
[51] A. Chen, D. M. Dohan, and D. R. So, “Evoprompting: Language models for code-level neural architecture search,” arXiv preprint arXiv: 2302.14838, 2023.
A. Chen, D. M. Dohan, and D. R. So, "Evoprompting：用于代码级神经架构搜索的语言模型"，arXiv preprint arXiv: 2302.14838, 2023.
[52] J. Lehman, J. Gordon, S. Jain, K. Ndousse, C. Yeh, and K. O. Stanley, “Evolution through large models,” arXiv preprint arXiv: 2206.08896, 2022.
J. Lehman, J. Gordon, S. Jain, K. Ndousse, C. Yeh, and K. O. Stanley, "Evolution through large models," arXiv preprint arXiv: 2206.08896, 2022.
[53] M. Pluhacek, A. Kazikova, T. Kadavy, A. Viktorin, and R. Senkerik, “Leveraging large language models for the generation of novel metaheuristic optimization algorithms,” in Companion Proceedings of the 2023 Conference on Genetic and Evolutionary Computation, GECCO’2023, S. Silva and L. Paquete, Eds., 2023, pp. 1812–1820.
M. Pluhacek, A. Kazikova, T. Kadavy, A. Viktorin, and R. Senkerik, "Leveraging large language models for the generation of novel metaheuristic optimization algorithms," in Companion Proceedings of the 2023 Conference on Genetic and Evolutionary Computation, GECCO'2023, S. Silva and L. Paquete, Eds., 2023, pp.
[54] Q. Guo, R. Wang, J. Guo, B. Li, K. Song, X. Tan, G. Liu, J. Bian, and Y. Yang, “Connecting large language models with evolutionary algorithms yields powerful prompt optimizers,” arXiv preprint arXiv: 2309.08532, 2023.
Q. Guo, R. Wang, J. Guo, B. Li, K. Song, X. Tan, G. Liu, J. Bian, and Y. Yang, "Connecting large language models with evolutionary algorithms yields powerful prompt optimizers," arXiv preprint arXiv: 2309.08532, 2023.
[55] C. Yang, X. Wang, Y. Lu, H. Liu, Q. V. Le, D. Zhou, and X. Chen, “Large language models as optimizers,” arXiv preprint arXiv: 2309.03409, 2023.
C. Yang, X. Wang, Y. Lu, H. Liu, Q. V. Le, D. Zhou, and X. Chen, "Large language models as optimizers," arXiv preprint arXiv: 2309.03409, 2023.
[56] F. Liu, X. Lin, Z. Wang, S. Yao, X. Tong, M. Yuan, and Q. Zhang, “Large language model for multi-objective evolutionary optimization,” arXiv preprint arXiv: 2310.12541, 2023.
F. Liu, X. Lin, Z. Wang, S. Yao, X. Tong, M. Yuan, and Q. Zhang, "Large language model for multi-objective evolutionary optimization," arXiv preprint arXiv: 2310.12541, 2023.
[57] G. Gutin and A. P. Punnen, The traveling salesman problem and its variations. Springer Science & Business Media, 2006, vol. 12.
G. Gutin and A. P. Punnen, The traveling salesman problem and its variations.Springer Science & Business Media, 2006, vol. 12.
[58] D. Applegate, R. Bixby, V. Chvatal, and W. Cook, “Concorde tsp solver,” 2006.
D.Applegate, R. Bixby, V. Chvatal, and W. Cook, "Concorde tsp solver," 2006.
[59] N. Shinn, B. Labash, and A. Gopinath, “Reflexion: an autonomous agent with dynamic memory and self-reflection,” arXiv preprint arXiv:2303.11366, 2023.
N. Shinn, B. Labash, and A. Gopinath, "Reflexion: an autonomous agent with dynamic memory and self-reflection," arXiv preprint arXiv:2303.11366, 2023.

Large Language Models as Evolutionary Optimizers作为进化优化器的大型语言模型

Abstract 摘要

Index Terms:

I Introduction I 引言

II Related Works II 相关作品

II-A EAs for Combinatorial OptimizationII-A 用于组合优化的 EAs

II-B Large Language Models (LLMs) and PromptsII-B 大型语言模型（LLMs）和提示

II-C Intersection between EAs and LLMsII-C 《环境协定》与 LLMs 的交叉点

III LMEA for Combinatorial OptimizationIII 用于组合优化的 LMEA

III-A Algorithm FrameworkIII-A 算法框架

III-B LLMs as Evolutionary OptimizersIII-B 作为进化优化器的LLMs

III-C Self-Adaptation of the LLM’s TemperatureIII-C LLM的温度自适应