Surrogate-Assisted Search with Competitive Knowledge Transfer for Expensive Optimization

Expensive optimization problems (EOPs) have attracted increasing research attention over the decades due to their ubiquity in a variety of practical applications. Despite many sophisticated surrogate-assisted evolutionary algorithms (SAEAs) that have been developed for solving such problems, most of them lack the ability to transfer knowledge from previously-solved tasks and always start their search from scratch, making them troubled by the notorious cold-start issue. A few preliminary studies that integrate transfer learning into SAEAs still face some issues, such as defective similarity quantification that is prone to underestimate promising knowledge, surrogate-dependency that makes the transfer methods not coherent with the state-of-the-art in SAEAs, etc. In light of the above, a plug and play competitive knowledge transfer method is proposed to boost various SAEAs in this paper. Specifically, both the optimized solutions from the source tasks and the promising solutions acquired by the target surrogate are treated as task-solving knowledge, enabling them to compete with each other to elect the winner for expensive evaluation, thus boosting the search speed on the target task. Moreover, the lower bound of the convergence gain brought by the knowledge competition is mathematically analyzed, which is expected to strengthen the theoretical foundation of sequential transfer optimization. Experimental studies conducted on a series of benchmark problems and a practical application from the petroleum industry verify the efficacy of the proposed method. The source code of the competitive knowledge transfer is available at https://github.com/XmingHsueh/SAS-CKT.

Benchmarking LLMs for Optimization Modeling and Enhancing Reasoning via Reverse Socratic Synthesis

Large language models (LLMs) have exhibited their problem-solving ability in mathematical reasoning. Solving realistic optimization (OPT) problems in industrial application scenarios requires advanced and applied math ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose E-OPT, a benchmark for end-to-end optimization problem-solving with human-readable inputs and outputs. E-OPT contains rich optimization problems, including linear/nonlinear programming with/without table data, which can comprehensively evaluate LLMs' solving ability. In our benchmark, LLMs are required to correctly understand the problem in E-OPT and call code solver to get precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-2-7b and Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a novel data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, ReSocratic first incrementally synthesizes optimization scenarios with mathematical formulations step by step and then back-translates the generated scenarios into questions. In such a way, we construct the ReSocratic-29k dataset from a small seed sample pool with the powerful open-source large model DeepSeek-V2. To demonstrate the effectiveness of ReSocratic, we conduct supervised fine-tuning with ReSocratic-29k on multiple open-source models. The results show that Llama3-8b is significantly improved from 13.6% to 51.7% on E-OPT, while DeepSeek-V2 reaches 61.0%, approaching 65.5% of GPT-4.

Learning to Transfer for Evolutionary Multitasking

Evolutionary multitasking (EMT) is an emerging approach for solving multitask optimization problems (MTOPs) and has garnered considerable research interest. The implicit EMT is a significant research branch that utilizes evolution operators to enable knowledge transfer (KT) between tasks. However, current approaches in implicit EMT face challenges in adaptability, due to the use of a limited number of evolution operators and insufficient utilization of evolutionary states for performing KT. This results in suboptimal exploitation of implicit KT's potential to tackle a variety of MTOPs. To overcome these limitations, we propose a novel Learning to Transfer (L2T) framework to automatically discover efficient KT policies for the MTOPs at hand. Our framework conceptualizes the KT process as a learning agent's sequence of strategic decisions within the EMT process. We propose an action formulation for deciding when and how to transfer, a state representation with informative features of evolution states, a reward formulation concerning convergence and transfer efficiency gain, and the environment for the agent to interact with MTOPs. We employ an actor-critic network structure for the agent and learn it via proximal policy optimization. This learned agent can be integrated with various evolutionary algorithms, enhancing their ability to address a range of new MTOPs. Comprehensive empirical studies on both synthetic and real-world MTOPs, encompassing diverse inter-task relationships, function classes, and task distributions are conducted to validate the proposed L2T framework. The results show a marked improvement in the adaptability and performance of implicit EMT when solving a wide spectrum of unseen MTOPs.

Towards Next Era of Multi-objective Optimization: Large Language Models as Architects of Evolutionary Operators

Multi-objective optimization problems (MOPs) are prevalent in various real-world applications, necessitating sophisticated solutions that balance conflicting objectives. Traditional evolutionary algorithms (EAs), while effective, often rely on domain-specific expert knowledge and iterative tuning, which can impede innovation when encountering novel MOPs. Very recently, the emergence of Large Language Models (LLMs) has revolutionized software engineering by enabling the autonomous development and refinement of programs. Capitalizing on this advancement, we propose a new LLM-based framework for evolving EA operators, designed to address a wide array of MOPs. This framework facilitates the production of EA operators without the extensive demands for expert intervention, thereby streamlining the design process. To validate the efficacy of our approach, we have conducted extensive empirical studies across various categories of MOPs. The results demonstrate the robustness and superior performance of our LLM-evolved operators.

Exploring the True Potential: Evaluating the Black-box Optimization Capability of Large Language Models

Large language models (LLMs) have gained widespread popularity and demonstrated exceptional performance not only in natural language processing (NLP) tasks but also in non-linguistic domains. Their potential as artificial general intelligence extends beyond NLP, showcasing promising capabilities in diverse optimization scenarios. Despite this rising trend, whether the integration of LLMs into these black-box optimization problems is genuinely beneficial remains unexplored. This paper endeavors to tackle this issue by offering deeper insights into the potential of LLMs in optimization tasks through a comprehensive investigation. Our approach involves a comprehensive evaluation, covering both discrete and continuous optimization problems, aiming to assess the efficacy and distinctive characteristics that LLMs bring to the realm of optimization. Our findings reveal both the limitations and advantages of LLMs in optimization. On one hand, despite consuming the significant power required to run the model, LLMs exhibit subpar performance and lack desirable properties in pure numerical tasks, primarily due to a mismatch between the problem domain and their processing capabilities. On the other hand, although LLMs may not be ideal for traditional numerical optimization, their potential in broader optimization contexts remains promising. LLMs exhibit the ability to solve problems in non-numerical domains and can leverage heuristics from the prompt to enhance their performance. To the best of our knowledge, this work presents the first systematic evaluation of LLMs for numerical optimization, offering a progressive, wide-coverage, and behavioral analysis. Our findings pave the way for a deeper understanding of LLMs' role in optimization and guide future application in diverse scenarios for LLMs.

CausalBench: A Comprehensive Benchmark for Causal Learning Capability of Large Language Models

Causality reveals fundamental principles behind data distributions in real-world scenarios, and the capability of large language models (LLMs) to understand causality directly impacts their efficacy across explaining outputs, adapting to new evidence, and generating counterfactuals. With the proliferation of LLMs, the evaluation of this capacity is increasingly garnering attention. However, the absence of a comprehensive benchmark has rendered existing evaluation studies being straightforward, undiversified, and homogeneous. To address these challenges, this paper proposes a comprehensive benchmark, namely CausalBench, to evaluate the causality understanding capabilities of LLMs. Originating from the causal research community, CausalBench encompasses three causal learning-related tasks, which facilitate a convenient comparison of LLMs' performance with classic causal learning algorithms. Meanwhile, causal networks of varying scales and densities are integrated in CausalBench, to explore the upper limits of LLMs' capabilities across task scenarios of varying difficulty. Notably, background knowledge and structured data are also incorporated into CausalBench to thoroughly unlock the underlying potential of LLMs for long-text comprehension and prior information utilization. Based on CausalBench, this paper evaluates nineteen leading LLMs and unveils insightful conclusions in diverse aspects. Firstly, we present the strengths and weaknesses of LLMs and quantitatively explore the upper limits of their capabilities across various scenarios. Meanwhile, we further discern the adaptability and abilities of LLMs to specific structural networks and complex chain of thought structures. Moreover, this paper quantitatively presents the differences across diverse information sources and uncovers the gap between LLMs' capabilities in causal understanding within textual contexts and numerical domains.

How Multimodal Integration Boost the Performance of LLM for Optimization: Case Study on Capacitated Vehicle Routing Problems

Recently, large language models (LLMs) have notably positioned them as capable tools for addressing complex optimization challenges. Despite this recognition, a predominant limitation of existing LLM-based optimization methods is their struggle to capture the relationships among decision variables when relying exclusively on numerical text prompts, especially in high-dimensional problems. Keeping this in mind, we first propose to enhance the optimization performance using multimodal LLM capable of processing both textual and visual prompts for deeper insights of the processed optimization problem. This integration allows for a more comprehensive understanding of optimization problems, akin to human cognitive processes. We have developed a multimodal LLM-based optimization framework that simulates human problem-solving workflows, thereby offering a more nuanced and effective analysis. The efficacy of this method is evaluated through extensive empirical studies focused on a well-known combinatorial optimization problem, i.e., capacitated vehicle routing problem. The results are compared against those obtained from the LLM-based optimization algorithms that rely solely on textual prompts, demonstrating the significant advantages of our multimodal approach.

Evolutionary Computation in the Era of Large Language Model: Survey and Roadmap

Large Language Models (LLMs), built upon Transformer-based architectures with massive pretraining on diverse data, have not only revolutionized natural language processing but also extended their prowess to various domains, marking a significant stride towards artificial general intelligence. The interplay between LLMs and Evolutionary Algorithms (EAs), despite differing in objectives and methodologies, reveals intriguing parallels, especially in their shared optimization nature, black-box characteristics, and proficiency in handling complex problems. Meanwhile, EA can not only provide an optimization framework for LLM's further enhancement under black-box settings but also empower LLM with flexible global search and iterative mechanism in applications. On the other hand, LLM's abundant domain knowledge enables EA to perform smarter searches, while its text processing capability assist in deploying EA across various tasks. Based on their complementary advantages, this paper presents a comprehensive review and forward-looking roadmap, categorizing their mutual inspiration into LLM-enhanced evolutionary optimization and EA-enhanced LLM. Some integrated synergy methods are further introduced to exemplify the amalgamation of LLMs and EAs in various application scenarios, including neural architecture search, code generation, software engineering, and text generation. As the first comprehensive review specifically focused on the EA research in the era of LLMs, this paper provides a foundational stepping stone for understanding and harnessing the collaborative potential of LLMs and EAs. By presenting a comprehensive review, categorization, and critical analysis, we contribute to the ongoing discourse on the cross-disciplinary study of these two powerful paradigms. The identified challenges and future directions offer guidance to unlock the full potential of this innovative collaboration.

Towards Multi-Objective High-Dimensional Feature Selection via Evolutionary Multitasking

Evolutionary Multitasking (EMT) paradigm, an emerging research topic in evolutionary computation, has been successfully applied in solving high-dimensional feature selection (FS) problems recently. However, existing EMT-based FS methods suffer from several limitations, such as a single mode of multitask generation, conducting the same generic evolutionary search for all tasks, relying on implicit transfer mechanisms through sole solution encodings, and employing single-objective transformation, which result in inadequate knowledge acquisition, exploitation, and transfer. To this end, this paper develops a novel EMT framework for multiobjective high-dimensional feature selection problems, namely MO-FSEMT. In particular, multiple auxiliary tasks are constructed by distinct formulation methods to provide diverse search spaces and information representations and then simultaneously addressed with the original task through a multi-slover-based multitask optimization scheme. Each task has an independent population with task-specific representations and is solved using separate evolutionary solvers with different biases and search preferences. A task-specific knowledge transfer mechanism is designed to leverage the advantage information of each task, enabling the discovery and effective transmission of high-quality solutions during the search process. Comprehensive experimental results demonstrate that our MO-FSEMT framework can achieve overall superior performance compared to the state-of-the-art FS methods on 26 datasets. Moreover, the ablation studies verify the contributions of different components of the proposed MO-FSEMT.

Solving Expensive Optimization Problems in Dynamic Environments with Meta-learning

Dynamic environments pose great challenges for expensive optimization problems, as the objective functions of these problems change over time and thus require remarkable computational resources to track the optimal solutions. Although data-driven evolutionary optimization and Bayesian optimization (BO) approaches have shown promise in solving expensive optimization problems in static environments, the attempts to develop such approaches in dynamic environments remain rarely unexplored. In this paper, we propose a simple yet effective meta-learning-based optimization framework for solving expensive dynamic optimization problems. This framework is flexible, allowing any off-the-shelf continuously differentiable surrogate model to be used in a plug-in manner, either in data-driven evolutionary optimization or BO approaches. In particular, the framework consists of two unique components: 1) the meta-learning component, in which a gradient-based meta-learning approach is adopted to learn experience (effective model parameters) across different dynamics along the optimization process. 2) the adaptation component, where the learned experience (model parameters) is used as the initial parameters for fast adaptation in the dynamic environment based on few shot samples. By doing so, the optimization process is able to quickly initiate the search in a new environment within a strictly restricted computational budget. Experiments demonstrate the effectiveness of the proposed algorithm framework compared to several state-of-the-art algorithms on common benchmark test problems under different dynamic characteristics.