This systematic review explores the application of Large Language Models (LLMs) in Combinatorial Optimization (CO). We report our findings using the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines. We conduct a literature search via Scopus and Google Scholar, examining over 2,000 publications. We assess publications against four inclusion and four exclusion criteria related to their language, research focus, publication year, and type. Eventually, we select 103 studies. We classify these studies into semantic categories and topics to provide a comprehensive overview of the field, including the tasks performed by LLMs, the architectures of LLMs, the existing datasets specifically designed for evaluating LLMs in CO, and the field of application. Finally, we identify future directions for leveraging LLMs in this field.

Automated Program Repair (APR) for introductory programming assignments (IPAs) is motivated by the large number of student enrollments in programming courses each year. Since providing feedback on IPAs requires substantial time and effort from faculty, personalized feedback often involves suggesting fixes to students' programs. Formal Methods (FM)-based semantic repair approaches, check a program's execution against a test suite or reference solution, are effective but limited. These tools excel at identifying buggy parts but can only fix programs if the correct implementation and the faulty one share the same control flow graph. Conversely, Large Language Models (LLMs) are used for APR but often make extensive instead of minimal rewrites. This leads to more invasive fixes, making it harder for students to learn from their mistakes. In summary, LLMs excel at completing strings, while FM-based fault localization excel at identifying buggy parts of a program. In this paper, we propose a novel approach that combines the strengths of both FM-based fault localization and LLMs, via zero-shot learning, to enhance APR for IPAs. Our method uses MaxSAT-based fault localization to identify buggy parts of a program, then presents the LLM with a program sketch devoid of these buggy statements. This hybrid approach follows a CEGIS loop to iteratively refine the program. We ask the LLM to synthesize the missing parts, which are then checked against a test suite. If the suggested program is incorrect, a counterexample from the test suite is fed back to the LLM. Our experiments show that our counterexample guided approach, using MaxSAT-based bug-free program sketches, significantly improves the repair capabilities of all six evaluated LLMs. This method allows LLMs to repair more programs with smaller fixes, outperforming other configurations and state-of-the-art symbolic program repair tools.

Recent advances in deep learning have shown significant potential for solving combinatorial optimization problems in real-time. Unlike traditional methods, deep learning can generate high-quality solutions efficiently, which is crucial for applications like routing and scheduling. However, existing approaches like deep reinforcement learning (RL) and behavioral cloning have notable limitations, with deep RL suffering from slow learning and behavioral cloning relying solely on expert actions, which can lead to generalization issues and neglect of the optimization objective. This paper introduces a novel offline RL method designed for combinatorial optimization problems with complex constraints, where the state is represented as a heterogeneous graph and the action space is variable. Our approach encodes actions in edge attributes and balances expected rewards with the imitation of expert solutions. We demonstrate the effectiveness of this method on job-shop scheduling and flexible job-shop scheduling benchmarks, achieving superior performance compared to state-of-the-art techniques.

In this paper, we present a novel nonlinear programming-based approach to fine-tune pre-trained neural networks to improve robustness against adversarial attacks while maintaining high accuracy on clean data. Our method introduces adversary-correction constraints to ensure correct classification of adversarial data and minimizes changes to the model parameters. We propose an efficient cutting-plane-based algorithm to iteratively solve the large-scale nonconvex optimization problem by approximating the feasible region through polyhedral cuts and balancing between robustness and accuracy. Computational experiments on standard datasets such as MNIST and CIFAR10 demonstrate that the proposed approach significantly improves robustness, even with a very small set of adversarial data, while maintaining minimal impact on accuracy.

Though current long-context large language models (LLMs) have demonstrated impressive capacities in answering user questions based on extensive text, the lack of citations in their responses makes user verification difficult, leading to concerns about their trustworthiness due to their potential hallucinations. In this work, we aim to enable long-context LLMs to generate responses with fine-grained sentence-level citations, improving their faithfulness and verifiability. We first introduce LongBench-Cite, an automated benchmark for assessing current LLMs' performance in Long-Context Question Answering with Citations (LQAC), revealing considerable room for improvement. To this end, we propose CoF (Coarse to Fine), a novel pipeline that utilizes off-the-shelf LLMs to automatically generate long-context QA instances with precise sentence-level citations, and leverage this pipeline to construct LongCite-45k, a large-scale SFT dataset for LQAC. Finally, we train LongCite-8B and LongCite-9B using the LongCite-45k dataset, successfully enabling their generation of accurate responses and fine-grained sentence-level citations in a single output. The evaluation results on LongBench-Cite show that our trained models achieve state-of-the-art citation quality, surpassing advanced proprietary models including GPT-4o.

We develop hard clustering based on likelihood rather than distance and prove convergence. We also provide simulations and real data examples.

Streamliners are certain constraints added to a constraint model to reduce the search space, thereby improving the feasibility and speed of finding solutions to complex constraint satisfaction problems. By incorporating domain-specific knowledge, streamliners can guide the constraint solver, allowing it to bypass less promising areas of the search space. Gomes and Sellmann (2004a) introduced streamliners to speed up the constrained-based search for hard combinatorial design problems. Today, streamliners are a standard tool for speeding up constrained-based search. Streamliners are closely related to implied/redundant constraints, symmetry-breaking constraints, and dominance-breaking constraints; however, adding a streamliner may even cause the constraint model to become inconsistent. Originally, streamliners were hand-crafted by researchers who used their theoretical insight to analyze the constrained model. However, progress has also been made on the automated generation of streamliners (Spracklen et al. 2023) by systematically trying the effect of some atomic constraints, such as imposing specific constraints on integer and function domains, like enforcing odd or even values, monotonicity, and properties like commutativity, as well as facilitating specific attributes in binary relations. These atomic restrictions are tested on thousands of problem instances, and those that show a good streamlining effect are systematically combined.

Trend following and momentum investing are common strategies employed by asset managers. Even though they can be helpful in the proper situations, they are limited in the sense that they work just by looking at past, as if we were driving with our focus on the rearview mirror. In this paper, we advocate for the use of Artificial Intelligence and Machine Learning techniques to predict future market trends. These predictions, when done properly, can improve the performance of asset managers by increasing returns and reducing drawdowns.

The evolution of data architecture has seen the rise of data lakes, aiming to solve the bottlenecks of data management and promote intelligent decision-making. However, this centralized architecture is limited by the proliferation of data sources and the growing demand for timely analysis and processing. A new data paradigm, Data Mesh, is proposed to overcome these challenges. Data Mesh treats domains as a first-class concern by distributing the data ownership from the central team to each data domain, while keeping the federated governance to monitor domains and their data products. Many multi-million dollar organizations like Paypal, Netflix, and Zalando have already transformed their data analysis pipelines based on this new architecture. In this decentralized architecture where data is locally preserved by each domain team, traditional centralized machine learning is incapable of conducting effective analysis across multiple domains, especially for security-sensitive organizations. To this end, we introduce a pioneering approach that incorporates Federated Learning into Data Mesh. To the best of our knowledge, this is the first open-source applied work that represents a critical advancement toward the integration of federated learning methods into the Data Mesh paradigm, underscoring the promising prospects for privacy-preserving and decentralized data analysis strategies within Data Mesh architecture.

Recent advancements in the flexible job-shop scheduling problem (FJSSP) are primarily based on deep reinforcement learning (DRL) due to its ability to generate high-quality, real-time solutions. However, DRL approaches often fail to fully harness the strengths of existing techniques such as exact methods or constraint programming (CP), which can excel at finding optimal or near-optimal solutions for smaller instances. This paper aims to integrate CP within a deep learning (DL) based methodology, leveraging the benefits of both. In this paper, we introduce a method that involves training a DL model using optimal solutions generated by CP, ensuring the model learns from high-quality data, thereby eliminating the need for the extensive exploration typical in DRL and enhancing overall performance. Further, we integrate CP into our DL framework to jointly construct solutions, utilizing DL for the initial complex stages and transitioning to CP for optimal resolution as the problem is simplified. Our hybrid approach has been extensively tested on three public FJSSP benchmarks, demonstrating superior performance over five state-of-the-art DRL approaches and a widely-used CP solver. Additionally, with the objective of exploring the application to other combinatorial optimization problems, promising preliminary results are presented on applying our hybrid approach to the traveling salesman problem, combining an exact method with a well-known DRL method.