Offline policy learning from logged data is a critical paradigm for enabling effective decision-making without costly online exploration. However, its application has been largely confined to single-objective problems, a stark contrast to real-world scenarios where decision-making inherently involves navigating multiple, often conflicting, objectives. This paper introduces a comprehensive framework for Offline Multi-Objective Bandits (OffMOB), providing a principled solution to the fundamental challenge of learning Pareto-optimal policies from a static dataset. Our core contribution is a novel algorithm that uniquely integrates the pessimism principle with multi-objective optimization to safely learn from off-policy data. Crucially, our approach transcends the primary limitation of scalarization techniques, which are restricted to finding a single policy for a pre-defined preference. Instead, OffMOB directly approximates the entire Pareto front, learning a single, flexible policy model capable of generating an optimal action for any desired trade-off. To rigorously evaluate performance, we introduce the Tchebycheff sub-optimality metric and establish the first finite-sample generalization bounds for this problem class, proving that our algorithm converges to the true Pareto front under practical data coverage assumptions. Extensive experiments on complex benchmarks demonstrate that OffMOB significantly outperforms existing methods, identifying the complete set of optimal trade-offs where naive extensions fail.

Heuristics are commonly used to tackle various search and optimization problems. Design heuristics usually require tedious manual crafting with domain knowledge. Recent works have incorporated Large Language Models (LLMs) into automatic heuristic search, leveraging their powerful language and coding capacity. However, existing research focuses on the optimal performance on the target problem as the sole objective, neglecting other criteria such as efficiency and scalability, which are vital in practice. To tackle this challenge, we propose to model the heuristic search as a multi-objective optimization problem and consider introducing additional practical criteria beyond optimal performance. Due to the complexity of the search space, conventional multi-objective optimization methods struggle to effectively handle LLM-based multi-objective heuristic search. We propose the first LLM-based multi-objective heuristic search framework, Multi-objective Evolution of Heuristic (MEoH), which integrates LLMs in a zero-shot manner to generate a non-dominated set of heuristics to meet multiple design criteria. We design a new dominance-dissimilarity mechanism for effective population management and selection, which incorporates both code dissimilarity in the search space and dominance in the objective space. MEoH is demonstrated in two well-known combinatorial optimization problems: the online Bin Packing Problem (BPP) and the Traveling Salesman Problem (TSP). The results indicate that a variety of elite heuristics are automatically generated in a single run, offering more trade-off options than the existing methods. It successfully achieves competitive or superior performance while improving efficiency up to 10 times. Moreover, we also observe that the multi-objective search introduces novel insights into heuristic design and leads to the discovery of diverse heuristics.

Neural combinatorial optimization (NCO) is a promising learning-based approach to solving complex combinatorial optimization problems such as the traveling salesman problem (TSP), the vehicle routing problem (VRP), and the orienteering problem (OP). However, how to efficiently train a powerful NCO solver for routing problems remains a crucial challenge. The widely used reinforcement learning method suffers from sparse rewards and low data efficiency, while the supervised learning approach requires a large number of high-quality solutions (i.e., labels) that could be costly to obtain. In this work, we find that simple data augmentation operations can drastically reduce the number of required high-quality solutions for supervised learning. Moreover, simple boosting strategies that leverage the property of multiple optima can significantly improve training efficiency. With only a small set of \(50{,}000\) labeled instances, supervised learning can achieve a competitive in-distribution performance with the widely used reinforcement learning counterpart. Furthermore, we also investigate the generalization ability for larger out-of-distribution problems. We believe the findings from this work may lead to a rethinking of the value of data-efficient supervised learning for NCO solver training.

The min-max vehicle routing problem (min-max VRP) traverses all given customers by assigning several routes and aims to minimize the length of the longest route. Recently, reinforcement learning (RL)-based sequential planning methods have exhibited advantages in solving efficiency and optimality. However, these methods fail to exploit the problem-specific properties in learning representations, resulting in less effective features for decoding optimal routes. This paper considers the sequential planning process of min-max VRPs as two coupled optimization tasks: customer partition for different routes and customer navigation in each route (i.e., partition and navigation). To effectively process min-max VRP instances, we present a novel attention-based Partition-and-Navigation encoder (P&N Encoder) that learns distinct embeddings for partition and navigation. Furthermore, we utilize an inherent symmetry in decoding routes and develop an effective agent-permutation-symmetric (APS) loss function. Experimental results demonstrate that the proposed Decoupling-Partition-Navigation (DPN) method significantly surpasses existing learning-based methods in both single-depot and multi-depot min-max VRPs. Our code is available at

This article proposes utilizing a single deep reinforcement learning model to solve combinatorial multiobjective optimization problems. We use the well-known multiobjective traveling salesman problem (MOTSP) as an example. Our proposed method employs an encoder-decoder framework to learn the mapping from the MOTSP instance to its Pareto-optimal set. Specifically, it leverages a novel routing encoder to extract information for both the entire multiobjective aspect and every individual objective from the MOTSP instance. The global embeddings and each objective's embeddings are adaptively aggregated via a routing network to form the subproblems' embedding that can well represent the MOTSP features. Using a modified context embedding, the subproblems' embeddings are fed into a decoder to produce a set of approximate Pareto-optimal solutions in parallel. Additionally, we develop a Top-k baseline to enable more efficient data utilization and lightweight training for our proposed method. We compare our method with heuristic-based and learning-based ones on various types of MOTSP instances, and the experimental results show that our method can solve MOTSP instances in real-time and outperform the other algorithms, especially on large-scale problem instances.

As a research hotspot across logistics, operations research, and artificial intelligence, route planning has become a key technology for intelligent transportation systems. Recently, data-driven machine learning heuristics, including learning construction methods and learning improvement methods, have achieved remarkable success in solving single-objective route planning problems. However, many practical route planning scenarios must simultaneously consider multiple conflict objectives. For example, modern logistics companies often need to simultaneously minimize time budget, transportation cost, and vehicle pollution. Several learning construction methods are proposed for solving classical multi-objective route planning (MORP) problems, yet no learning improvement heuristics have been developed so far, even though they are acknowledged to be more efficient in narrowing the optimality gap. To fill this gap, this paper proposes a learning improvement MORP method, Pareto Improver (PI). PI employs a population-based mechanism to approximate the Pareto front with a single deep reinforcement learning model. The experimental results on various MORP problems show that PI can significantly outperform other state-of-the-art methods.

In dealing with the expensive multiobjective optimization problem, some algorithms convert it into a number of single-objective subproblems for optimization. At each iteration, these algorithms conduct surrogate-assisted optimization on one or multiple subproblems. However, these subproblems may be unnecessary or resolved. Operating on such subproblems can cause server inefficiencies, especially in the case of expensive optimization. To overcome this shortcoming, we propose an adaptive subproblem selection (ASS) strategy to identify the most promising subproblems for further modeling. To better leverage the cross information between the subproblems, we use the collaborative multioutput Gaussian process surrogate to model them jointly. Moreover, the commonly used acquisition functions (also known as infill criteria) are investigated in this article. Our analysis reveals that these acquisition functions may cause severe imbalances between exploitation and exploration in multiobjective optimization scenarios. Consequently, we develop a new acquisition function, namely, adaptive lower confidence bound (ALCB), to cope with it. The experimental results on three different sets of benchmark problems indicate that our proposed algorithm is competitive. Beyond that, we also quantitatively validate the effectiveness of the ASS strategy, the CoMOGP model, and the ALCB acquisition function.