The discovery of symbolic solutions—mathematical expressions, logical rules, and algorithmic structures—is fundamental to advancing scientific and engineering progress. However, traditional methods often struggle with search efficiency and fail to integrate knowledge effectively. While recent large language model-based (LLM-based) approaches have demonstrated improvements in search efficiency, they lack the ability to continually refine and expand upon discovered solutions and their underlying knowledge, limiting their potential for \textit{open-ended innovation}. To address these limitations, we introduce CoEvo, a novel framework that leverages large language models within an evolutionary search methodology to continually generate and refine symbolic solutions. CoEvo integrates a dynamic knowledge library, enabling open-ended innovation of solutions through effective knowledge management. Additionally, CoEvo leverages multiple representations of solutions—including natural language, mathematical expressions, and code—to further enhance search efficiency. By combining the reasoning capabilities of LLMs with the exploratory power of evolutionary algorithms, CoEvo significantly improves the efficiency and scope of symbolic discovery. Our experimental results demonstrate that this method not only enhances the efficiency of searching for symbolic solutions but also supports the ongoing discovery process, akin to human scientific endeavors. This study represents a first effort in conceptualizing the search for symbolic solutions as a lifelong, iterative process, marking a significant step towards harnessing LLMs in the perpetual pursuit of scientific and engineering breakthroughs.

Timing-driven detailed placement is a critical stage in very large scale integrated (VLSI) design, aiming to locally adjust cell positions to further improve circuit timing performance. Existing methods commonly adopt proxy metrics as optimization objectives, such as weighted wirelength and approximate delay. However, these surrogate metrics are not fully aligned with the final timing metrics obtained through static timing analysis (STA), often leading to suboptimal timing results. Besides, methods based directly on STA tools suffer from very low search efficiency, making the cost of timing optimization prohibitive. To address these issues, we propose an effective timing-driven detailed placement method via TimingMask-guided path-level optimization. One core of our method is the TimingMask guidance mechanism, which integrates both arc delay and path slack information based on the RC timing model, thereby providing more targeted and effective guidance for refinement of critical cells. Meanwhile, our method adopts a path-level timing evaluation strategy with incremental updates, accelerating the optimization process while preserving timing accuracy. Experimental results on the ICCAD 2015 contest benchmarks demonstrate that our method significantly outperforms state-of-the-art detailed placement methods such as DREAMPlace4.0 DP, achieving an average improvement of 25.3% in total negative slack (TNS) and 21.7% in worst negative slack (WNS).

Disagreement optimization is an emerging research issue in social networks. Although leaders are often selected to shape the opinions of followers, little effort has been made to investigate how to optimize the disagreement between them. This paper aims to address this issue by proposing and solving the following subset selection problem called leader-follower disagreement minimization (LFDMP): given a social network with n vertices and m edges, how to select k (kn) vertices as leaders such that the disagreement between leaders and followers is minimized. We show that the objective function of LFDMP is monotone and supermodular, and the state-of-the-art algorithm called Pareto optimization for subset selection (POSS) can solve this problem with (1-1/e) approximation in O(k2n4) expected time. To address the computational challenge faced by POSS while maintaining its theoretical advantage, we further develop a fast algorithm called Pareto optimization for leader selection (POLS) within its framework. We demonstrate that POLS can solve LFDMP with -related approximation in O(k2mn) expected time, where is an error parameter. Extensive experiments on various real-world social networks illustrate both the effectiveness and efficiency of POLS.

In modern intelligent transportation systems (ITS), particularly in freight transportation and logistics, real-time route planning is crucial. It presents unique challenges driven by high uncertainty in service requests, where the number of service customers can vary drastically, ranging from hundreds to thousands. Existing neural methods struggle to maintain performance under such significant variations, which severely limits their practical applicability. To address this crucial shortcoming, this work proposes a novel Instance-Conditioned Adaptation Model (ICAM) designed for better large-scale generalization. In particular, we design a simple yet efficient instance-conditioned adaptation function that adjusts the policy based on the specific geometry and density of the current traffic scenario to improve model adaptability with minimal computational overhead. Furthermore, we propose a powerful yet low-complexity instance-conditioned adaptation module to generate better solutions for instances across various scales. Extensive experiments on synthetic, benchmark, and real-world instances demonstrate that ICAM can consistently achieve promising generalization performance across four widely studied large-scale route planning scenarios. Notably, our proposed method delivers high-quality solutions with remarkably fast inference speed, providing a scalable and efficient solution for real-time intelligent transportation operations. Our code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/CIAM-Group/ICAM</uri>

This paper studies lexicographic online learning within the framework of multiobjective stochastic linear bandits (MOSLB), where the agent aims to simultaneously maximize multiple objectives in a hierarchical manner. Previous literature has investigated lexicographic online learning in multiobjective multi-armed bandits, a special case of MOSLB. They provided a suboptimal algorithm whose regret bound is approximately O(T^(2/3)) based on a priority-based regret metric. In this paper, we propose an algorithm for lexicographic online learning in the MOSLB model, achieving an almost optimal regret bound of approximately O(dT^(1/2)) when evaluated by the general regret metric. Here, d is the dimension of arm vectors, and T is the time horizon. Our method introduces a new arm filter and a multiple trade-offs approach to effectively balance exploration and exploitation across different objectives. Experiments confirm the merits of our algorithms and provide compelling evidence to support our analysis.

Continual learning aims to learn multiple tasks sequentially. A key challenge in continual learning is balancing between two objectives: retaining knowledge from old tasks (stability) and adapting to new tasks (plasticity). Experience replay methods, which store and replay past data alongside new data, have become a widely adopted approach to mitigate catastrophic forgetting. However, these methods neglect the dynamic nature of the stability-plasticity trade-off and aim to find a fixed and unchanging balance, resulting in suboptimal adaptation during training and inference. In this paper, we propose Pareto Continual Learning (ParetoCL), a novel framework that reformulates the stability-plasticity trade-off in continual learning as a multi-objective optimization (MOO) problem. ParetoCL introduces a preference-conditioned model to efficiently learn a set of Pareto optimal solutions representing different trade-offs and enables dynamic adaptation during inference. From a generalization perspective, ParetoCL can be seen as an objective augmentation approach that learns from different objective combinations of stability and plasticity. Extensive experiments across multiple datasets and settings demonstrate that ParetoCL outperforms state-of-the-art methods and adapts to diverse continual learning scenarios.

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.

Crafting adversarial examples is crucial for evaluating and enhancing the robustness of Deep Neural Networks (DNNs), presenting a challenge equivalent to maximizing a non-differentiable 0-1 loss function. However, existing single objective methods, namely adversarial attacks focus on a surrogate loss function, do not fully harness the benefits of engaging multiple loss functions, as a result of insufficient understanding of their synergistic and conflicting nature. To overcome these limitations, we propose the Multi-Objective Set-Based Attack (MOS Attack), a novel adversarial attack framework leveraging multiple loss functions and automatically uncovering their interrelations. The MOS Attack adopts a set-based multi-objective optimization strategy, enabling the incorporation of numerous loss functions without additional parameters. It also automatically mines synergistic patterns among various losses, facilitating the generation of potent adversarial attacks with fewer objectives. Extensive experiments have shown that our MOS Attack outperforms single-objective attacks. Furthermore, by harnessing the identified synergistic patterns, MOS Attack continues to show superior results with a reduced number of loss functions. Our code is available at https://github.com/pgg3/MOS-Attack.

We introduce the ReMaP framework, which generates expert-quality macro placements through recursively prototyping and periphery-guided relocating. A key innovation is ABPlace, an angle-based analytical method that arranges macros along an ellipse to facilitate a rough distribution near the periphery, while optimizing dataflow, minimizing overlap, and ensuring convergence. Based on the results of ABPlace, an efficient heuristic is proposed to position macros along the chip’s periphery, mirroring practices often employed by experts. Our framework outperforms three leading macro placers in both WNS and TNS across eight test cases, achieving improvements up to 34.15% in WNS and 65.39% in TNS, as tested on the popular OpenROAD-flow-scripts infrastructure. Additionally, our parameter autotuning method further improves timing by 8.75%.

Timing optimization during the global placement of integrated circuits has been a significant focus for decades, yet it remains a complex, unresolved issue. Recent analytical methods typically use pin-level timing information to adjust net weights, which is fast and simple but neglects the path-based nature of the timing graph. The existing path-based methods, however, cannot balance the accuracy and efficiency due to the exponential growth of number of critical paths. In this work, we propose a GPU-accelerated timing-driven global placement framework, integrating accurate path-level information into the efficient DREAMPlace infrastructure. It optimizes the fine-grained pin-to-pin attraction objective and is facilitated by efficient critical path extraction. We also design a quadratic distance loss function specifically to align with the RC timing model. Experimental results demonstrate that our method significantly outperforms the current leading timing-driven placers, achieving an average improvement of 40.5% in total negative slack (TNS) and 8.3% in worst negative slack (WNS), as well as an improvement in half-perimeter wirelength (HPWL).

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

Neural Combinatorial Optimization (NCO) methods have exhibited promising performance in solving Vehicle Routing Problems (VRPs). However, most NCO methods rely on the conventional self-attention mechanism that induces excessive computational complexity, thereby struggling to contend with large-scale VRPs and hindering their practical applicability. In this paper, we propose a lightweight cross-attention mechanism with linear complexity, by which a Transformer network is developed to learn efficient and favorable solutions for large-scale VRPs. We also propose a Self-Improved Training (SIT) algorithm that enables direct model training on large-scale VRP instances, bypassing extensive computational overhead for attaining labels. By iterating solution reconstruction, the Transformer network itself can generate improved partial solutions as pseudo-labels to guide the model training. Experimental results on the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 100K nodes indicate that our method consistently achieves superior performance for synthetic and real-world benchmarks, significantly boosting the scalability of NCO methods.

This paper studies lexicographic multi-armed bandits (MAB), where after selecting an arm, the agent observes a reward vector including multiple objectives, each with a different level of importance. Although previous literature has proposed the algorithm for lexicographic MAB, their algorithm suffers from several limitations: (1) it exhibits poor adversarial robustness due to its reliance on stochastic rewards, (2) its regret bound is suboptimal compared to single-objective counterparts, and (3) the regret bound does not adapt to specific problem instances. To address these limitations, we study lexicographic MAB with adversarial corruptions, where an adversary might corrupt the stochastic rewards with a corruption budget of C. First, when the value of C is known, we propose an algorithm achieving a problem-dependent regret bound of O(∑(log T / Δⁱ(a) + C)) for the i-th objective (i ∈ [M]), where Δⁱ(a) is the reward gap for arm a on the i-th objective, and M is the number of objectives. In the purely stochastic setting (C=0), this regret bound approaches optimality. Second, we introduce another algorithm that does not require value of C but incurs a less favorable regret bound of O(∑(γ_T / Δⁱ(a) + γ_T)) for the i-th objective, where γ_T = O((log T)² + KC(log T)²). Finally, we conduct experiments on both synthetic and real-world datasets to verify the effectiveness of our algorithms.

Online Continual Learning (OCL) requires models to learn sequentially from data streams with limited memory. Rehearsal-based methods, particularly Experience Replay (ER), are commonly used in OCL scenarios. This paper revisits ER through the lens of $\epsilon$-constraint optimization, revealing that ER implicitly employs a soft constraint on past task performance, with its weighting parameter post-hoc defining a slack variable. While effective, ER's implicit and fixed slack strategy has limitations: it can inadvertently lead to updates that negatively impact generalization, and its fixed trade-off between plasticity and stability may not optimally balance current streaming with memory retention, potentially overfitting to the memory buffer. To address these shortcomings, we propose the \textbf{G}radient-Guided \textbf{E}psilon \textbf{C}onstraint (\textbf{GEC}) method for online continual learning. GEC explicitly formulates the OCL update as an $\epsilon$-constraint optimization problem, which minimize the loss on the current task data and transform the stability objective as constraints and propose a gradient-guided method to dynamically adjusts the update direction based on whether the performance on memory samples violates a predefined slack tolerance $\bar{\varepsilon}$: if forgetting exceeds this tolerance, GEC prioritizes constraint satisfaction; otherwise, it focuses on the current task while controlling the rate of increase in memory loss. Empirical evaluations on standard OCL benchmarks demonstrate GEC's ability to achieve a superior trade-off, leading to improved overall performance. Code is available at https://github.com/laisong-22004009/GEC_OCL.

Multi-objective optimization problems (MOPs) are prevalent in numerous real-world applications. Recently, Pareto Set Learning (PSL) has emerged as a powerful paradigm for solving MOPs. PSL can produce a neural network for modeling the set of all Pareto optimal solutions. However, applying PSL to black-box objectives, particularly those exhibiting non-separability, high dimensionality, and/or other complex properties, remains very challenging. To address this issue, we propose leveraging evolution strategies (ESs), a class of specialized black-box optimization algorithms, within the PSL paradigm. Traditional ESs capture the complex dimensional dependencies less efficiently, which can significantly hinder their performance in PSL. To tackle this issue, we suggest encapsulating the dependencies within a neural network, which is then trained using a novel gradient estimation method. The proposed method, termed Neural-ES, is evaluated using a bespoke benchmark suite for black-box PSL. Experimental comparisons with other methods demonstrate the efficiency of Neural-ES, underscoring its ability to learn the Pareto sets of challenging black-box MOPs.

Neural Combinatorial Optimisation (NCO) is a promising learning-based approach for solving Vehicle Routing Problems (VRPs) without extensive manual design. While existing constructive NCO methods typically follow an appending-based paradigm that sequentially adds unvisited nodes to partial solutions, this rigid approach often leads to suboptimal results. To overcome this limitation, we explore the idea of the insertion-based paradigm and propose Learning to Construct with Insertion-based Paradigm (L2C-Insert), a novel learning-based method for constructive NCO. Unlike traditional approaches, L2C-Insert builds solutions by strategically inserting unvisited nodes at any valid position in the current partial solution, which can significantly enhance the flexibility and solution quality. The proposed framework introduces three key components: a novel model architecture for precise insertion position prediction, an efficient training scheme for model optimization, and an advanced inference technique that fully exploits the insertion paradigm's flexibility. Extensive experiments on both synthetic and real-world instances of the Travelling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) demonstrate that L2C-Insert consistently achieves superior performance across various problem sizes. The code is available at [https://github.com/CIAM-Group/L2C\_Insert](https://github.com/CIAM-Group/L2C\_Insert).

Long-Tailed Recognition (LTR) presents a significant challenge due to extreme class imbalance, where existing methods often struggle to balance performance across head and tail classes. Directly applying multi-objective optimization (MOO) to leverage multiple LTR strategies can be complex and unstable. To address this, we propose TS-MOF (Two-Stage Multi-Objective Fine-tuning), a novel framework that strategically decouples feature learning from classifier adaptation. After standard pre-training, TS-MOF freezes the feature backbone and focuses on an efficient multi-objective fine-tuning of specialized classifier heads. The core of TS-MOF's second stage lies in two innovations: Refined Performance Level Agreement for adaptive task weighting based on real-time per-class performance, and Robust Deterministic Projective Conflict Gradient for stable gradient conflict resolution and constructive fusion. This approach enables effective synergy between diverse LTR strategies, leading to significant and balanced performance improvements. Extensive experiments on CIFAR100-LT, ImageNet-LT, and iNaturalist 2018 demonstrate that TS-MOF achieves state-of-the-art results, particularly enhancing tail class accuracy (e.g., +3.3\% on CIFAR100-LT IR=100 tail) while improving head class performance, all within a remarkably short fine-tuning period of 20 epochs.

In the past few decades, many multiobjective evolutionary optimization algorithms (MOEAs) have been proposed to find a finite set of approximate Pareto solutions for a given problem in a single run. However, in many real-world applications, it could be desirable to have structure constraints on the entire optimal solution set, which define the patterns shared among all solutions. The current population-based MOEAs cannot properly handle such requirements. In this work, we make a first attempt to incorporate the structure constraints into the whole solution set. Specifically, we propose to model such a multiobjective optimization problem as a set optimization problem with structure constraints. The structure constraints define some patterns that all the solutions are required to share. Such patterns can be fixed components shared by all solutions, specific relations among decision variables, and the required shape of the Pareto set. In addition, we develop a simple yet efficient evolutionary stochastic optimization method to learn the set model, which only requires a low computational budget similar to classic MOEAs. With our proposed method, the decision-makers can easily tradeoff the Pareto optimality with preferred structures, which is not supported by other MOEAs. A set of experiments on benchmark test suites and real-world application problems demonstrates that our proposed method is effective.

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.

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a lightweight and efficient smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method.

Heuristics are widely used for dealing with complex search and optimization problems. However, manual design of heuristics can be often very labour extensive and requires rich working experience and knowledge. This paper proposes Evolution of Heuristic (EoH), a novel evolutionary paradigm that leverages both Large Language Models (LLMs) and Evolutionary Computation (EC) methods for Automatic Heuristic Design (AHD). EoH represents the ideas of heuristics in natural language, termed thoughts. They are then translated into executable codes by LLMs. The evolution of both thoughts and codes in an evolutionary search framework makes it very effective and efficient for generating high-performance heuristics. Experiments on three widely studied combinatorial optimization benchmark problems demonstrate that EoH outperforms commonly used handcrafted heuristics and other recent AHD methods including FunSearch. Particularly, the heuristic produced by EoH with a low computational budget (in terms of the number of queries to LLMs) significantly outperforms widely-used human hand-crafted baseline algorithms for the online bin packing problem.

Vehicle routing problems (VRP) are very important in many realworld applications and has been studied for several decades.Recently, neural combinatorial optimization (NCO) has attracted growing research effort.NCO is to train a neural network model to solve an optimization problem in question.However, existing NCO methods often build a different model for each routing problem, which significantly hinders their application in some areas where there are many different VRP variants to solve.In this work, we make a first attempt to tackle the crucial challenge of cross-problem generalization in NCO.We formulate VRPs as different combinations of a set of shared underlying attributes and solve them simultaneously via a single model through attribute composition.In this way, our proposed model can successfully solve VRPs with unseen attribute combinations in a zero-shot generalization manner.In our experiments, the neural model is trained on five VRP variants and its performance is tested on eleven VRP variants.The experimental results show that the model demonstrates superior performance on these eleven VRP variants, reducing the average gap to around 5% from over 20% and achieving a notable performance boost on both benchmark datasets and real-world logistics scenarios.

In modern chip design, placement aims at placing millions of circuit modules, which is an essential step that significantly influences power, performance, and area (PPA) metrics. Recently, reinforcement learning (RL) has emerged as a promising technique for improving placement quality, especially macro placement. However, current RL-based placement methods suffer from long training times, low generalization ability, and inability to guarantee PPA results. A key issue lies in the problem formulation, i.e., using RL to place from scratch, which results in limits useful information and inaccurate rewards during the training process. In this work, we propose an approach that utilizes RL for the refinement stage, which allows the RL policy to learn how to adjust existing placement layouts, thereby receiving sufficient information for the policy to act and obtain relatively dense and precise rewards. Additionally, we introduce the concept of regularity during training, which is considered an important metric in the chip design industry but is often overlooked in current RL placement methods. We evaluate our approach on the ISPD 2005 and ICCAD 2015 benchmark, comparing the global half-perimeter wirelength and regularity of our proposed method against several competitive approaches. Besides, we test the PPA performance using commercial software, showing that RL as a regulator can achieve significant PPA improvements. Our RL regulator can fine-tune placements from any method and enhance their quality. Our work opens up new possibilities for the application of RL in placement, providing a more effective and efficient approach to optimizing chip design. Our code is available at \url{https://github.com/lamda-bbo/macro-regulator}.

Multiobjective optimization (MOO) plays a critical role in various real-world domains. A major challenge therein is generating $K$ uniform Pareto-optimal solutions to represent the entire Pareto front. To address this issue, this paper firstly introduces \emph{fill distance} to evaluate the $K$ design points, which provides a quantitative metric for the representativeness of the design. However, directly specifying the optimal design that minimizes the fill distance is nearly intractable due to the nested $\min-\max-\min$ optimization problem. To address this, we propose a surrogate ``max-packing'' design for the fill distance design, which is easier to optimize and leads to a rate-optimal design with a fill distance at most $4\times$ the minimum value. Extensive experiments on synthetic and real-world benchmarks demonstrate that our proposed paradigm efficiently produces high-quality, representative solutions and outperforms baseline methods.

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands to millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order or meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above challenges, this paper introduces \algoname, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair and comprehensive benchmark, and is open-sourced for the community.

Homotopy optimization is a traditional method to deal with a complicated optimization problem by solving a sequence of easy-to-hard surrogate subproblems. However, this method can be very sensitive to the continuation schedule design and might lead to a suboptimal solution to the original problem. In addition, the intermediate solutions, often ignored by classic homotopy optimization, could be useful for many real-world applications. In this work, we propose a novel model-based approach to learn the whole continuation path for homotopy optimization, which contains infinite intermediate solutions for any surrogate subproblems. Rather than the classic unidirectional easy-to-hard optimization, our method can simultaneously optimize the original problem and all surrogate subproblems in a collaborative manner. The proposed model also supports the real-time generation of any intermediate solution, which could be desirable for many applications. Experimental studies on different problems show that our proposed method can significantly improve the performance of homotopy optimization and provide extra helpful information to support better decision-making.

Neural combinatorial optimization (NCO) is a promising learning-based approach for solving challenging combinatorial optimization problems without specialized algorithm design by experts. However, most constructive NCO methods cannot solve problems with large-scale instance sizes, which significantly diminishes their usefulness for real-world applications. In this work, we propose a novel Light Encoder and Heavy Decoder (LEHD) model with a strong generalization ability to address this critical issue. The LEHD model can learn to dynamically capture the relationships between all available nodes of varying sizes, which is beneficial for model generalization to problems of various scales. Moreover, we develop a data-efficient training scheme and a flexible solution construction mechanism for the proposed LEHD model. By training on small-scale problem instances, the LEHD model can generate nearly optimal solutions for the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 1000 nodes, and also generalizes well to solve real-world TSPLib and CVRPLib problems. These results confirm our proposed LEHD model can significantly improve the state-of-the-art performance for constructive NCO.

This paper presents a novel approach to multiobjective algorithms aimed at modeling the Pareto set using neural networks. Whereas previous methods mainly focused on identifying a finite number of solutions, our approach allows for the direct modeling of the entire Pareto set. Furthermore, we establish an equivalence between learning the complete Pareto set and maximizing the associated hypervolume, which enables the convergence analysis of hypervolume (as a new metric) for Pareto set learning. Specifically, our new analysis framework reveals the connection between the learned Pareto solution and its representation in a polar coordinate system. We evaluate our proposed approach on various benchmark problems and real-world problems, and the encouraging results make it a potentially viable alternative to existing multiobjective algorithms. Code is available at \url{https://github.com/xzhang2523/hvpsl/tree/master}.

Expensive multi-objective optimization problems can be found in many real-world applications, where their objective function evaluations involve expensive computations or physical experiments. It is desirable to obtain an approximate Pareto front with a limited evaluation budget. Multi-objective Bayesian optimization (MOBO) has been widely used for finding a finite set of Pareto optimal solutions. However, it is well-known that the whole Pareto set is on a continuous manifold and can contain infinite solutions. The structural properties of the Pareto set are not well exploited in existing MOBO methods, and the finite-set approximation may not contain the most preferred solution(s) for decision-makers. This paper develops a novel learning-based method to approximate the whole Pareto set for MOBO, which generalizes the decomposition-based multi-objective optimization algorithm (MOEA/D) from finite populations to models. We design a simple and powerful acquisition search method based on the learned Pareto set, which naturally supports batch evaluation. In addition, with our proposed model, decision-makers can readily explore any trade-off area in the approximate Pareto set for flexible decision-making. This work represents the first attempt to model the Pareto set for expensive multi-objective optimization. Experimental results on different synthetic and real-world problems demonstrate the effectiveness of our proposed method.

Covariance matrix adaptation evolution strategy (CMA-ES) is a successful gradient-free optimization algorithm. Yet, it can hardly scale to handle high-dimensional problems. In this paper, we propose a fast variant of CMA-ES (Fast CMA-ES) to handle large-scale black-box optimization problems. We approximate the covariance matrix by a low-rank matrix with a few vectors and use two of them to generate each new solution. The algorithm achieves linear internal complexity on the dimension of search space. We illustrate that the covariance matrix of the underlying distribution can be considered as an ensemble of simple models constructed by two vectors. We experimentally investigate the algorithm's behaviors and performances. It is more efficient than the CMA-ES in terms of running time. It outperforms or performs comparatively to the variant limited memory CMA-ES on large-scale problems. Finally, we evaluate the algorithm's performance with a restart strategy on the CEC'2010 large-scale global optimization benchmarks, and it shows remarkable performance and outperforms the large-scale variants of the CMA-ES.

Multi-task learning is a powerful method for solving multiple correlated tasks simultaneously. However, it is often impossible to find one single solution to optimize all the tasks, since different tasks might conflict with each other. Recently, a novel method is proposed to find one single Pareto optimal solution with good trade-off among different tasks by casting multi-task learning as multiobjective optimization. In this paper, we generalize this idea and propose a novel Pareto multi-task learning algorithm (Pareto MTL) to find a set of well-distributed Pareto solutions which can represent different trade-offs among different tasks. The proposed algorithm first formulates a multi-task learning problem as a multiobjective optimization problem, and then decomposes the multiobjective optimization problem into a set of constrained subproblems with different trade-off preferences. By solving these subproblems in parallel, Pareto MTL can find a set of well-representative Pareto optimal solutions with different trade-off among all tasks. Practitioners can easily select their preferred solution from these Pareto solutions, or use different trade-off solutions for different situations. Experimental results confirm that the proposed algorithm can generate well-representative solutions and outperform some state-of-the-art algorithms on many multi-task learning applications.

This paper proposes a novel surrogate-model-based multi-objective evolutionary algorithm, which is called Multi-objective Bayesian Optimization Algorithm based on Decomposition (MOBO/D). In this algorithm, a multi-objective problem is decomposed into several subproblems which will be solved simultaneously. MOBO/D builds Gaussian process model for each objective to learn the optimization surface, and defines utility function for each subproblem to guide the searching process. At each generation, MOEA/D algorithm is called to locate a set of candidate solutions which maximize all utility functions respectively, and a subset of those candidate solutions is selected for parallel batch evaluation. Experimental study on different test instances validates that MOBO/D can efficiently solve expensive multi-objective problems in parallel. The performance of MOBO/D is also better than several classical expensive optimization methods.

This paper investigates how to use a pre-selection approach to improve the performance of the multiobjective evolutionary algorithm based on decomposition (MOEA/D). It proposes a novel MOEA/D algorithm with classification to serve this purpose. The proposed algorithm builds a classification model on the search space to filter all new generated solutions, and mainly evaluates those promising solutions for reducing real function evaluation costs during the search process. Experimental study on different test instances validates that the pre-selection approach can significantly improve the performance of a classical MOEA/D.