LGOct 11, 2022Code
MARLlib: A Scalable and Efficient Multi-agent Reinforcement Learning LibrarySiyi Hu, Yifan Zhong, Minquan Gao et al.
A significant challenge facing researchers in the area of multi-agent reinforcement learning (MARL) pertains to the identification of a library that can offer fast and compatible development for multi-agent tasks and algorithm combinations, while obviating the need to consider compatibility issues. In this paper, we present MARLlib, a library designed to address the aforementioned challenge by leveraging three key mechanisms: 1) a standardized multi-agent environment wrapper, 2) an agent-level algorithm implementation, and 3) a flexible policy mapping strategy. By utilizing these mechanisms, MARLlib can effectively disentangle the intertwined nature of the multi-agent task and the learning process of the algorithm, with the ability to automatically alter the training strategy based on the current task's attributes. The MARLlib library's source code is publicly accessible on GitHub: \url{https://github.com/Replicable-MARL/MARLlib}.
AIAug 22, 2023
ProAgent: Building Proactive Cooperative Agents with Large Language ModelsCeyao Zhang, Kaijie Yang, Siyi Hu et al. · pku
Building agents with adaptive behavior in cooperative tasks stands as a paramount goal in the realm of multi-agent systems. Current approaches to developing cooperative agents rely primarily on learning-based methods, whose policy generalization depends heavily on the diversity of teammates they interact with during the training phase. Such reliance, however, constrains the agents' capacity for strategic adaptation when cooperating with unfamiliar teammates, which becomes a significant challenge in zero-shot coordination scenarios. To address this challenge, we propose ProAgent, a novel framework that harnesses large language models (LLMs) to create proactive agents capable of dynamically adapting their behavior to enhance cooperation with teammates. ProAgent can analyze the present state, and infer the intentions of teammates from observations. It then updates its beliefs in alignment with the teammates' subsequent actual behaviors. Moreover, ProAgent exhibits a high degree of modularity and interpretability, making it easily integrated into various of coordination scenarios. Experimental evaluations conducted within the Overcooked-AI environment unveil the remarkable performance superiority of ProAgent, outperforming five methods based on self-play and population-based training when cooperating with AI agents. Furthermore, in partnered with human proxy models, its performance exhibits an average improvement exceeding 10% compared to the current state-of-the-art method. For more information about our project, please visit~\url{https://pku-proagent.github.io}.
LGApr 19, 2023
Heterogeneous-Agent Reinforcement LearningYifan Zhong, Jakub Grudzien Kuba, Xidong Feng et al.
The necessity for cooperation among intelligent machines has popularised cooperative multi-agent reinforcement learning (MARL) in AI research. However, many research endeavours heavily rely on parameter sharing among agents, which confines them to only homogeneous-agent setting and leads to training instability and lack of convergence guarantees. To achieve effective cooperation in the general heterogeneous-agent setting, we propose Heterogeneous-Agent Reinforcement Learning (HARL) algorithms that resolve the aforementioned issues. Central to our findings are the multi-agent advantage decomposition lemma and the sequential update scheme. Based on these, we develop the provably correct Heterogeneous-Agent Trust Region Learning (HATRL), and derive HATRPO and HAPPO by tractable approximations. Furthermore, we discover a novel framework named Heterogeneous-Agent Mirror Learning (HAML), which strengthens theoretical guarantees for HATRPO and HAPPO and provides a general template for cooperative MARL algorithmic designs. We prove that all algorithms derived from HAML inherently enjoy monotonic improvement of joint return and convergence to Nash Equilibrium. As its natural outcome, HAML validates more novel algorithms in addition to HATRPO and HAPPO, including HAA2C, HADDPG, and HATD3, which generally outperform their existing MA-counterparts. We comprehensively test HARL algorithms on six challenging benchmarks and demonstrate their superior effectiveness and stability for coordinating heterogeneous agents compared to strong baselines such as MAPPO and QMIX.
MAJun 1, 2022
Policy Diagnosis via Measuring Role Diversity in Cooperative Multi-agent RLSiyi Hu, Chuanlong Xie, Xiaodan Liang et al.
Cooperative multi-agent reinforcement learning (MARL) is making rapid progress for solving tasks in a grid world and real-world scenarios, in which agents are given different attributes and goals, resulting in different behavior through the whole multi-agent task. In this study, we quantify the agent's behavior difference and build its relationship with the policy performance via {\bf Role Diversity}, a metric to measure the characteristics of MARL tasks. We define role diversity from three perspectives: action-based, trajectory-based, and contribution-based to fully measure a multi-agent task. Through theoretical analysis, we find that the error bound in MARL can be decomposed into three parts that have a strong relation to the role diversity. The decomposed factors can significantly impact policy optimization on three popular directions including parameter sharing, communication mechanism, and credit assignment. The main experimental platforms are based on {\bf Multiagent Particle Environment (MPE)} and {\bf The StarCraft Multi-Agent Challenge (SMAC). Extensive experiments} clearly show that role diversity can serve as a robust measurement for the characteristics of a multi-agent cooperation task and help diagnose whether the policy fits the current multi-agent system for a better policy performance.
MAJun 19, 2023
Maximum Entropy Heterogeneous-Agent Reinforcement LearningJiarong Liu, Yifan Zhong, Siyi Hu et al.
Multi-agent reinforcement learning (MARL) has been shown effective for cooperative games in recent years. However, existing state-of-the-art methods face challenges related to sample complexity, training instability, and the risk of converging to a suboptimal Nash Equilibrium. In this paper, we propose a unified framework for learning stochastic policies to resolve these issues. We embed cooperative MARL problems into probabilistic graphical models, from which we derive the maximum entropy (MaxEnt) objective for MARL. Based on the MaxEnt framework, we propose Heterogeneous-Agent Soft Actor-Critic (HASAC) algorithm. Theoretically, we prove the monotonic improvement and convergence to quantal response equilibrium (QRE) properties of HASAC. Furthermore, we generalize a unified template for MaxEnt algorithmic design named Maximum Entropy Heterogeneous-Agent Mirror Learning (MEHAML), which provides any induced method with the same guarantees as HASAC. We evaluate HASAC on six benchmarks: Bi-DexHands, Multi-Agent MuJoCo, StarCraft Multi-Agent Challenge, Google Research Football, Multi-Agent Particle Environment, and Light Aircraft Game. Results show that HASAC consistently outperforms strong baselines, exhibiting better sample efficiency, robustness, and sufficient exploration. See our page at https://sites.google.com/view/meharl.
48.2AIMay 29
HADT: A Heterogeneous Multi-Agent Differential Transformer for Autonomous Earth Observation Satellite ClusterMohamad A. Hady, Muhammad Anwar Masum, Siyi Hu et al.
This work addresses the problem of autonomous resource management in heterogeneous satellite cluster conducting Earth Observation (EO) missions including optical and Synthetic Aperture Radar (SAR) satellites. In autonomous operation mode, satellites are equipped with intelligent capabilities enabling real-time decision-making based on the latest conditions, while requiring minimal interaction with ground operators. Traditional scheduling approaches typically rely on mathematical models to represent satellite mission and resource management. Then, this problem is solved by using optimization algorithms. However, such solutions become less effective when the underlying models are not available, over complex, and inaccurate due to dynamic changes and uncertainties inherent in the space mission environment. A promising alternative is to reformulate the problem as a sequential decision-making process and apply model-free reinforcement learning techniques to enable adaptive and real-time resource management. To this end, we propose a novel transformer-based architecture tailored for heterogeneous satellite cluster autonomous EO Mission with relational observations-actions tokenization and differential attention mechanism. Our experimental results demonstrate significant performance improvements compared to the available baselines. Moreover, the proposed architecture exhibits strong adaptability and transferability with respect to varying numbers of satellite clusters.
ROJun 9, 2022
Extracting Zero-shot Common Sense from Large Language Models for Robot 3D Scene UnderstandingWilliam Chen, Siyi Hu, Rajat Talak et al.
Semantic 3D scene understanding is a problem of critical importance in robotics. While significant advances have been made in simultaneous localization and mapping algorithms, robots are still far from having the common sense knowledge about household objects and their locations of an average human. We introduce a novel method for leveraging common sense embedded within large language models for labelling rooms given the objects contained within. This algorithm has the added benefits of (i) requiring no task-specific pre-training (operating entirely in the zero-shot regime) and (ii) generalizing to arbitrary room and object labels, including previously-unseen ones -- both of which are highly desirable traits in robotic scene understanding algorithms. The proposed algorithm operates on 3D scene graphs produced by modern spatial perception systems, and we hope it will pave the way to more generalizable and scalable high-level 3D scene understanding for robotics.
AIAug 30, 2023
Debunking Disinformation: Revolutionizing Truth with NLP in Fake News DetectionLi He, Siyi Hu, Ailun Pei
The Internet and social media have altered how individuals access news in the age of instantaneous information distribution. While this development has increased access to information, it has also created a significant problem: the spread of fake news and information. Fake news is rapidly spreading on digital platforms, which has a negative impact on the media ecosystem, public opinion, decision-making, and social cohesion. Natural Language Processing(NLP), which offers a variety of approaches to identify content as authentic, has emerged as a potent weapon in the growing war against disinformation. This paper takes an in-depth look at how NLP technology can be used to detect fake news and reveals the challenges and opportunities it presents.
ROSep 12, 2022
Leveraging Large (Visual) Language Models for Robot 3D Scene UnderstandingWilliam Chen, Siyi Hu, Rajat Talak et al.
Abstract semantic 3D scene understanding is a problem of critical importance in robotics. As robots still lack the common-sense knowledge about household objects and locations of an average human, we investigate the use of pre-trained language models to impart common sense for scene understanding. We introduce and compare a wide range of scene classification paradigms that leverage language only (zero-shot, embedding-based, and structured-language) or vision and language (zero-shot and fine-tuned). We find that the best approaches in both categories yield $\sim 70\%$ room classification accuracy, exceeding the performance of pure-vision and graph classifiers. We also find such methods demonstrate notable generalization and transfer capabilities stemming from their use of language.
14.3AIApr 8
KD-MARL: Resource-Aware Knowledge Distillation in Multi-Agent Reinforcement LearningMonirul Islam Pavel, Siyi Hu, Muhammad Anwar Masum et al.
Real world deployment of multi agent reinforcement learning MARL systems is fundamentally constrained by limited compute memory and inference time. While expert policies achieve high performance they rely on costly decision cycles and large scale models that are impractical for edge devices or embedded platforms. Knowledge distillation KD offers a promising path toward resource aware execution but existing KD methods in MARL focus narrowly on action imitation often neglecting coordination structure and assuming uniform agent capabilities. We propose resource aware Knowledge Distillation for Multi Agent Reinforcement Learning KD MARL a two stage framework that transfers coordinated behavior from a centralized expert to lightweight decentralized student agents. The student policies are trained without a critic relying instead on distilled advantage signals and structured policy supervision to preserve coordination under heterogeneous and limited observations. Our approach transfers both action level behavior and structural coordination patterns from expert policies while supporting heterogeneous student architectures allowing each agent model capacity to match its observation complexity which is crucial for efficient execution under partial or limited observability and limited onboard resources. Extensive experiments on SMAC and MPE benchmarks demonstrate that KD MARL achieves high performance retention while substantially reducing computational cost. Across standard multi agent benchmarks KD MARL retains over 90 percent of expert performance while reducing computational cost by up to 28.6 times FLOPs. The proposed approach achieves expert level coordination and preserves it through structured distillation enabling practical MARL deployment across resource constrained onboard platforms.
MAApr 29, 2025
Multi-Agent Reinforcement Learning for Resources Allocation Optimization: A SurveyMohamad A. Hady, Siyi Hu, Mahardhika Pratama et al.
Multi-Agent Reinforcement Learning (MARL) has become a powerful framework for numerous real-world applications, modeling distributed decision-making and learning from interactions with complex environments. Resource Allocation Optimization (RAO) benefits significantly from MARL's ability to tackle dynamic and decentralized contexts. MARL-based approaches are increasingly applied to RAO challenges across sectors playing pivotal roles to Industry 4.0 developments. This survey provides a comprehensive review of recent MARL algorithms for RAO, encompassing core concepts, classifications, and a structured taxonomy. By outlining the current research landscape and identifying primary challenges and future directions, this survey aims to support researchers and practitioners in leveraging MARL's potential to advance resource allocation solutions.
LGMay 7, 2025
Onboard Optimization and Learning: A SurveyMonirul Islam Pavel, Siyi Hu, Mahardhika Pratama et al.
Onboard learning is a transformative approach in edge AI, enabling real-time data processing, decision-making, and adaptive model training directly on resource-constrained devices without relying on centralized servers. This paradigm is crucial for applications demanding low latency, enhanced privacy, and energy efficiency. However, onboard learning faces challenges such as limited computational resources, high inference costs, and security vulnerabilities. This survey explores a comprehensive range of methodologies that address these challenges, focusing on techniques that optimize model efficiency, accelerate inference, and support collaborative learning across distributed devices. Approaches for reducing model complexity, improving inference speed, and ensuring privacy-preserving computation are examined alongside emerging strategies that enhance scalability and adaptability in dynamic environments. By bridging advancements in hardware-software co-design, model compression, and decentralized learning, this survey provides insights into the current state of onboard learning to enable robust, efficient, and secure AI deployment at the edge.
MAMar 1, 2025
Nucleolus Credit Assignment for Effective Coalitions in Multi-agent Reinforcement LearningYugu Li, Zehong Cao, Jianglin Qiao et al.
In cooperative multi-agent reinforcement learning (MARL), agents typically form a single grand coalition based on credit assignment to tackle a composite task, often resulting in suboptimal performance. This paper proposed a nucleolus-based credit assignment grounded in cooperative game theory, enabling the autonomous partitioning of agents into multiple small coalitions that can effectively identify and complete subtasks within a larger composite task. Specifically, our designed nucleolus Q-learning could assign fair credits to each agent, and the nucleolus Q-operator provides theoretical guarantees with interpretability for both learning convergence and the stability of the formed small coalitions. Through experiments on Predator-Prey and StarCraft scenarios across varying difficulty levels, our approach demonstrated the emergence of multiple effective coalitions during MARL training, leading to faster learning and superior performance in terms of win rate and cumulative rewards especially in hard and super-hard environments, compared to four baseline methods. Our nucleolus-based credit assignment showed the promise for complex composite tasks requiring effective subteams of agents.
AINov 16, 2025
Multi-Agent Reinforcement Learning for Heterogeneous Satellite Cluster Resources OptimizationMohamad A. Hady, Siyi Hu, Mahardhika Pratama et al.
This work investigates resource optimization in heterogeneous satellite clusters performing autonomous Earth Observation (EO) missions using Reinforcement Learning (RL). In the proposed setting, two optical satellites and one Synthetic Aperture Radar (SAR) satellite operate cooperatively in low Earth orbit to capture ground targets and manage their limited onboard resources efficiently. Traditional optimization methods struggle to handle the real-time, uncertain, and decentralized nature of EO operations, motivating the use of RL and Multi-Agent Reinforcement Learning (MARL) for adaptive decision-making. This study systematically formulates the optimization problem from single-satellite to multi-satellite scenarios, addressing key challenges including energy and memory constraints, partial observability, and agent heterogeneity arising from diverse payload capabilities. Using a near-realistic simulation environment built on the Basilisk and BSK-RL frameworks, we evaluate the performance and stability of state-of-the-art MARL algorithms such as MAPPO, HAPPO, and HATRPO. Results show that MARL enables effective coordination across heterogeneous satellites, balancing imaging performance and resource utilization while mitigating non-stationarity and inter-agent reward coupling. The findings provide practical insights into scalable, autonomous satellite operations and contribute a foundation for future research on intelligent EO mission planning under heterogeneous and dynamic conditions.
AIJul 14, 2025
Adaptability in Multi-Agent Reinforcement Learning: A Framework and Unified ReviewSiyi Hu, Mohamad A Hady, Jianglin Qiao et al.
Multi-Agent Reinforcement Learning (MARL) has shown clear effectiveness in coordinating multiple agents across simulated benchmarks and constrained scenarios. However, its deployment in real-world multi-agent systems (MAS) remains limited, primarily due to the complex and dynamic nature of such environments. These challenges arise from multiple interacting sources of variability, including fluctuating agent populations, evolving task goals, and inconsistent execution conditions. Together, these factors demand that MARL algorithms remain effective under continuously changing system configurations and operational demands. To better capture and assess this capacity for adjustment, we introduce the concept of \textit{adaptability} as a unified and practically grounded lens through which to evaluate the reliability of MARL algorithms under shifting conditions, broadly referring to any changes in the environment dynamics that may occur during learning or execution. Centred on the notion of adaptability, we propose a structured framework comprising three key dimensions: learning adaptability, policy adaptability, and scenario-driven adaptability. By adopting this adaptability perspective, we aim to support more principled assessments of MARL performance beyond narrowly defined benchmarks. Ultimately, this survey contributes to the development of algorithms that are better suited for deployment in dynamic, real-world multi-agent systems.
AIJun 18, 2025
Multi-Agent Reinforcement Learning for Autonomous Multi-Satellite Earth Observation: A Realistic Case StudyMohamad A. Hady, Siyi Hu, Mahardhika Pratama et al.
The exponential growth of Low Earth Orbit (LEO) satellites has revolutionised Earth Observation (EO) missions, addressing challenges in climate monitoring, disaster management, and more. However, autonomous coordination in multi-satellite systems remains a fundamental challenge. Traditional optimisation approaches struggle to handle the real-time decision-making demands of dynamic EO missions, necessitating the use of Reinforcement Learning (RL) and Multi-Agent Reinforcement Learning (MARL). In this paper, we investigate RL-based autonomous EO mission planning by modelling single-satellite operations and extending to multi-satellite constellations using MARL frameworks. We address key challenges, including energy and data storage limitations, uncertainties in satellite observations, and the complexities of decentralised coordination under partial observability. By leveraging a near-realistic satellite simulation environment, we evaluate the training stability and performance of state-of-the-art MARL algorithms, including PPO, IPPO, MAPPO, and HAPPO. Our results demonstrate that MARL can effectively balance imaging and resource management while addressing non-stationarity and reward interdependency in multi-satellite coordination. The insights gained from this study provide a foundation for autonomous satellite operations, offering practical guidelines for improving policy learning in decentralised EO missions.
LGMay 15, 2021
Neural Trees for Learning on GraphsRajat Talak, Siyi Hu, Lisa Peng et al.
Graph Neural Networks (GNNs) have emerged as a flexible and powerful approach for learning over graphs. Despite this success, existing GNNs are constrained by their local message-passing architecture and are provably limited in their expressive power. In this work, we propose a new GNN architecture -- the Neural Tree. The neural tree architecture does not perform message passing on the input graph, but on a tree-structured graph, called the H-tree, that is constructed from the input graph. Nodes in the H-tree correspond to subgraphs in the input graph, and they are reorganized in a hierarchical manner such that the parent of a node in the H-tree always corresponds to a larger subgraph in the input graph. We show that the neural tree architecture can approximate any smooth probability distribution function over an undirected graph. We also prove that the number of parameters needed to achieve an $ε$-approximation of the distribution function is exponential in the treewidth of the input graph, but linear in its size. We prove that any continuous $\mathcal{G}$-invariant/equivariant function can be approximated by a nonlinear combination of such probability distribution functions over $\mathcal{G}$. We apply the neural tree to semi-supervised node classification in 3D scene graphs, and show that these theoretical properties translate into significant gains in prediction accuracy, over the more traditional GNN architectures. We also show the applicability of the neural tree architecture to citation networks with large treewidth, by using a graph sub-sampling technique.
LGJan 20, 2021
UPDeT: Universal Multi-agent Reinforcement Learning via Policy Decoupling with TransformersSiyi Hu, Fengda Zhu, Xiaojun Chang et al.
Recent advances in multi-agent reinforcement learning have been largely limited in training one model from scratch for every new task. The limitation is due to the restricted model architecture related to fixed input and output dimensions. This hinders the experience accumulation and transfer of the learned agent over tasks with diverse levels of difficulty (e.g. 3 vs 3 or 5 vs 6 multi-agent games). In this paper, we make the first attempt to explore a universal multi-agent reinforcement learning pipeline, designing one single architecture to fit tasks with the requirement of different observation and action configurations. Unlike previous RNN-based models, we utilize a transformer-based model to generate a flexible policy by decoupling the policy distribution from the intertwined input observation with an importance weight measured by the merits of the self-attention mechanism. Compared to a standard transformer block, the proposed model, named as Universal Policy Decoupling Transformer (UPDeT), further relaxes the action restriction and makes the multi-agent task's decision process more explainable. UPDeT is general enough to be plugged into any multi-agent reinforcement learning pipeline and equip them with strong generalization abilities that enables the handling of multiple tasks at a time. Extensive experiments on large-scale SMAC multi-agent competitive games demonstrate that the proposed UPDeT-based multi-agent reinforcement learning achieves significant results relative to state-of-the-art approaches, demonstrating advantageous transfer capability in terms of both performance and training speed (10 times faster).
CVJun 23, 2020
Multi-view Drone-based Geo-localization via Style and Spatial AlignmentSiyi Hu, Xiaojun Chang
In this paper, we focus on the task of multi-view multi-source geo-localization, which serves as an important auxiliary method of GPS positioning by matching drone-view image and satellite-view image with pre-annotated GPS tag. To solve this problem, most existing methods adopt metric loss with an weighted classification block to force the generation of common feature space shared by different view points and view sources. However, these methods fail to pay sufficient attention to spatial information (especially viewpoint variances). To address this drawback, we propose an elegant orientation-based method to align the patterns and introduce a new branch to extract aligned partial feature. Moreover, we provide a style alignment strategy to reduce the variance in image style and enhance the feature unification. To demonstrate the performance of the proposed approach, we conduct extensive experiments on the large-scale benchmark dataset. The experimental results confirm the superiority of the proposed approach compared to state-of-the-art alternatives.
ROOct 27, 2018
Modeling Perceptual Aliasing in SLAM via Discrete-Continuous Graphical ModelsPierre-Yves Lajoie, Siyi Hu, Giovanni Beltrame et al.
Perceptual aliasing is one of the main causes of failure for Simultaneous Localization and Mapping (SLAM) systems operating in the wild. Perceptual aliasing is the phenomenon where different places generate a similar visual (or, in general, perceptual) footprint. This causes spurious measurements to be fed to the SLAM estimator, which typically results in incorrect localization and mapping results. The problem is exacerbated by the fact that those outliers are highly correlated, in the sense that perceptual aliasing creates a large number of mutually-consistent outliers. Another issue stems from the fact that most state-of-the-art techniques rely on a given trajectory guess (e.g., from odometry) to discern between inliers and outliers and this makes the resulting pipeline brittle, since the accumulation of error may result in incorrect choices and recovery from failures is far from trivial. This work provides a unified framework to model perceptual aliasing in SLAM and provides practical algorithms that can cope with outliers without relying on any initial guess. We present two main contributions. The first is a Discrete-Continuous Graphical Model (DC-GM) for SLAM: the continuous portion of the DC-GM captures the standard SLAM problem, while the discrete portion describes the selection of the outliers and models their correlation. The second contribution is a semidefinite relaxation to perform inference in the DC-GM that returns estimates with provable sub-optimality guarantees. Experimental results on standard benchmarking datasets show that the proposed technique compares favorably with state-of-the-art methods while not relying on an initial guess for optimization.
CVOct 27, 2018
Accelerated Inference in Markov Random Fields via Smooth Riemannian OptimizationSiyi Hu, Luca Carlone
Markov Random Fields (MRFs) are a popular model for several pattern recognition and reconstruction problems in robotics and computer vision. Inference in MRFs is intractable in general and related work resorts to approximation algorithms. Among those techniques, semidefinite programming (SDP) relaxations have been shown to provide accurate estimates while scaling poorly with the problem size and being typically slow for practical applications. Our first contribution is to design a dual ascent method to solve standard SDP relaxations that takes advantage of the geometric structure of the problem to speed up computation. This technique, named Dual Ascent Riemannian Staircase (DARS), is able to solve large problem instances in seconds. Our second contribution is to develop a second and faster approach. The backbone of this second approach is a novel SDP relaxation combined with a fast and scalable solver based on smooth Riemannian optimization. We show that this approach, named Fast Unconstrained SEmidefinite Solver (FUSES), can solve large problems in milliseconds. Contrarily to local MRF solvers, e.g., loopy belief propagation, our approaches do not require an initial guess. Moreover, we leverage recent results from optimization theory to provide per-instance sub-optimality guarantees. We demonstrate the proposed approaches in multi-class image segmentation problems. Extensive experimental evidence shows that (i) FUSES and DARS produce near-optimal solutions, attaining an objective within 0.1% of the optimum, (ii) FUSES and DARS are remarkably faster than general-purpose SDP solvers, and FUSES is more than two orders of magnitude faster than DARS while attaining similar solution quality, (iii) FUSES is faster than local search methods while being a global solver.