This paper presents a novel learning-based approach for online estimation of maximal safe sets for local motion planning tasks in mobile robotics. We leverage the idea of hypernetworks to achieve good generalization properties and real-time performance simultaneously. As the source of supervision, we employ the Hamilton-Jacobi (HJ) reachability analysis, allowing us to consider general nonlinear dynamics and arbitrary constraints. We integrate our model into a model predictive control (MPC) local planner as a safety constraint and compare the performance with relevant baselines in realistic 3D simulations for different environments and robot dynamics. The results show the advantages of our approach in terms of a significantly higher success rate: 2 to 18 percent over the best baseline, while achieving real-time performance.

Participatory budgeting (PB) is a voting paradigm for distributing a divisible resource, usually called a budget, among a set of projects by aggregating the preferences of individuals over these projects. It is implemented quite extensively for purposes such as government allocating funds to public projects and funding agencies selecting research proposals to support. This PhD dissertation studies the welfare-related and fairness-related objectives for different PB models. Our contribution lies in proposing and exploring novel PB rules that maximize welfare and promote fairness, as well as, in introducing and investigating a range of novel utility notions, axiomatic properties, and fairness notions, effectively filling the gaps in the existing literature for each PB model. The thesis is divided into two main parts, the first focusing on dichotomous and the second focusing on ordinal preferences. Each part considers two cases: (i) the cost of each project is restricted to a single value and partial funding is not permitted and (ii) the cost of each project is flexible and may assume multiple values.

Keyphrase ranking plays a crucial role in information retrieval and summarization by indexing and retrieving relevant information efficiently. Advances in natural language processing, especially large language models (LLMs), have improved keyphrase extraction and ranking. However, traditional methods often overlook diversity, resulting in redundant keyphrases. We propose a novel approach using Submodular Function Optimization (SFO) to balance relevance and diversity in keyphrase ranking. By framing the task as submodular maximization, our method selects diverse and representative keyphrases. Experiments on benchmark datasets show that our approach outperforms existing methods in both relevance and diversity metrics, achieving SOTA performance in execution time. Our code is available online.

In this work, we present FRTree planner, a novel robot navigation framework that leverages a tree structure of free regions, specifically designed for navigation in cluttered and unknown environments with narrow passages. The framework continuously incorporates real-time perceptive information to identify distinct navigation options and dynamically expands the tree toward explorable and traversable directions. This dynamically constructed tree incrementally encodes the geometric and topological information of the collision-free space, enabling efficient selection of the intermediate goals, navigating around dead-end situations, and avoidance of dynamic obstacles without a prior map. Crucially, our method performs a comprehensive analysis of the geometric relationship between free regions and the robot during online replanning. In particular, the planner assesses the accessibility of candidate passages based on the robot's geometries, facilitating the effective selection of the most viable intermediate goals through accessible narrow passages while minimizing unnecessary detours. By combining the free region information with a bi-level trajectory optimization tailored for robots with specific geometries, our approach generates robust and adaptable obstacle avoidance strategies in confined spaces. Through extensive simulations and real-world experiments, FRTree demonstrates its superiority over benchmark methods in generating safe, efficient motion plans through highly cluttered and unknown terrains with narrow gaps.

This work introduces an unconventional inexact augmented Lagrangian method, where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems, that involve nonlinear equality constraints over a convex set under a mild regularity condition. First, we conduct a full complexity analysis of the method, leveraging an accelerated first-order algorithm for solving the H\"older-smooth subproblems. Next, we present an inexact proximal point method to tackle these subproblems, demonstrating that it achieves an improved convergence rate. Notably, this rate reduces to the best-known convergence rate for first-order methods when the augmenting term is a squared Euclidean norm. Our worst-case complexity results further show that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease in the dual residual. Numerical experiments support our theoretical findings, illustrating that this trade-off between constraint satisfaction and cost minimization is advantageous for certain practical problems.

Continuum manipulators have gained significant attention as a promising alternative to rigid manipulators, offering notable advantages in terms of flexibility and adaptability within intricate workspace. However, the broader application of high degree-of-freedom (DoF) continuum manipulators in intricate environments with multiple obstacles necessitates the development of an efficient inverse kinematics (IK) solver specifically tailored for such scenarios. Existing IK methods face challenges in terms of computational cost and solution guarantees for high DoF continuum manipulators, particularly within intricate workspace that obstacle avoidance is needed. To address these challenges, we have developed a novel IK solver for continuum manipulators that incorporates obstacle avoidance and other constraints like length, orientation, etc., in intricate environments, drawing inspiration from optimization-based path planning methods. Through simulations, our proposed method showcases superior flexibility, efficiency with increasing DoF, and robust performance within highly unstructured workspace, achieved with acceptable latency.

Updates of extensive Internet of Things (IoT) data are critical to the immersion of vehicular metaverse services. However, providing high-quality and sustainable data in unstable and resource-constrained vehicular networks remains a significant challenge. To address this problem, we put forth a novel immersion-aware model trading framework that incentivizes metaverse users (MUs) to contribute learning models trained by their latest local data for augmented reality (AR) services in the vehicular metaverse, while preserving their privacy through federated learning. To comprehensively evaluate the contribution of locally trained learning models provided by MUs to AR services, we design a new immersion metric that captures service immersion by considering the freshness and accuracy of learning models, as well as the amount and potential value of raw data used for training. We model the trading interactions between metaverse service providers (MSPs) and MUs as an equilibrium problem with equilibrium constraints (EPEC) to analyze and balance their costs and gains. Moreover, considering dynamic network conditions and privacy concerns, we formulate the reward decisions of MSPs as a multi-agent Markov decision process. Then, a fully distributed dynamic reward method based on deep reinforcement learning is presented, which operates without any private information about MUs and other MSPs. Experimental results demonstrate that the proposed framework can effectively provide higher-value models for object detection and classification in AR services on real AR-related vehicle datasets compared to benchmark schemes.

In aerospace and mechanical engineering, the design process for new products relies on hierarchies of mathematical models, the physical complexity of which may be imposed by computational costs or dictated by regulations. These models typically incorporate parameters to account for various operating conditions and configurations. In the framework of optimization, for example, hundreds of parameters (design variables) may be required to define the configuration of a system. Similarly, uncertainty propagation may necessitate defining a complex parameter space to account for variations in geometrical imperfections, material properties, or flow conditions. The design process, especially during optimization and uncertainty quantification, often involves numerous evaluations of the system's mathematical models across a wide range of points in the parameter space. Computational costs vary with the level of model fidelity: lower fidelity models are traditionally used for computationally intensive evaluations, while higher fidelity models - often involving nonlinear partial differential equations discretized on fine grids - are typically reserved for later stages of the design process.

We study stochastic second-order methods for solving general non-convex optimization problems. We propose using a special version of momentum to stabilize the stochastic gradient and Hessian estimates in Newton's method. We show that momentum provably improves the variance of stochastic estimates and allows the method to converge for any noise level. Using the cubic regularization technique, we prove a global convergence rate for our method on general non-convex problems to a second-order stationary point, even when using only a single stochastic data sample per iteration. This starkly contrasts with all existing stochastic second-order methods for non-convex problems, which typically require large batches. Therefore, we are the first to demonstrate global convergence for batches of arbitrary size in the non-convex case for the Stochastic Cubic Newton. Additionally, we show improved speed on convex stochastic problems for our regularized Newton methods with momentum.

In this work, we study the problem of finding Pareto optimal policies in multi-agent reinforcement learning problems with cooperative reward structures. We show that any algorithm where each agent only optimizes their reward is subject to suboptimal convergence. Therefore, to achieve Pareto optimality, agents have to act altruistically by considering the rewards of others. This observation bridges the multi-objective optimization framework and multi-agent reinforcement learning together. We first propose a framework for applying the Multiple Gradient Descent algorithm (MGDA) for learning in multi-agent settings. We further show that standard MGDA is subjected to weak Pareto convergence, a problem that is often overlooked in other learning settings but is prevalent in multi-agent reinforcement learning. To mitigate this issue, we propose MGDA++, an improvement of the existing algorithm to handle the weakly optimal convergence of MGDA properly. Theoretically, we prove that MGDA++ converges to strong Pareto optimal solutions in convex, smooth bi-objective problems. We further demonstrate the superiority of our MGDA++ in cooperative settings in the Gridworld benchmark. The results highlight that our proposed method can converge efficiently and outperform the other methods in terms of the optimality of the convergent policies. The source code is available at \url{https://github.com/giangbang/Strong-Pareto-MARL}.