We address the issue of safety in reinforcement learning. We pose the problem in an episodic framework of a constrained Markov decision process. Existing results have shown that it is possible to achieve a reward regret of \tilde{\mathcal{O}}(\sqrt{K}) while allowing an \tilde{\mathcal{O}}(\sqrt{K}) constraint violation in K episodes. A critical question that arises is whether it is possible to keep the constraint violation even smaller. We show that when a strictly safe policy is known, then one can confine the system to zero constraint violation with arbitrarily high probability while keeping the reward regret of order \tilde{\mathcal{O}}(\sqrt{K}) .

While deep learning techniques have become extremely popular for solving a broad range of optimization problems, methods to enforce hard constraints during optimization, particularly on deep neural networks, remain underdeveloped. Inspired by the rich literature on meshless interpolation and its extension to spectral collocation methods in scientific computing, we develop a series of approaches for enforcing hard constraints on neural fields, which we refer to as Constrained Neural Fields (CNF). The constraints can be specified as a linear operator applied to the neural field and its derivatives. We also design specific model representations and training strategies for problems where standard models may encounter difficulties, such as conditioning of the system, memory consumption, and capacity of the network when being constrained. Our approaches are demonstrated in a wide range of real-world applications.

We present GeoManip, a framework to enable generalist robots to leverage essential conditions derived from object and part relationships, as geometric constraints, for robot manipulation. For example, cutting the carrot requires adhering to a geometric constraint: the blade of the knife should be perpendicular to the carrot's direction. By interpreting these constraints through symbolic language representations and translating them into low-level actions, GeoManip bridges the gap between natural language and robotic execution, enabling greater generalizability across diverse even unseen tasks, objects, and scenarios. Unlike vision-language-action models that require extensive training, operates training-free by utilizing large foundational models: a constraint generation module that predicts stage-specific geometric constraints and a geometry parser that identifies object parts involved in these constraints. A solver then optimizes trajectories to satisfy inferred constraints from task descriptions and the scene. Furthermore, GeoManip learns in-context and provides five appealing human-robot interaction features: on-the-fly policy adaptation, learning from human demonstrations, learning from failure cases, long-horizon action planning, and efficient data collection for imitation learning. Extensive evaluations on both simulations and real-world scenarios demonstrate GeoManip's state-of-the-art performance, with superior out-of-distribution generalization while avoiding costly model training.

We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first introduce the "extra fields'' from the mixed finite element method to reformulate the PDEs so as to equivalently transform the three types of BCs into linear forms. Based on the reformulation, we derive the general solutions of the BCs analytically, which are employed to construct an ansatz that automatically satisfies the BCs. With such a framework, we can train the neural networks without adding extra loss terms and thus efficiently handle geometrically complex PDEs, alleviating the unbalanced competition between the loss terms corresponding to the BCs and PDEs. We theoretically demonstrate that the "extra fields'' can stabilize the training process. Experimental results on real-world geometrically complex PDEs showcase the effectiveness of our method compared with state-of-the-art baselines.

Neuro-symbolic systems combine the abilities of neural perception and logical reasoning. However, end-to-end learning of neuro-symbolic systems is still an unsolved challenge. This paper proposes a natural framework that fuses neural network training, symbol grounding, and logical constraint synthesis into a coherent and efficient end-to-end learning process. The capability of this framework comes from the improved interactions between the neural and the symbolic parts of the system in both the training and inference stages. Technically, to bridge the gap between the continuous neural network and the discrete logical constraint, we introduce a difference-of-convex programming technique to relax the logical constraints while maintaining their precision.

Navigating safely in dynamic and uncertain environments is challenging due to uncertainties in perception and motion. This letter presents C2U-MPPI, a robust sampling-based Model Predictive Control (MPC) framework that addresses these challenges by leveraging the Unscented Model Predictive Path Integral (U-MPPI) control strategy with integrated probabilistic chance constraints, ensuring more reliable and efficient navigation under uncertainty. Unlike gradient-based MPC methods, our approach (i) avoids linearization of system dynamics and directly applies non-convex and nonlinear chance constraints, enabling more accurate and flexible optimization, and (ii) enhances computational efficiency by reformulating probabilistic constraints into a deterministic form and employing a layered dynamic obstacle representation, enabling real-time handling of multiple obstacles. Extensive experiments in simulated and real-world human-shared environments validate the effectiveness of our algorithm against baseline methods, showcasing its capability to generate feasible trajectories and control inputs that adhere to system dynamics and constraints in dynamic settings, enabled by unscented-based sampling strategy and risk-sensitive trajectory evaluation. A supplementary video is available at: https://youtu.be/FptAhvJlQm8

The Implicit Hitting Set (HS) approach has shown to be very effective for MaxSAT, Pseudo-boolean optimization and other boolean frameworks. Very recently, it has also shown its potential in the very similar Weighted CSP framework by means of the so-called cost-function merging. The original formulation of the HS approach focuses on obtaining increasingly better lower bounds (HS-lb). However, and as shown for Pseudo-Boolean Optimization, this approach can also be adapted to compute increasingly better upper bounds (HS-ub). In this paper we consider both HS approaches and show how they can be easily combined in a multithread architecture where cores discovered by either component are available by the other which, interestingly, generates synergy between them. We show that the resulting algorithm (HS-lub) is consistently superior to either HS-lb and HS-ub in isolation. Most importantly, HS-lub has an effective anytime behaviour with which the optimality gap is reduced during the execution. We tested our approach on the Weighted CSP framework and show on three different benchmarks that our very simple implementation sometimes outperforms the parallel hybrid best-first search implementation of the far more developed state-of-the-art Toulbar2.

We consider online convex optimization with time-varying constraints and conduct performance analysis using two stringent metrics: dynamic regret with respect to the online solution benchmark, and hard constraint violation that does not allow any compensated violation over time. We propose an efficient algorithm called Constrained Online Learning with Doubly-bounded Queue (COLDQ), which introduces a novel virtual queue that is both lower and upper bounded, allowing tight control of the constraint violation without the need for the Slater condition. We prove via a new Lyapunov drift analysis that COLDQ achieves $O(T^\frac{1+V_x}{2})$ dynamic regret and $O(T^{V_g})$ hard constraint violation, where $V_x$ and $V_g$ capture the dynamics of the loss and constraint functions. For the first time, the two bounds smoothly approach to the best-known $O(T^\frac{1}{2})$ regret and $O(1)$ violation, as the dynamics of the losses and constraints diminish. For strongly convex loss functions, COLDQ matches the best-known $O(\log{T})$ static regret while maintaining the $O(T^{V_g})$ hard constraint violation. We further introduce an expert-tracking variation of COLDQ, which achieves the same performance bounds without any prior knowledge of the system dynamics. Simulation results demonstrate that COLDQ outperforms the state-of-the-art approaches.

Modern SMT solvers have revolutionized the approach to constraint satisfaction problems by integrating advanced theory reasoning and encoding techniques. In this work, we evaluate the performance of modern SMT solvers in Z3, CVC5 and DPLL(T) against a standard SAT solver in DPLL. By benchmarking these solvers on novel, diverse 25x25 Sudoku puzzles of various difficulty levels created by our improved Sudoku generator, we examine the impact of advanced theory reasoning and encoding techniques. Our findings demonstrate that modern SMT solvers significantly outperform classical SAT solvers. This work highlights the evolution of logical solvers and exemplifies the utility of SMT solvers in addressing large-scale constraint satisfaction problems.

This paper proposes a dual divide-and-optimize algorithm (DualOpt) for solving the large-scale traveling salesman problem (TSP). DualOpt combines two complementary strategies to improve both solution quality and computational efficiency. The first strategy is a grid-based divide-and-conquer procedure that partitions the TSP into smaller sub-problems, solving them in parallel and iteratively refining the solution by merging nodes and partial routes. The process continues until only one grid remains, yielding a high-quality initial solution. The second strategy involves a path-based divide-and-optimize procedure that further optimizes the solution by dividing it into sub-paths, optimizing each using a neural solver, and merging them back to progressively improve the overall solution. Extensive experiments conducted on two groups of TSP benchmark instances, including randomly generated instances with up to 100,000 nodes and real-world datasets from TSPLIB, demonstrate the effectiveness of DualOpt. The proposed DualOpt achieves highly competitive results compared to 10 state-of-the-art algorithms in the literature. In particular, DualOpt achieves an improvement gap up to 1.40% for the largest instance TSP100K with a remarkable 104x speed-up over the leading heuristic solver LKH3. Additionally, DualOpt demonstrates strong generalization on TSPLIB benchmarks, confirming its capability to tackle diverse real-world TSP applications.