We present PDLP, a practical first-order method for linear programming (LP) that can solve to the high levels of accuracy that are expected in traditional LP applications.

These forachievingT requires onthepriorB> 1. 2.1 Generalizing Choosing Follow [Kalaiand worst-case FPLalgorithm Follow (FPML).

As illustrated by the success of integer linear programming, linear integer arithmetics is a powerful tool for modelling combinatorial problems. Furthermore, the probabilistic extension of linear programming has been used to formulate problems in neurosymbolic AI. However, two key problems persist that prevent the adoption of neurosymbolic techniques beyond toy problems. First, probabilistic inference is inherently hard, #P-hard to be precise. Second, the discrete nature of integers renders the construction of meaningful gradients challenging, which is problematic for learning. In order to mitigate these issues, we formulate linear arithmetics over integer-valued random variables as tensor manipulations that can be implemented in a straightforward fashion using modern deep learning libraries. At the core of our formulation lies the observation that the addition of two integer-valued random variables can be performed by adapting the fast Fourier transform to probabilities in the log-domain. By relying on tensor operations we obtain a differentiable data structure, which unlocks, virtually for free, gradient-based learning. In our experimental validation we show that tensorising probabilistic integer linear arithmetics and leveraging the fast Fourier transform allows us to push the state of the art by several orders of magnitude in terms of inference and learning times.

However, such a rate is related to the problem dimension and the algorithm exhibits a slow and fluctuating ``tail convergence'' in practice. In this paper, we propose a new variable splitting method of LP and prove that our method has a convergence rate of $O(\|\mathbf{A}\|^2\log(1/\epsilon))$. The proof is based on simultaneously estimating the distance from a pair of primal dual iterates to the optimal primal and dual solution set by certain residuals. In practice, we result in a new first-order LP solver that can exploit both the sparsity and the specific structure of matrix $\mathbf{A}$ and a significant speedup for important problems such as basis pursuit, inverse covariance matrix estimation, L1 SVM and nonnegative matrix factorization problem compared with current fastest LP solvers.

Many machine learning applications require outputs that satisfy complex, dynamic constraints. This task is particularly challenging in Graph Neural Network models due to the variable output sizes of graph-structured data. In this paper, we introduce ProjNet, a Graph Neural Network framework which satisfies input-dependant constraints. ProjNet combines a sparse vector clipping method with the Component-Averaged Dykstra (CAD) algorithm, an iterative scheme for solving the best-approximation problem. We establish a convergence result for CAD and develop a GPU-accelerated implementation capable of handling large-scale inputs efficiently. To enable end-to-end training, we introduce a surrogate gradient for CAD that is both computationally efficient and better suited for optimization than the exact gradient. We validate ProjNet on four classes of constrained optimisation problems: linear programming, two classes of non-convex quadratic programs, and radio transmit power optimization, demonstrating its effectiveness across diverse problem settings.

Solving optimization problems is the key to decision making in many real-life analytics applications. However, the coefficients of the optimization problems are often uncertain and dependent on external factors, such as future demand or energy or stock prices. Machine learning (ML) models, especially neural networks, are increasingly being used to estimate these coefficients in a data-driven way.

Over the past decades, Linear Programming (LP) has been widely used in different areas and considered as one of the mature technologies in numerical optimization. However, the complexity offered by state-of-the-art algorithms (i.e.