An Overview of GPU-based First-Order Methods for Linear Programming and Extensions

The rapid progress in GPU computing has revolutionized many fields, yet its potential in mathematical programming, such as linear programming (LP), has only recently begun to be realized. This survey aims to provide a comprehensive overview of recent advancements in GPU-based first-order methods for LP, with a particular focus on the design and development of cuPDLP. We begin by presenting the design principles and algorithmic foundation of the primal-dual hybrid gradient (PDHG) method, which forms the core of the solver. Practical enhancements, such as adaptive restarts, preconditioning, Halpern-type acceleration and infeasibility detection, are discussed in detail, along with empirical comparisons against industrial-grade solvers, highlighting the scalability and efficiency of cuPDLP. We also provide a unified theoretical framework for understanding PDHG, covering both classical and recent results on sublinear and linear convergence under sharpness conditions. Finally, we extend the discussion to GPU-based optimization beyond LP, including quadratic, semidefinite, conic, and nonlinear programming.