From Sequential to Parallel: Reformulating Dynamic Programming as GPU Kernels for Large-Scale Stochastic Combinatorial Optimization

Dynamic programming (DP) is central to combinatorial optimization, optimal control, and reinforcement learning, yet its perceived sequentiality has long hindered scalability. We introduce a general-purpose GPU framework that reformulates broad classes of forward DP recursions as batched min--plus matrix--vector products over layered DAGs, collapsing actions into masked state-to-state transitions that map directly to GPU kernels. This approach removes a major bottleneck in scenario-based stochastic programming (SP), where the use of DP has traditionally restricted the number of scenarios due to excessive computational cost. Our framework exposes massive parallelism across scenarios, transition layers, and, when applicable, route or action options, via self-designed GPU kernels that implement Bellman updates with warp-/block-level reductions and numerically safe masking. In a single GPU pass, these kernels can process over $10^6$ uncertainty realizations, far beyond the capacity of prior scenario-based methods. We demonstrate the approach in two canonical SP applications: (i) a vectorized split operator for the capacitated vehicle routing problem with stochastic demand, exploiting 2D parallelism (scenarios $\times$ transitions); and (ii) a forward inventory reinsertion DP under an order-up-to policy, exploiting 3D parallelism (scenarios $\times$ inventory transitions $\times$ route options). Across benchmarks, the implementation scales nearly linearly in the number of scenarios and achieves one to three orders of magnitude speedups over multithreaded CPU baselines, yielding tighter SAA estimates and consistently stronger first-stage decisions under identical wall-clock budgets. Viewed as hardware-aware software primitives, our min--plus DP kernels offer a drop-in path to scalable, GPU-accelerated stochastic discrete optimization.