Linear program scientists have found that changing the way a problem is written can make a big difference for a class of solvers called primal–dual hybrid gradient (PDHG) methods. Unlike interior‑point solvers, which usually see little benefit from formulation changes, PDHG solvers – such as the open‑source PDLP – react strongly to how constraints are presented. The new work shows that a general recipe can absorb many constraints into the solver’s projection step. This reduces the number of dual variables and changes four key constants that control the algorithm’s restart, termination, distance, and sharpness.

When the authors applied this recipe to two widely studied problems – the quadratic unconstrained binary optimisation (QUBO) relaxation and the joint‑replenishment problem (JRP) – the reformulated solvers ran between 6.4 and 9.6 times faster on QUBO instances and between 6.1 and 28.9 times faster on JRP instances compared with a GPU LP solver that solves the original formulations. The reformulated solvers also required far fewer iterations: 1.7–4.3 times fewer on QUBO and 13.6–52.3 times fewer on JRP.

The QUBO speedups are illustrated in Table 4, which compares solve times for each instance size and shows the dramatic 6.1–9.6× acceleration that the reformulated solver gains over the GPU solver. The JRP speedups appear in Table 10, where the reformulated solver outperforms the GPU solver by factors of 6.1–28.9, demonstrating that absorbing constraints can also reduce the number of instances that time out.