Faster first-order primal-dual methods for linear programming using restarts and sharpness

First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they often slow at finding high accuracy solutions, which creates a barrier to use in traditional linear programming (LP) applications. This paper exploits the sharpness of formulations LP instances achieve convergence using restarts general setting that applies alternating direction method multipliers (ADMM), hybrid gradient (PDHG) extragradient (EGM). In special case PDHG, without we show an iteration count lower bound $$\Omega (\kappa ^2 \log (1/\epsilon ))$$ , while with upper $$O(\kappa where $$\kappa $$ is condition number $$\epsilon desired accuracy. Moreover, optimal wide class methods, strictly more sharp problems. We develop adaptive restart scheme verify significantly improve ability EGM, ADMM find solutions