Energy Minimization via a Primal-Dual Algorithm for a Convex Program

We present an optimal primal-dual algorithm for the energy minimization preemptive open-shop problem in the speed-scaling setting. Our algorithm uses the approach of Devanur et al. [JACM 2008], by applying the primal-dual method in the setting of convex programs and KKT conditions. We prove that our algorithm converges and that it returns an optimal solution, but we were unable to prove that it converges in polynomial time. For this reason, we conducted a series of experiments showing that the number of iterations of our algorithm increases linearly with the number of jobs, n, when n is greater than the number of machines, m. We also compared the speed of our method with respect to the time spent by a commercial solver to directly solve the corresponding convex program. The computational results give evidence that for n > m, our algorithm is clearly faster. However, for the special family of instances where n = m, our method is slower.