Local Linear Time Convergence of Primal-Dual Energy Minimization Algorithm for Parallel Processing

We consider energy minimization by speed-scaling of an open shop multiprocessor with n jobs and m machines. The paper studies the complexity of a primal-dual solution algorithm of [4], which was an open question in that paper. We prove that in a neighbourhood of the solution the complexity of the algorithm is O(mnlog 1/ε) if n ≠ m and e is the round off error of the computer. The paper demonstrates how linearization can be used to investigate the complexity of an algorithm close to the optimum. An estimate of the size of the neighbourhood where the linearization error is smaller than the computer's round off error is also given.