  We consider Lagrangian duality based approaches to design and analyze
algorithms for online energy-efficient scheduling. First, we present a
primal-dual framework. Our approach makes use of the Lagrangian weak duality
and convexity to derive dual programs for problems which could be formulated as
convex assignment problems. The duals have intuitive structures as the ones in
linear programming. The constraints of the duals explicitly indicate the online
decisions and naturally lead to competitive algorithms. Second, we use a
dual-fitting approach, which also based on the weak duality, to study problems
which are unlikely to admit convex relaxations. Through the analysis, we show
an interesting feature in which primal-dual gives idea for designing algorithms
while the analysis is done by dual-fitting.
  We illustrate the advantages and the flexibility of the approaches through
problems in different setting: from single machine to unrelated machine
environments, from typical competitive analysis to the one with resource
augmentation, from convex relaxations to non-convex relaxations.
