  We reinterpret some online greedy algorithms for a class of nonlinear
"load-balancing" problems as solving a mathematical program online. For
example, we consider the problem of assigning jobs to (unrelated) machines to
minimize the sum of the alpha^{th}-powers of the loads plus assignment costs
(the online Generalized Assignment Problem); or choosing paths to connect
terminal pairs to minimize the alpha^{th}-powers of the edge loads (online
routing with speed-scalable routers). We give analyses of these online
algorithms using the dual of the primal program as a lower bound for the
optimal algorithm, much in the spirit of online primal-dual results for linear
problems.
  We then observe that a wide class of uni-processor speed scaling problems
(with essentially arbitrary scheduling objectives) can be viewed as such load
balancing problems with linear assignment costs. This connection gives new
algorithms for problems that had resisted solutions using the dominant
potential function approaches used in the speed scaling literature, as well as
alternate, cleaner proofs for other known results.
