Online primal-dual for non-linear optimization with applications to speed scaling

We give a principled method to design online algorithms (for potentially non-linear problems) using a mathematical programming formulation of the problem, and also to analyze the competitiveness of the resulting algorithm using the dual program. This method can be viewed as an extension of the online primal-dual method for linear programming problems, to nonlinear programs. We show the application of this method to two online speed-scaling problems: one involving scheduling jobs on a speed scalable processor so as to minimize energy plus an arbitrary sum scheduling objective, and one involving routing virtual circuit connection requests in a network of speed scalable routers so as to minimize the aggregate power or energy used by the routers. This analysis shows that competitive algorithms exist for problems that had resisted analysis using the dominant potential function approach in the speed-scaling literature, and provides alternate cleaner analysis for other known results. This gives us another tool in the design and analysis of primal-dual algorithms for online problems.