A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs

We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time t is a location xt on the real line. At each integer time t, a convex function ft(x) arrives online. In response, the online algorithm picks a new location xt. The cost paid by the online algorithm for this response is the distance moved, namely |xt − xt−1|, plus the value of the function at the final destination, namely ft(xt). The objective is then to minimize the aggregate cost over all time, namely ∑ t (|xt − xt−1|+ ft(xt)). The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems: Sequencing and Scheduling