We give an algorithmic framework for minimizing general convex objectives (that are differentiable and monotone non-decreasing) over a set of covering constraints that arrive online. This substantially extends previous work on online covering for linear objectives (Alon et al., STOC 2003) and online covering with offline packing constraints (Azar et al., SODA 2013). To the best of our knowledge, this is the first result in online optimization for generic non-linear objectives; special cases of such objectives have previously been considered, particularly for energy minimization. As a specific problem in this genre, we consider the unrelated machine scheduling problem with startup costs and arbitrary ℓ p norms on machine loads (including the surprisingly non-trivial ℓ 1 norm representing total machine load). This problem was studied earlier for the makespan norm in both the offline (Khuller et al., SODA 2010; Li and Khuller, SODA 2011) and online settings (Azar et al., SODA 2013). We adapt the two-phase approach of obtaining a fractional solution and then rounding it online (used successfully to many linear objectives) to the non-linear objective. The fractional algorithm uses ideas from our general framework that we described above (but does not fit the framework exactly because of non-positive entries in the constraint matrix). The rounding algorithm uses ideas from offline rounding of LPs with non-linear objectives (Azar and Epstein, STOC 2005; Kumar et al., FOCS 2005). Our competitive ratio is tight up to a logarithmic factor. Finally, for the important special case of total load (ℓ 1 norm), we give a different rounding algorithm that obtains a better competitive ratio than the generic rounding algorithm for ℓ p norms. We show that this competitive ratio is asymptotically tight.
Due to a lower bound result of Azar et al. REF , our competitive ratio is tight.
Online Covering with Convex Objectives and Applications
ArXiv
ArXiv
Computer Science Mathematics