Online Covering with Convex Objectives and Applications

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 lp norms on machine loads (including the surprisingly non-trivial l1 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 (l1 norm), we give a different rounding algorithm that obtains a better competitive ratio than the generic rounding algorithm for lp norms. We show that this competitive ratio is asymptotically tight. ∗Email: [email protected]. Supported in part by the Israel Science Foundation (grant No. 1404/10) and by the Israeli Centers of Research Excellence (I-CORE) program, (Center No.4/11). Part of this work was done while the author was visiting Microsoft Research, Redmond. †Email: [email protected]. Supported in part by the Israeli Centers of Research Excellence (I-CORE) program. ‡Email: [email protected]. Supported in part by a Duke University startup grant and a Google Faculty Research Award. Part of this work was done while the author was visiting Microsoft Research, Redmond.