Online Covering with Convex Objectives and Applications
Yossi Azar, Ilan Reuven Cohen, Debmalya Panigrahi

TL;DR
This paper introduces a novel online algorithmic framework for minimizing general convex objectives under covering constraints, extending previous linear-focused work to non-linear objectives, with applications in machine scheduling.
Contribution
It is the first to address online covering problems with generic non-linear convex objectives, providing new algorithms and competitive ratio bounds, especially for machine scheduling with load norms.
Findings
First online framework for convex objectives in covering problems
Achieves near-optimal competitive ratios for non-linear objectives
Improves rounding techniques for total load ($$ norm) in online settings
Abstract
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 {\em et al.}, STOC 2003) and online covering with offline packing constraints (Azar {\em 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 norms on machine loads (including the surprisingly non-trivial norm representing total machine load). This problem was studied earlier for the makespan norm…
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