A Tight Lower Bound for Online Convex Optimization with Switching Costs

Abstract

We investigate online convex optimization with switching costs (OCO; Lin et al., INFOCOM 2011), a natural online problem arising when rightsizing data centers: A server initially located at $p_0$ on the real line is presented with an online sequence of non-negative convex functions $f_1,f_2,\dots ,f_n: \mathbb {R}\rightarrow \mathbb {R}_+$. In response to each function $f_i$, the server moves to a new position $p_i$ on the real line, resulting in cost $|p_i-p_{i-1}|+f_i(p_i)$. The total cost is the sum of costs of all steps. One is interested in designing competitive algorithms.

In this paper, we solve the problem in the classical sense: We give a lower bound of 2 on the competitive ratio of any possibly randomized online algorithm, matching the competitive ratio of previously known deterministic online algorithms (Andrew et al., COLT 2013/arXiv 2015; Bansal et al., APPROX 2015). It has been previously conjectured that $(2-\epsilon )$-competitive algorithms exist for some $\epsilon >0$ (Bansal et al., APPROX 2015).

A. Antoniadis—Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under AN 1262/1-1.

K. Schewior—Supported by CONICYT grant PCI PII 20150140 and the Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003.