A Dynamic Near-Optimal Algorithm for Online Linear Programming

Abstract

A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) where the constraint matrix is revealed column by column along with the objective function. We provide a near-optimal algorithm for this surprisingly general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from revealed columns in the previous period are used to determine the sequential decisions in the current period. Our algorithm has a feature of "learning by doing", and the prices are updated at a carefully chosen pace that is neither too fast nor too slow. In particular, our algorithm doesn't assume any distribution information on the input itself, thus is robust to data uncertainty and variations due to its dynamic learning capability. Applications of our algorithm include many online multi-resource allocation and multi-product revenue management problems such as online routing and packing, online combinatorial auctions, adwords matching, inventory control and yield management.

Yield Optimization of Display Advertising with Ad Exchange

“…Roughly speaking, in [6] (and similarly in other work [7,25,1], it is shown that as long as the sample is of size ≈ n, the overall assignment will be ≈ close to the optimal offline solution.…”

“…Our work is closely related to online ad allocation problems, including the Display Ads Allocation (DA) problem [8,7,1,25], and the AdWords (AW) problem [16,6]. In both of these problems, the publisher must assign online impressions to an inventory of ads, optimizing efficiency or revenue of the allocation while re- [6].…”

“…The stochastic model of the DA problem is more related to our problem. Following a training-based dual algorithm by Devanur and Hayes [6], training-based (1 − )-competitive algorithms have been developed for the DA problem and its generalization to various packing linear programs [7,25,1]. These papers develop a (1 − )-competitive algorithm for online stochastic packing problems in which OPT wij ≥ O( m log n ε 3 ) (or OPT wij ≥ O( m log n ε 2 ) applying the technique of [1]) and the demand of each advertiser is large, in the randomorder and the i.i.d model.…”

“…In fact one can make such a reduction go through by considering online decisions on pairs of (reserve price, advertiser)s, and formalize the problem as an online allocation problem with general packing constraints. After a couple more steps, one can apply the techniques of [6,7,25,1] to derive an online algorithm for the combined problem. However this approach discretizes the price space into multiples of δ; thus (a) we lose 1+δ in the yield, (b) we increase the running time by a factor of 1 δ , and (c) for the the competitiveness proof to hold, the OPT be larger than O( 1 δ m log n ε 3 ).…”

“…Proposition 8 Suppose that the user type-advertiser graph admits a feasible flow, and that qualities and bids from AdX are bounded by 1 A min a τ a . Then, the stochastic control policy does not assign any impressions outside of the targeting criteria, except perhaps for the left-over regime.…”