Function Design for Improved Competitive Ratio in Online Resource Allocation with Procurement Costs

TL;DR

This paper develops new primal-dual algorithms with optimized surrogate functions to improve the competitive ratio in online resource allocation problems involving procurement costs, applicable to polynomial and general cost functions.

Contribution

It introduces a novel optimization framework for designing surrogate functions that enhance the competitive ratio of primal-dual algorithms in online resource allocation with procurement costs.

Findings

Refined competitive ratio bounds for polynomial procurement costs.

Optimal parameters for a broad class of cost functions via quasiconvex optimization.

Extension to posted pricing mechanisms without customer preference disclosure.

Abstract

We study the problem of online resource allocation, where multiple customers arrive sequentially and the seller must irrevocably allocate resources to each incoming customer while also facing a procurement cost for the total allocation. Assuming resource procurement follows an a priori known marginally increasing cost function, the objective is to maximize the reward obtained from fulfilling the customers' requests sans the cumulative procurement cost. We analyze the competitive ratio of a primal-dual algorithm in this setting, and develop an optimization framework for synthesizing a surrogate function for the procurement cost function to be used by the algorithm, in order to improve the competitive ratio of the primal-dual algorithm. Our first design method focuses on polynomial procurement cost functions and uses the optimal surrogate function to provide a more refined bound than the state of the art. Our second design method uses quasiconvex optimization to find optimal design parameters for a general class of procurement cost functions. Numerical examples are used to illustrate the design techniques. We conclude by extending the analysis to devise a posted pricing mechanism in which the algorithm does not require the customers' preferences to be revealed.