Fast Algorithms for Packing Proportional Fairness and its Dual

TL;DR

This paper introduces faster distributed algorithms for the packing proportional fairness problem and its dual, with applications to linear programming, improving efficiency and width-independence.

Contribution

It presents a novel distributed accelerated first-order method for packing proportional fairness and an algorithm for its dual, both width-independent, with applications to LP volume reduction.

Findings

Improved convergence rates for the primal and dual algorithms.

Width-independence of the proposed algorithms.

Enhanced volume reduction in linear programming applications.

Abstract

The proportional fair resource allocation problem is a major problem studied in flow control of networks, operations research, and economic theory, where it has found numerous applications. This problem, defined as the constrained maximization of $\sum_i \log x_i$, is known as the packing proportional fairness problem when the feasible set is defined by positive linear constraints and $x \in \mathbb{R}^{n}_{\geq 0}$. In this work, we present a distributed accelerated first-order method for this problem which improves upon previous approaches. We also design an algorithm for the optimization of its dual problem. Both algorithms are width-independent. Finally, we show the latter problem has applications to the volume reduction of bounding simplices in an old linear programming algorithm of (YL82), and we obtain some improvements as a result.