  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.
