SIAM Journal on Optimization, Volume 30, Issue 4, Page 3284-3314, January 2020.
Fair resource allocation is a fundamental optimization problem with applications in operations research, networking, and economic and game theory. Research in these areas has led to the general acceptance of a class of $\alpha$-fair utility functions parameterized by $\alpha \in [0, \infty]$. We consider $\alpha$-fair packing---the problem of maximizing $\alpha$-fair utilities under positive linear constraints---and provide a simple first-order method for solving it with relative-error guarantees. The method has a significantly lower convergence time than the state of the art, and to analyze it, we leverage the approximate duality gap technique, which provides an intuitive interpretation of the convergence argument. Finally, we introduce a natural counterpart of $\alpha$-fairness for minimization problems and motivate its usage in the context of fair task allocation. This generalization yields $\alpha$-fair covering problems, for which we provide the first near-linear-time solvers with relative-error guarantees.

中文翻译：

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相对包装的合理包装和覆盖

SIAM优化杂志，第30卷，第4期，第3284-3314页，2020年1月。
公平的资源分配是一个基本的优化问题，在运筹学，网络，经济和博弈论中都有应用。在这些领域的研究已导致普遍接受由$ \ alpha \ in [0，\ infty] $参数化的$ \ alpha $ -fair实用函数类。我们考虑$α-公平打包-在正线性约束下最大化$α公平效用的问题-并提供一种简单的一阶方法来用相对误差保证来解决它。该方法的收敛时间比现有技术要短得多，并且为了对其进行分析，我们利用了近似对偶间隙技术，该技术对收敛参数提供了直观的解释。最后，我们为最小化问题引入了自然的\\ alpha $ -fairness对应项，并在公平任务分配的背景下激励了它的使用。这种概括产生了\\ alpha $-公平的覆盖问题，为此，我们为第一批具有相对误差保证的近线性时间求解器提供了解决方案。