The Frank-Wolfe algorithm is a method for constrained optimization that relies on linear minimizations, as opposed to projections. Therefore, a motivation put forward in a large body of work on the Frank-Wolfe algorithm is the computational advantage of solving linear minimizations instead of projections. However, the discussions supporting this advantage are often too succinct or incomplete. In this paper, we review the complexity bounds for both tasks on several sets commonly used in optimization. Projection methods onto the $\ell_p$-ball, $p\in\left]1,2\right[\cup\left]2,+\infty\right[$, and the Birkhoff polytope are also proposed.

翻译：Frank-Wolfe 算法是一种限制优化的方法,它依赖于线性最小化,而不是预测。 因此,在大量关于Frank-Wolfe 算法的工作中提出的一个动机是解决线性最小化而不是预测的计算优势。 但是,支持这一优势的讨论往往过于简洁或不完整。 在本文件中,我们审查了在优化中常用的数组任务中这两种任务的复杂性界限。 在$\ell_p$- ball, $p\in\left] 1,2\right[\cup\left]2,\\\infty\right[$, 和Birkhoff plitope也提出了投影法。