Aimed at both casual spectators and active participants of optimization on manifolds alike, this introductory article presents a wide range of information I would have liked to have been told when I first entered the field. Several arguments are put forth: 1) it is not true that nonconvex implies difficult, 2) many optimization problems in signal processing are approached from the wrong perspective (once-off versus realtime optimization), and 3) the geometry of a manifold should not be used simply for the sake of it. This article also predicts that there is considerable potential for future work on optimization on compact manifolds; large classes of such problems are no harder than convex problems.

中文翻译：

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几何、流形和非凸优化：几何如何帮助优化

这篇介绍性文章针对的是普通观众和流形优化的积极参与者，它提供了我第一次进入该领域时希望得到的广泛信息。提出了几个论点：1）非凸意味着困难是不正确的，2）从错误的角度（一次性优化与实时优化）处理信号处理中的许多优化问题，以及3）流形的几何形状不应该只是为了它而使用。本文还预测，未来在紧凑流形优化方面的工作具有相当大的潜力；这类问题的大类并不比凸问题难。