约化齐性空间上的直接搜索法
Direct Search Methods on Reductive Homogeneous Spaces
摘要：
直接搜索法主要设计用于没有等式约束的问题。然而，有许多情况下，可行集在周围空间中的测度为零，并且没有网格点位于其中。有一些方法可以处理（黎曼）流形的可行集，但并非所有流形都是平等的。特别是，约化齐性空间似乎是最一般的空间，可以方便地优化。其原因是，在可行域上的“运动定律”也被给出。例子包括\Rn{n}和它的线性子空间，李群和陪集流形，如格拉斯曼流形和Stiefel流形。这些都是优化的重要领域，例如，在图像处理和数据挖掘领域。我们展示了一般约化齐性空间利用映射从切空间的流形上的优化程序。给出了概率下降直接搜索法的具体实现。然后将其扩展到只与流形元素一起工作的过程，从而消除了使用切空间的需要。
英文摘要：
Direct search methods are mainly designed for use in problems with no equality constraints. However, there are many instances where the feasible set is of measure zero in the ambient space and no mesh point lies within it. There are methods for working with feasible sets that are (Riemannian) manifolds, but not all manifolds are created equal. In particular, reductive homogeneous spaces seem to be the most general space that can be conveniently optimized over. The reason is that a `law of motion' over the feasible region is also given. Examples include \Rn{n} and it's linear subspaces, Lie groups and coset manifolds such as Grassmannians and Stiefel manifolds. These are important arenas for optimization, for example, in the areas of image processing and data mining. We demonstrate optimization procedures over general reductive homogeneous spaces utilizing maps from the tangent space to the manifold. A concrete implementation of the probabilistic descent direct search method is shown. This is then extended to a procedure that works solely with the manifold elements, eliminating the need for the use of the tangent space.
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