带约束的黎曼流形上optimization的简单算法
Simple algorithms for optimization on Riemannian manifolds with constraints
摘要：
我们考虑流形上的等式和不等式约束的最优化问题。大量的工作处理约束优化欧几里得空间。在这项工作中，我们考虑将现有算法从欧几里得情况扩展到黎曼情况。因此，变量存在于已知的光滑流形上，并且被进一步约束。在这样做时，我们利用不断增长的文献无约束黎曼优化。对于流形本身由等式约束描述的特殊情况，原则上可以将整个问题视为欧几里得空间中的约束问题。我们在这里测试的主要假设是，有时利用约束的几何形状是否更好，即使只是其中的一个子集。具体来说，本文扩展了增广拉格朗日方法和光滑版本的精确惩罚法的黎曼的情况下，连同一些基本的收敛性结果。数值实验表明，在某些制度的最小平衡割，非负PCA和$k$ -手段，特别是在高维的计算效率和精度的一些收益。
英文摘要：
We consider optimization problems on manifolds with equality and inequality constraints. A large body of work treats constrained optimization in Euclidean spaces. In this work, we consider extensions of existing algorithms from the Euclidean case to the Riemannian case. Thus, the variable lives on a known smooth manifold and is further constrained. In doing so, we exploit the growing literature on unconstrained Riemannian optimization. For the special case where the manifold is itself described by equality constraints, one could in principle treat the whole problem as a constrained problem in a Euclidean space. The main hypothesis we test here is whether it is sometimes better to exploit the geometry of the constraints, even if only for a subset of them. Specifically, this paper extends an augmented Lagrangian method and smoothed versions of an exact penalty method to the Riemannian case, together with some fundamental convergence results. Numerical experiments indicate some gains in computational efficiency and accuracy in some regimes for minimum balanced cut, non-negative PCA and $k$-means, especially in high dimensions.
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