Direct Search Methods on Reductive Homogeneous Spaces

被引:0
作者
David W. Dreisigmeyer
机构
[1] United States Census Bureau,Center for Economic Studies
[2] Colorado State University,Department of Electrical and Computer Engineering
来源
Journal of Optimization Theory and Applications | 2018年 / 176卷
关键词
Direct search method; Manifold; Nonlinear optimization; 65K10; 90C56;
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中图分类号
学科分类号
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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{n}$$\end{document} and its 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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页码:585 / 604
页数:19
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