@Article{CSIAM-AM-2-585,
author = {Xin Liu , Zaiwen Wen ,  and Yuan , Ya-Xiang},
title = {Subspace Methods for Nonlinear Optimization},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2021},
volume = {2},
number = {4},
pages = {585--651},
abstract = {<p style="text-align: justify;">Subspace techniques such as Krylov subspace methods have been well
known and extensively used in numerical linear algebra. They are also ubiquitous and
becoming indispensable tools in nonlinear optimization due to their ability to handle large scale problems. There are generally two types of principals: i) the decision
variable is updated in a lower dimensional subspace; ii) the objective function or constraints are approximated in a certain smaller subspace of their domain. The key ingredients are the constructions of suitable subspaces and subproblems according to the
specific structures of the variables and functions such that either the exact or inexact
solutions of subproblems are readily available and the corresponding computational
cost is significantly reduced. A few relevant techniques include but not limited to direct combinations, block coordinate descent, active sets, limited-memory, Anderson
acceleration, subspace correction, sampling and sketching. This paper gives a comprehensive survey on the subspace methods and their recipes in unconstrained and
constrained optimization, nonlinear least squares problem, sparse and low rank optimization, linear and nonlinear eigenvalue computation, semidefinite programming,
stochastic optimization and etc. In order to provide helpful guidelines, we emphasize on high level concepts for the development and implementation of practical algorithms from the subspace framework.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2021-0016},
url = {http://global-sci.org/intro/article_detail/csiam-am/19986.html}
}