TY - JOUR
T1 - A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems
AU - Cores , Debora
AU - Figueroa , Johanna
JO - Journal of Computational Mathematics
VL - 4
SP - 932
EP - 954
PY - 2024
DA - 2024/04
SN - 42
DO - http://doi.org/10.4208/jcm.2301-m2021-0313
UR - https://global-sci.org/intro/article_detail/jcm/23041.html
KW - Nonlinear convex optimization, Gradient-type methods, Spectral gradient method, Minimum norm solution linear systems, Linear least-squares solution.
AB - <p style="text-align: justify;">Recently, the authors proposed a low-cost approach, named Optimization Approach for
Linear Systems (OPALS) for solving any kind of a consistent linear system regarding the
structure, characteristics, and dimension of the coefficient matrix $A.$ The results obtained
by this approach for matrices with no structure and with indefinite symmetric part were
encouraging when compare with other recent and well-known techniques. In this work,
we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem
(LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices $A^TA$ or $AA^T,$ and
taking full advantage of the structure and form of the gradient of the proposed nonlinear
objective function in the gradient direction. The combination of those conditions together
with the choice of the initial iterate allow us to produce a novel and efficient low-cost
numerical scheme for solving both problems. Moreover, the scheme presented in this
work can also be used and extended for the weighted minimum norm linear systems and
minimum norm linear least-squares problems. We include encouraging numerical results
to illustrate the practical behavior of the proposed schemes.</p>