TY - JOUR
T1 - Two Novel Gradient Methods with Optimal Step Sizes
AU - Oviedo , Harry
AU - Dalmau , Oscar
AU - Herrera , Rafael
JO - Journal of Computational Mathematics
VL - 3
SP - 375
EP - 391
PY - 2021
DA - 2021/04
SN - 39
DO - http://doi.org/10.4208/jcm.2001-m2018-0205
UR - https://global-sci.org/intro/article_detail/jcm/18746.html
KW - Gradient methods, Convex quadratic optimization, Hessian spectral properties,
Steplength selection.
AB - <p style="text-align: justify;">In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed
step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve
approximate information of the Hessian of the objective function. A convergence analysis of the proposed algorithm is provided. Some numerical experiments are performed
in order to compare the efficiency and effectiveness of the proposed methods with similar
methods in the literature. Experimentally, it is observed that our proposals accelerate the
gradient method at nearly no extra computational cost, which makes our proposal a good
alternative to solve large-scale problems.</p>