  It is widely accepted that the stepsize is of great significance to gradient
method. Two efficient gradient methods with approximately optimal stepsizes
mainly based on regularization models are proposed for unconstrained
optimization. More exactly, if the objective function is not close to a
quadratic function on the line segment between the current and latest iterates,
regularization models are exploited carefully to generate approximately optimal
stepsizes. Otherwise, quadratic approximation models are used. In particular,
when the curvature is non-positive, special regularization models are
developed. The convergence of the proposed methods is established under the
weak conditions. Extensive numerical experiments indicated the proposed method
is superior to the BBQ method (SIAM J. Optim. 2021,31(4), 3068-3096) and other
efficient gradient methods, and is competitive to two famous and efficient
conjugate gradient software packages CG$ \_ $DESCENT (5.0) (SIAM J. Optim.
16(1), 170-192, 2005) and CGOPT (1.0) (SIAM J. Optim. 23(1), 296-320, 2013) for
the CUTEr library. Due to the surprising efficiency, we believe that gradient
methods with approximately optimal stepsizes can become strong candidates for
large-scale unconstrained optimization.
