  A new type of stepsize, which was recently introduced by Liu and Liu
(Optimization, 67(3), 427-440, 2018), is called approximately optimal stepsize
and is quit efficient for gradient method. Interestingly, all gradient methods
can be regarded as gradient methods with approximately optimal stepsizes. In
this paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), we
present an improved gradient method with approximately optimal stepsize based
on conic model for unconstrained optimization. If the objective function $ f $
is not close to a quadratic on the line segment between the current and latest
iterates, we construct a conic model to generate approximately optimal stepsize
for gradient method if the conic model can be used; otherwise, we construct
some quadratic models to generate approximately optimal stepsizes for gradient
method. The convergence of the proposed method is analyzed under suitable
conditions. Numerical comparisons with some well-known conjugate gradient
software packages such as CG$ \_ $DESCENT (SIAM J. Optim. 16(1), 170-192, 2005)
and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method is
very promising.
