  This paper applies an idea of adaptive momentum for the nonlinear conjugate
gradient to accelerate optimization problems in sparse recovery. Specifically,
we consider two types of minimization problems: a (single) differentiable
function and the sum of a non-smooth function and a differentiable function. In
the first case, we adopt a fixed step size to avoid the traditional line search
and establish the convergence analysis of the proposed algorithm for a
quadratic problem. This acceleration is further incorporated with an operator
splitting technique to deal with the non-smooth function in the second case. We
use the convex $\ell_1$ and the nonconvex $\ell_1-\ell_2$ functionals as two
case studies to demonstrate the efficiency of the proposed approaches over
traditional methods.
