In this paper, we propose new inexact adaptive proximal gradient algorithms for solving
nonconvex composite optimization problems, where the objective is the sum of a differentiable
nonconvex function and a convex non-differentiable function. A new relative error criterion to
compute the proximal operator inexactly has been proposed together with new adaptive strate-
gies for selecting stepsizes used in inexact proximal gradient steps. In particular, both of relative
error criterion and adaptive stepsizes in our new method are dynamically adjusted quickly at
each iteration by using information of the current step. Standard theoretical convergence results
for our proposed algorithms are obtained for both nonconvex and convex cases of composite
optimization models. Specifically, for a wide class of nonconvex objective functions we show
that any cluster point of the iterates is a stationary point of the considered problem. For the con-
vex case, we prove that the sequence generated by our algorithms converge to a global optimal
solution of the considered problem with sublinear convergence rate (from a fixed iteration). To
the best of our knowledge, no inexact proximal gradient algorithms with adaptive stepsizes have
been proposed for nonconvex composite optimization problems to date. The efficiency of our new
algorithms is verified throughout extensive numerical experiments for applicative composite op-
timization problems with benchmark and synthetic datasets.