Efficient Inexact Proximal Gradient Algorithm for Nonconvex Problems
  The proximal gradient algorithm has been popularly used for convex optimization. Recently, it has also been extended for nonconvex problems, and the current state-of-the-art is the nonmonotone accelerated proximal gradient algorithm. However, it typically requires two exact proximal steps in each iteration, and can be inefficient when the proximal step is expensive. In this paper, we propose an efficient proximal gradient algorithm that requires only one inexact (and thus less expensive) proximal step in each iteration. Convergence to a critical point %of the nonconvex problem is still guaranteed and has a $O(1/k)$ convergence rate, which is the best rate for nonconvex problems with first-order methods. Experiments on a number of problems demonstrate that the proposed algorithm has comparable performance as the state-of-the-art, but is much faster. 
Quanming Yao and James T. Kwok and Fei Gao and Wei Chen and Tie-Yan   Liu
math.OC