Forward-Backward Envelope for the Sum of Two Nonconvex Functions: Further Properties and Nonmonotone Linesearch Algorithms
提出ZeroFPR算法，用于最小化两个非凸函数之和，其中一个是光滑的，另一个可能非光滑。该算法仅需前向后向分裂的黑箱操作，在温和条件下实现超线性收敛，数值实验表明在大规模问题上优于传统方法。
We propose \sf ZeroFPR, a nonmonotone linesearch algorithm for minimizing the sum of two nonconvex functions, one of which is smooth and the other possibly nonsmooth. \sf ZeroFPR is the first algorithm that, despite being fit for fully nonconvex problems and requiring only the black-box oracle of forward-backward splitting (FBS)---namely evaluations of the gradient of the smooth term and of the proximity operator of the nonsmooth one---achieves superlinear convergence rates under mild assumptions at the limit point when the linesearch directions satisfy a Dennis--Moré condition, and we show that this is the case for Broyden's quasi-Newton directions. Our approach is based on the forward-backward envelope (FBE), an exact and strictly continuous penalty function for the original cost. Extending previous results we show that, despite being nonsmooth for fully nonconvex problems, the FBE still enjoys favorable first- and second-order properties which are key for the convergence results of \sf ZeroFPR. Our theoretical results are backed up by promising numerical simulations. On large-scale problems, by computing linesearch directions using limited-memory quasi-Newton updates our algorithm greatly outperforms FBS and its accelerated variant (AFBS).