A nested primal–dual FISTA-like scheme for composite convex optimization problems
提出一种带外推的嵌套原始-对偶算法，用于最小化两个凸函数之和（其中一个连续可微），通过惯性参数选择证明收敛到鞍点并给出O(1/n)收敛率，图像恢复实验表明热启动策略对收敛至关重要。
Abstract We propose a nested primal–dual algorithm with extrapolation on the primal variable suited for minimizing the sum of two convex functions, one of which is continuously differentiable. The proposed algorithm can be interpreted as an inexact inertial forward–backward algorithm equipped with a prefixed number of inner primal–dual iterations for the proximal evaluation and a “warm–start” strategy for starting the inner loop, and generalizes several nested primal–dual algorithms already available in the literature. By appropriately choosing the inertial parameters, we prove the convergence of the iterates to a saddle point of the problem, and provide an O (1/ n ) convergence rate on the primal–dual gap evaluated at the corresponding ergodic sequences. Numerical experiments on some image restoration problems show that the combination of the “warm–start” strategy with an appropriate choice of the inertial parameters is strictly required in order to guarantee the convergence to the real minimum point of the objective function.