We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of \({\mathcal {O}}(\epsilon ^{-3})\) to achieve an \(\epsilon \)-approximate solution. This bound interpolates between the \({\mathcal {O}}(\epsilon ^{-2})\) bound for the smooth case and the \({\mathcal {O}}(\epsilon ^{-4})\) bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.

我们研究结构化目标函数的最小化，该函数是平滑函数和具有线性算子的弱凸函数的和。应用包括带有正则化器的图像重建问题，该问题比标准的凸形正则化器引入的偏差要小。我们基于具有递减顺序的平滑参数的Moreau包络开发了变量平滑算法，并证明了\（{\ mathcal {O}}（\ epsilon ^ {-3}）\）的复杂度以实现\（ \ epsilon \） -近似解。此边界在光滑情况的\（{\ mathcal {O}}（\ epsilon ^ { -2}）\）和\（{\ mathcal {O}}（\ epsilon ^ {-4}）之间插值\）绑定到次梯度方法。我们的复杂性界限与其他处理弱凸函数的结构化非光滑性的作品是一致的。