论文标题
复合非平滑度最小化的利用结构
Harnessing structure in composite nonsmooth minimization
论文作者
论文摘要
我们考虑在可以明确计算非滑动函数的接近性操作员的情况下，将非平滑函数的组成最小化，并具有平滑的映射。我们首先表明，此接近操作员不仅可以提供最小化功能的精确平滑子结构，还可以提供完整的复合函数的精确平滑子结构。然后，我们通过提出一种将近端步骤与顺序二次编程步骤相结合的算法来利用此近端识别。我们表明，我们的方法在局部识别最佳平滑子结构，然后四次收敛。我们在两个问题上说明了它的行为：最小化二次函数的最小化以及参数化矩阵的最大特征值的最小化。
We consider the problem of minimizing the composition of a nonsmooth function with a smooth mapping in the case where the proximity operator of the nonsmooth function can be explicitly computed. We first show that this proximity operator can provide the exact smooth substructure of minimizers, not only of the nonsmooth function, but also of the full composite function. We then exploit this proximal identification by proposing an algorithm which combines proximal steps with sequential quadratic programming steps. We show that our method locally identifies the optimal smooth substructure and then converges quadratically. We illustrate its behavior on two problems: the minimization of a maximum of quadratic functions and the minimization of the maximal eigenvalue of a parametrized matrix.
