Abstract

Let a weakly convex function (in the general case, nonconvex and nonsmooth) satisfy the quadratic growth condition. It is proved that the gradient projection method for minimizing such a function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold), provided that the constant of weak convexity of the function is less than the constant in the quadratic growth condition and the constant of proximal smoothness for the set is sufficiently large. The connection between the quadratic growth condition on the function and other conditions is discussed.

中文翻译：

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近似光滑集上弱凸函数的梯度投影方法

摘要

让一个弱凸函数（一般情况下，非凸和非光滑）满足二次增长条件。证明了最小化集合上的这种函数的梯度投影方法以线性速率收敛于特殊形式的近端光滑（非凸）集合（例如，在光滑流形上），条件是该集合的弱凸常数保持不变。函数小于二次生长条件下的常数，并且该组的近端平滑度常数足够大。讨论了函数的二次增长条件与其他条件之间的联系。