论文标题
非不符合复合函数的一阶变异分析
First-order variational analysis of non-amenable composite functions
论文作者
论文摘要
本文致力于研究非凸和非差异函数的一阶变异分析，这些函数可能不符合规则。为了实现这一目标，我们完全依靠两个定向衍生物的概念，称为亚衍生物和半衍生物。我们通过这些定向导数建立了此类函数的确切链和总和规则。这些演算规则提供了可实现的自动分化过程，例如复合函数中的后传播。后一个结石规则可用于识别由降级定义的定向固定点。我们表明，几何衍生约束集的距离函数是半差异的，这为设计一阶算法打开了针对非clarke常规约束优化问题的一阶算法。我们提出了一种一阶算法，以找到非clarke常规和可能非lipschitz函数的定向固定点。我们介绍了一种下降特性，在该特性下，我们以速率$ o（\ varepsilon^{ - 2}）$建立了方法的非反应收敛性，类似于梯度下降以进行平滑最小化。我们表明，后者的下降属性在一些有趣的不符合的复合函数中可以免费保留，特别是它具有任何有界底函数的莫罗包络。
This paper is devoted to studying the first-order variational analysis of non-convex and non-differentiable functions that may not be subdifferentially regular. To achieve this goal, we entirely rely on two concepts of directional derivatives known as subderivative and semi-derivative. We establish the exact chain and sum rules for this class of functions via these directional derivatives. These calculus rules provide an implementable auto-differentiation process such as back-propagation in composite functions. The latter calculus rules can be used to identify the directional stationary points defined by the subderivative. We show that the distance function of a geometrically derivable constraint set is semi-differentiable, which opens the door for designing first-order algorithms for non-Clarke regular constrained optimization problems. We propose a first-order algorithm to find a directional stationary point of non-Clarke regular and perhaps non-Lipschitz functions. We introduce a descent property under which we establish the non-asymptotic convergence of our method with rate $O(\varepsilon^{-2})$, akin to gradient descent for smooth minimization. We show that the latter descent property holds for free in some interesting non-amenable composite functions, in particular, it holds for the Moreau envelope of any bounded-below function.
