关于分段解析函数的收敛引理和收敛稳定性
On Convergence Lemma and Convergence Stability for Piecewise Analytic Functions
摘要：
本文证明了函数$f$是解析映射和极大算子的有限复合的一个收敛引理。引理表明，在孤立的局部极小点$x^*$附近的$δ$ -平稳点集收缩为$x^*$为$δ\to 0$。它是强凸$C^1$函数形式的自然扩展。然而，引理的正确性是微妙的。解析映射对于引理是必要的，因为用可微映射或$C^\infty$映射替换它会使引理为假。证明是基于分层定理的半解析集的Eschojasiewicz。这个证明的一个扩展提出了一个几何特征的一组不动点的$f$。最后，提出了稳定点的概念，称为收敛稳定性。它问，在小的数值误差下，是否一个合理的收敛优化方法开始附近的一个稳定点，最终应该收敛到同一个稳定点。只有当目标函数既非光滑又非凸时，收敛稳定性的概念才定性地变得非平凡。利用收敛引理，给出了$f$收敛稳定性的直观等价条件.这些结果为研究非光滑非凸优化的“何处收敛”问题提供了一个新的几何视角。
英文摘要：
In this work, a convergence lemma for function $f$ being finite compositions of analytic mappings and the maximum operator is proved. The lemma shows that the set of $δ$-stationary points near an isolated local minimum point $x^*$ is shrinking to $x^*$ as $δ\to 0$. It is a natural extension of the version for strongly convex $C^1$ functions. However, the correctness of the lemma is subtle. Analytic mappings are necessary for the lemma in the sense that replacing it with differentiable or $C^\infty$ mappings makes the lemma false. The proof is based on stratification theorems of semi-analytic sets by Łojasiewicz. An extension of this proof presents a geometric characterization of the set of stationary points of $f$. Finally, a notion of stability on stationary points, called convergence stability, is proposed. It asks, under small numerical errors, whether a reasonable convergent optimization method started near a stationary point should eventually converge to the same stationary point. The concept of convergence stability becomes nontrivial qualitatively only when the objective function is both nonsmooth and nonconvex. Via the convergence lemma, an intuitive equivalent condition for convergence stability of $f$ is proved. These results together provide a new geometric perspective to study the problem of "where-to-converge" in nonsmooth nonconvex optimization.
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