On Convergence Lemma and Convergence Stability for Piecewise Analytic Functions
Xiaotie Deng, Hanyu Li, Ningyuan Li

TL;DR
This paper establishes a convergence lemma for compositions of analytic functions and the maximum operator, introduces a geometric characterization of stationary points, and proposes a new concept of convergence stability for nonsmooth nonconvex optimization.
Contribution
It proves a convergence lemma specific to analytic compositions, extends geometric analysis of stationary points, and introduces the notion of convergence stability in nonsmooth nonconvex optimization.
Findings
Convergence points near a local minimum shrink as tolerance decreases.
Analyticity is essential for the convergence lemma's validity.
A geometric characterization of stationary points is provided.
Abstract
In this work, a convergence lemma for function being finite compositions of analytic mappings and the maximum operator is proved. The lemma shows that the set of -stationary points near an isolated local minimum point is shrinking to as . It is a natural extension of the version for strongly convex 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 mappings makes the lemma false. The proof is based on stratification theorems of semi-analytic sets by {\L}ojasiewicz. An extension of this proof presents a geometric characterization of the set of stationary points of . Finally, a notion of stability on stationary points, called convergence stability, is proposed. It asks, under small numerical errors, whether a reasonable…
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Taxonomy
TopicsOptimization and Variational Analysis · Advanced Optimization Algorithms Research · Mathematical Inequalities and Applications
