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The exact modulus of the generalized Kurdyka-Łojasiewicz property Wang, Ziyuan


This work aims to provide a self-contained analysis of the Kurdyka-Łojasiewicz (KL) property within the framework of nonsmooth analysis. Our work focuses on two aspects. On one hand, we introduce the generalized KL property, a new concept that generalizes the classic KL property by employing nonsmooth desingularizing functions. Examples and calculus rules for this generalized notion are given. Our results are new and extend the classic KL property. On the other hand, by introducing the exact modulus of the generalized KL property, we provide an answer to the open question: "What is the optimal desingularizing function?", which fills a gap in the current literature. The exact modulus is designed to be the smallest among all possible desingularizing functions. Examples are given to illustrate this pleasant property. We also provide ways to determine or at least estimate the exact modulus. In turn, we obtain explicit formulae for the optimal desingularizing function of locally convex continuously differentiable functions and polynomials on the line, which is usually considered to be challenging. Furthermore, by using the exact modulus, we find the sharpest upper bound for the trajectory of iterates generated the celebrated PALM algorithm.

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