2023/08/30 by Li, Minghua, Meng, Kaiwen, Yang, Xiaoqi · 1 citation
#49J53 #90C30 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2308.15760
The Kurdyka-Łojasiewicz (KŁ) property, exponent and modulus have played a very important role in the study of global convergence and rate of convergence for optimal algorithms. In this paper, at a stationary point of a locally lower semicontinuous function, we obtain complete characterizations of the KŁ property and the KŁ modulus via the outer limiting subdifferential of an auxilliary function and a newly-introduced subderivative function respectively. In particular, for a class of prox-regular, twice epi-differentiable and subdifferentially continuous functions, we show that the KŁ property and the KŁ modulus can be described by its Moreau envelopes and a quadratic growth condition. We apply the obtained results to establish the KŁ property with exponent \frac12 and to provide calculation of the modulus for a smooth function, the pointwise maximum of finitely many smooth functions and regularized functions respectively. These functions often appear in the modelling of structured optimization problems.