Variational Analysis of Kurdyka-Łojasiewicz Property, Exponent and Modulus

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 $\frac{1}{2}$ 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.