  We introduce a generalized version of the concave Kurdyka-\L ojasiewicz (KL)
property by employing nonsmooth desingularizing functions. We also present the
exact modulus of the generalized concave KL property, which provides an answer
to the open question regarding the optimal concave desingularizing function.
The exact modulus is designed to be the smallest among all possible concave
desingularizing functions. Examples are given to illustrate this pleasant
property. In turn, using the exact modulus we provide the sharpest upper bound
for the total length of iterates generated by the celebrated
Bolte-Sabach-Teboulle PALM algorithm.

Optimization and Control