  We have developed novel algorithms for optimizing nonsmooth, nonconvex
functions in which the nonsmoothness is caused by nonsmooth operators presented
in the analytical form of the objective. The algorithms are based on encoding
the active branch of each nonsmooth operator such that the active smooth
component function and its code can be extracted at any given point, and the
transition of the solution from one smooth piece to another can be detected via
tracking the change of active branches of all the operators. This mechanism
enables the possibility of collecting the information about the sequence of
active component functions encountered in the previous iterations (i.e., the
component transition information), and using it in the construction of a
current local model or identification of a descent direction in a very economic
and effective manner. Based on this novel idea, we have developed a
trust-region method and a joint gradient descent method driven by the component
information for optimizing the encodable piecewise-smooth, nonconvex functions.
It has further been shown that the joint gradient descent method using a
technique called proactive component function accessing can achieve a linear
rate of convergence if a so-called multi-component Polyak-Lojasiewicz
inequality and some other regularity conditions hold at a neighborhood of a
local minimizer.
