  Recent research has shown that piecewise smooth (PS) functions can be
approximated by piecewise linear functions with second order error in the
distance to a given reference point. A semismooth Newton type algorithm based
on successive application of these piecewise linearizations was subsequently
developed for the solution of PS equation systems. For local bijectivity of the
linearization at a root, a radius of quadratic convergence was explicitly
calculated in terms of local Lipschitz constants of the underlying PS function.
In the present work we relax the criterium of local bijectivity of the
linearization to local openness. For this purpose a weak implicit function
theorem is proved via local mapping degree theory. It is shown that there exist
PS functions $f:\mathbb R^2\rightarrow\mathbb R^2$ satisfying the weaker
criterium where every neighborhood of the root of $f$ contains a point $x$ such
that all elements of the Clarke Jacobian at $x$ are singular. In such
neighborhoods the steps of classical semismooth Newton are not defined, which
establishes the new method as an independent algorithm. To further clarify the
relation between a PS function and its piecewise linearization, several
statements about structure correspondences between the two are proved.
Moreover, the influence of the specific representation of the local piecewise
linear models on the robustness of our method is studied. An example
application from cardiovascular mathematics is given.
