Generalized Subdifferentials of the Sign Change Counting Function

The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for the sign change counting function is given where classical subdifferentials remain intractable. An attempt to prove global optimality at some point, for the 4-dimensional first non trivial example, is made by using a sufficient condition specially tailored among all the cases for this subdifferential.