The local Lipschitz continuity condition is standard in nonsmooth, nonconvex optimization -- see e.g. [this paper][2] and Sections "Nonsmooth, Nonconvex Optimization" and "The Clarke Subdifferential" in these [lecture notes][1].  


  [1]: https://cs.nyu.edu/overton/conv_ns_opt/notes/CourseLastLecture2018.pdf
  [2]: https://arxiv.org/pdf/1807.07554