Gutierrez, Alexander2020-10-262020-10-262020-08https://hdl.handle.net/11299/216884University of Minnesota Ph.D. dissertation. 2019. Major: Mathematics. Advisor: Gilad Lerman. 1 computer file (PDF); 54 pages.A recent paper by Klatte and Kummer provides a new characterization for the regularity of the $\argmin$ set of an optimization problem with respect to perturbations. In this thesis we prove that this characterization applies to a broad class of widely used minimization problems. In particular, this work applies to both $\ell_1$ and (anisotropic) discrete total variation minimization problems under linear constraints.enThesis or Dissertation