You Need to Calm Down: Calmness Regularity for a Class of Seminorm Optimization Problems

Abstract

Compressed sensing involves solving a minimization problem with objective function $\Omega(\boldsymbol{x}) = \|\boldsymbol{x}\|_1$ and linear constraints $\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}$. Previous work has explored robustness to errors in $\boldsymbol{A}$ and $\boldsymbol{b}$ under special assumptions. Motivated by these results, we explore robustness to errors in $\boldsymbol{A}$ for a wider class of objective functions $\Omega$ and for a more general setting, where the solution may not be unique. Similar results for errors in $\boldsymbol{b}$ are known and easier to prove. More precisely, for a seminorm $\Omega(\boldsymbol{x})$ with a polyhedral unit ball, we prove that the set-valued map $S(\boldsymbol{A}) = \arg \min_{\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}} \Omega(\boldsymbol{x})$ is calm in $\boldsymbol{A}$, where calmness is a kind of local Lipschitz regularity.