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

TL;DR

This paper investigates the robustness of solutions to a broad class of optimization problems involving seminorms, demonstrating that the solution set varies calmly with respect to changes in the linear operator, thus extending robustness results in compressed sensing.

Contribution

It generalizes robustness analysis to seminorm optimization problems with polyhedral unit balls, proving calmness of the solution set map under perturbations in the linear operator.

Findings

Established calmness of the solution set map for seminorms with polyhedral unit balls.

Extended robustness results to a wider class of objective functions beyond $oldsymbol{ ext{l}_1}$.

Provided theoretical guarantees for stability of solutions under operator errors.

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.