论文标题
您需要冷静下来：镇定性的平静规律性，以供一类半米级优化问题
You Need to Calm Down: Calmness Regularity for a Class of Seminorm Optimization Problems
论文作者
论文摘要
压缩传感涉及解决目标函数$ω（\ boldsymbol {x}）= \ | \ boldsymbol {x} \ | _1 $和线性约束$ \ boldsymbol {a} \ boldsymbol {x} = \ boldsymbol {b boldsymbol {b b} $。以前的工作已经在特殊假设下探索了$ \ boldsymbol {a} $和$ \ boldsymbol {b} $中错误的鲁棒性。在这些结果的激励下，我们探索了$ \ boldsymbol {a} $在更广泛的目标函数$ω$的情况下以及更通用的设置中的鲁棒性，而解决方案可能不是唯一的。 $ \ boldsymbol {b} $中错误的类似结果是已知的，更容易证明。更准确地说，对于带有多面体单元球的eminorm $ω（\ boldsymbol {x}）$，我们证明set-valued map $ s（\ boldsymbol {a}）= \ arg \ arg \ min _ {\ min _ { ω（\ boldsymbol {x}）$在$ \ boldsymbol {a} $中是平静的，而平静是一种本地lipschitz的规律性。
Compressed sensing involves solving a minimization problem with objective function $Ω(\boldsymbol{x}) = \|\boldsymbol{x}\|_1$ 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 $Ω$ 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 $Ω(\boldsymbol{x})$ with a polyhedral unit ball, we prove that the set-valued map $S(\boldsymbol{A}) = \arg \min_{\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}} Ω(\boldsymbol{x})$ is calm in $\boldsymbol{A}$, where calmness is a kind of local Lipschitz regularity.
