The problem of finding the sparsest solution to a linear underdetermined system of equations, as it often appears in data analysis, optimal control and system identification problems, is considered. This non-convex problem is commonly solved by convexification via $\ell_1$-norm minimization, also known as basis pursuit. In this work, a class of structured matrices, representing the system of equations, is introduced for which the basis pursuit approach tractably fails to recover the sparsest solution. In particular, we are able to identify matrix columns that correspond to unrecoverable non-zero entries of the sparsest solution, as well as to conclude the uniqueness of the sparsest solution in polynomial time. These deterministic guarantees contrast popular probabilistic ones, and as such, provide valuable insights into the a priori design of sparse optimization problems. As our matrix structure appears naturally in optimal control problems, we exemplify our findings by showing that it is possible to verify a priori that basis pursuit may fail in finding fuel optimal regulators for a class of discrete-time linear time-invariant systems.

翻译：本文研究了在数据分析和最优控制等领域中常见的问题：为线性欠定方程组寻找最稀疏解。这一非凸问题通常通过凸化方法——即$\ell_1$范数最小化（亦称基追踪）——求解。本研究引入一类具有特定结构的方程组矩阵，证明对于此类矩阵，基追踪方法可验证地无法恢复最稀疏解。具体而言，我们能够识别出对应于最稀疏解中不可恢复非零项的矩阵列，并可在多项式时间内判定最稀疏解的唯一性。这些确定性保证与常见的概率性保证形成对比，从而为稀疏优化问题的先验设计提供了重要见解。由于所研究的矩阵结构天然存在于最优控制问题中，我们通过示例表明：对于一类离散时间线性时不变系统，可以预先验证基追踪方法在寻找燃料最优调节器时可能失效。