Tractable downfall of basis pursuit in structured sparse optimization

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.