Tractable downfall of basis pursuit in structured sparse optimization

The problem of finding the sparsest solution to a linear underdetermined system of equations, often appearing, e.g., in data analysis, optimal control, system identification or sensor selection problems, is considered. This non-convex problem is commonly solved by convexification via $\ell_1$-norm minimization, known as basis pursuit (BP). In this work, a class of structured matrices, representing the system of equations, is introduced for which (BP) tractably fails to recover the sparsest solution. In particular, this enables efficient identification of matrix columns corresponding to unrecoverable non-zero entries of the sparsest solution and determination of the uniqueness of such a solution. These deterministic guarantees contrast with popular probabilistic ones and provide insights into the a priori design of sparse optimization problems. As our matrix structures appear naturally in optimal control problems, we exemplify our findings based on a fuel-optimal control problem for a class of discrete-time linear time-invariant systems.