Basis pursuit by inconsistent alternating projections

Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a famous convex relaxation of what the literature refers to as sparse affine feasibility problem, in which sparse solutions to underdetermined systems are sought. In addition to enjoying convexity, the basis pursuit can be trivially rewritten as a linear program, and thus standard tools in linear programming apply for solving it. In turn, we tackle the basis pursuit in its very original shape with a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Quite recently, inconsistency coming from infeasibility has been seen to work in favor of alternating projections and correspondent convergence rates. In our work, this feature is now suitably enforced in a new and numerically competitive method for solving the basis pursuit.