  Sparse level-set formulations allow practitioners to find the minimum 1-norm
solution subject to likelihood constraints. Prior art requires this constraint
to be convex. In this letter, we develop an efficient approach for nonconvex
likelihoods, using Regula Falsi root-finding techniques to solve the level-set
formulation. Regula Falsi methods are simple, derivative-free, and efficient,
and the approach provably extends level-set methods to the broader class of
nonconvex inverse problems. Practical performance is illustrated using
l1-regularized Student's t inversion, which is a nonconvex approach used to
develop outlier-robust formulations.
