  We consider the problem of optimizing the sum of a smooth, nonconvex function
for which derivatives are unavailable, and a convex, nonsmooth function with
easy-to-evaluate proximal operator. Of particular focus is the case where the
smooth part has a nonlinear least-squares structure. We adapt two existing
approaches for derivative-free optimization of nonsmooth compositions of smooth
functions to this setting. Our main contribution is adapting our algorithm to
handle inexactly computed stationary measures, where the inexactness is
adaptively adjusted as required by the algorithm (where previous approaches
assumed access to exact stationary measures, which is not realistic in this
setting). Numerically, we provide two extensions of the state-of-the-art DFO-LS
solver for nonlinear least-squares problems and demonstrate their strong
practical performance.
