  In large-scale optimization, the presence of nonsmooth and nonconvex terms in
a given problem typically makes it hard to solve. A popular approach to address
nonsmooth terms in convex optimization is to approximate them with their
respective Moreau envelopes. In this work, we study the use of Lasry-Lions
double envelopes to approximate nonsmooth terms that are also not convex. These
envelopes are an extension of the Moreau ones but exhibit an additional
smoothness property that makes them amenable to fast optimization algorithms.
Lasry-Lions envelopes can also be seen as an "intermediate" between a given
function and its convex envelope, and we make use of this property to develop a
method that builds a sequence of approximate subproblems that are easier to
solve than the original problem. We discuss convergence properties of this
method when used to address composite minimization problems; additionally,
based on a number of experiments, we discuss settings where it may be more
useful than classical alternatives in two domains: signal decoding and spectral
unmixing.
