A parameter-free unconstrained reformulation for nonsmooth problems with convex constraints

In the present paper we propose to rewrite a nonsmooth problem subjected to convex constraints as an unconstrained problem. We show that this novel formulation shares the same global and local minima with the original constrained problem. Moreover, the reformulation can be solved with standard nonsmooth optimization methods if we are able to make projections onto the feasible sets. Numerical evidence shows that the proposed formulation compares favorably against state-of-art approaches. Code can be found at https://github.com/jth3galv/dfppm.