Parametric Computation of the Legendre-Fenchel Conjugate with Application to the Computation of the Moreau Envelope

A new algorithm, named the Parametric Legendre Transform (PLT) algorithm, to compute the Legendre-Fenchel conjugate of a convex function of one variable is presented. It returns a parameterization of the graph of the conjugate except for some affine parts corresponding to nondifferentiable points of the function. The approach is extended to the computation of the Moreau envelope, resulting in a simple yet efficient algorithm. Theoretical results, the description (and extension) of the algorithm, its approximation error and the convergence, as well as the comparison with known algorithms are included.