Symbolic computation of Fenchel conjugates

Convex optimization deals with certain classes of mathematical optimization problems including least-squares and linear programming problems. This area has recently been the focus of considerable study and interest due to the facts that convex optimization problems can be solved efficiently by interior-point methods and that convex optimization problems are actually much more prevalent in practice that previously thought.Key notions in convex optimization are the Fenchel conjugate and the subdifferential of a convex function. In this paper, we build a new bridge between convex optimization and symbolic mathematics by describing the Maple package fenchel, which allows for the symbolic computation of these objects for numerous convex functions defined on the real line. We are able to symbolically reproduce computations for finding Fenchel conjugates and subdifferentials for numerous nontrivial examples found in the literature.