On convex envelopes for bivariate functions over polytopes

In this paper we discuss convex envelopes for bivariate functions, satisfying suitable assumptions, over polytopes. We first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two-dimensional polytope through the solution of a three-dimensional convex subproblem with continuously differentiable constraint functions. Then, for quadratic functions as well as for some polynomial and rational ones, again satisfying suitable assumptions, we show how the same computations can be carried out through the solution of a single semidefinite problem.