Computing uniform convex approximations for convex envelopes and convex hulls
We provide a numerical procedure to compute uniform (convex) approximations {f_{r}} of the convex envelope f of a rational fraction f, on a compact semi-algebraic set D. At each point x in K=co(D), computing f_{r}(x) reduces to solving a semidefinite program. We next characterize the convex hull K=co(D) in terms of the projection of a semi-infinite LMI, and provide outer convex approximations {K_{r}}"K. Testing whether x is not in K reduces to solving finitely many semidefinite programs.