I study set programming: the computation of sets satisfying given properties and maximizing some criteria like their volume. Various techniques can be employed depending on the family of sets considered:
Polyhedral computation or dual dynamic programming for polyhedra,
Semidefinite programming for ellipsoids and
Sum-of-Squares programming for sublevel sets of polynomials or sets with polynomial support function.
I am also interested in the computation of infeasibility certificates for set programs. I apply set programming to the stability analysis and control synthesis of hybrid systems and to the approximation of the Entropic Cone.