Title: LMI approximations for cones of positive semidefinite forms Abstract: An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, it has been shown that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. We address the approximability of cones of positive semidefinite forms (homogeneous polynomials). This perspective allows us to extract some key common ideas underlying the approximation schemes for polynomial optimization problems developed so far. At the same time, our approach suggests natural new approximation schemes. In particular, we show that for several interesting cones of positive semidefinite forms it is possible to construct polyhedral approximations, which opens the possibility of using linear programming technology in optimization problems over these cones. This is joint work with Luis Zuluaga and Juan Vera at Carnegie Mellon University.