  The Shannon capacity of a graph is an important graph invariant in
information theory that is extremely difficult to compute. The Lovasz number,
which is based on semidefinite programming relaxation, is a well-known upper
bound for the Shannon capacity. To improve this upper bound, previous
researches tried to generalize the Lovasz number using the ideas from the
sum-of-squares optimization. In this paper, we consider the possibility of
developing general conic programming upper bounds for the Shannon capacity,
which include the previous attempts as special cases, and show that it is
impossible to find better upper bounds for the Shannon capacity along this way.
