  Given graphs $X$ and $Y$, we define two conic feasibility programs which we
show have a solution over the completely positive cone if and only if there
exists a homomorphism from $X$ to $Y$. By varying the cone, we obtain similar
characterizations of quantum/entanglement-assisted homomorphisms and three
previously studied relaxations of these relations. Motivated by this, we
investigate the properties of these "conic homomorphisms" for general
(suitable) cones. We also consider two generalized versions of the Lov\'asz
theta function, and how they interact with these conic homomorphisms. We prove
analogs of several results on classical graph homomorphisms as well as some
monotonicity theorems. We also show that one of the generalized theta functions
is multiplicative on lexicographic and disjunctive graph products.
