  Lovasz theta function and the related theta body of graphs have been in the
center of the intersection of four research areas: combinatorial optimization,
graph theory, information theory, and semidefinite optimization. In this paper,
utilizing a modern convex optimization viewpoint, we provide a set of minimal
conditions (axioms) under which certain key, desired properties are
generalized, including the main equivalent characterizations of the theta
function, the theta body of graphs, and the corresponding antiblocking duality
relations. Our framework describes several semidefinite and polyhedral
relaxations of the stable set polytope of a graph as generalized theta bodies.
As a by-product of our approach, we introduce the notion of "Schur Lifting" of
cones which is dual to PSD Lifting (more commonly used in SDP relaxations of
combinatorial optimization problems) in our axiomatic generalization. We also
generalize the notion of complements of graphs to diagonally scaling-invariant
polyhedral cones. Finally, we provide a weighted generalization of the
copositive formulation of the fractional chromatic number by Dukanovic and
Rendl.
