  Strong (Lagrangian) duality of general conic optimization problems (COPs) has
long been studied and its profound and complicated results appear in different
forms in a wide range of literatures. As a result, characterizing the known and
unknown results can sometimes be difficult. The aim of this article is to
provide a unified and geometric view of strong duality of COPs for the known
results. For our framework, we employ a COP minimizing a linear function in a
vector variable $x$ subject to a single hyperplane constraint $x \in H$ and two
cone constraints $x \in K_1$, $x \in K_2$. It can be identically reformulated
as a simpler COP with the single hyperplane constraint $x \in H$ and the single
cone constraint $x \in K_1 \cap K_2$. This simple COP and its dual as well as
their duality relation can be represented geometrically, and they have no
duality gap without any constraint qualification. The dual of the original
target COP is equivalent to the dual of the reformulated COP if the Minkowski
sum of the duals of the two cones $K_1$ and $K_2$ is closed or if the dual of
the reformulated COP satisfies a certain Slater condition. Thus, these two
conditions make it possible to transfer all duality results, including the
existence and/or boundedness of optimal solutions, on the reformulated COP to
the ones on the original target COP, and further to the ones on a standard
primal-dual pair of COPs with symmetry.
