The Geometry of Semidefinite Programming

Consider the primal-dual pair of optimization problems $$ \begin{gathered} Min \left\langle {c,x} \right\rangle {\rm M}ax \left\langle {b,y} \right\rangle \hfill \\ (P) s.t. x \in K s.t. z \in K* (D) \hfill \\ Ax = b A*y + z = c \hfill \\ \end{gathered} $$ where X and Y are Euclidean spaces with dim X ≥ dim Y. A : X → Y is a linear operator, assumed to be onto. A* : Y → X is its adjoint. K is a closed, convex, facially exposed cone in X. K* := {z|〈z,x〉≤ 0 ∀x∈K} is the dual of K, also a closed, convex, facially exposed cone.