Information geometric analysis of semidefinite programming problems

In this paper, we consider semidefinite programming (SDP) problems from points of view of information geometry. Regarding the feasible region of an SDP problem as a submanifold imbedded in the set of positive definite matrices, we elucidate that a certain imbedding curvature of the submanifold, called the second fundamental form derived from information geometric structure, relates to computational cost to solve the SDP problem. Specifically, when the second fundamental form vanishes everywhere on the submanifold, in such case we call the feasible region doubly autoparallel, we show the corresponding SDP problem can be solved without any Newton iterations and give a formula for an optimal solution. In addition, we study the properties of doubly autoparallel submanifolds in the set of positive definite matrices. It turns out this specific structure has a relation with Jordan subalgebra of symmetric matrices. Using this fact, we construct some concrete examples of doubly autoparallel submanifold.