Uniqueness of local maximizers for some non-convex log-determinant optimization problems using information theory.

Certain families of non-convex optimization problems involving linear combinations of log-determinants of positive definite matrices are shown to have a unique local maximizer. These geometric results are established using information-theoretic arguments. We demonstrate these results for three different families: two of which arise in the study of the capacity region of the vector Gaussian broadcast channel and another one in the study of computing the optimal generalized Brascamp-Lieb constant.