We study the boundary of the region of weighted analytic centers for linear matrix inequality constraints. Let be the convex subset of defined by simultaneous linear matrix inequalities (LMIs) where are symmetric matrices and . Given a strictly positive vector , the weighted analytic center is the minimizer of the strictly convex function
over . The region of weighted analytic centers, , is a subset of . We give several examples for which has interesting topological properties. We show that every point on a central path in semidefinite programming is a weighted analytic center.
We introduce the concept of the frame of , which contains the boundary points of which are not boundary points of . The frame has the same dimension as the boundary of and is therefore easier to compute than itself. Furthermore, we develop a Newton-based algorithm that uses a Monte Carlo technique to compute the frame points of as well as the boundary points of that are also boundary points of .