The Boundary of Weighted Analytic Centers for Linear Matrix Inequalities

We study the boundary of the region of weighted analytic centers for linear matrix inequality constraints. Let be the convex subset of R defined by q simultaneous linear matrix inequalities (LMIs) A(x) := A 0 + n ∑ i=1 xiA (j) i 0, j = 1, 2, . . . , q, where A i are symmetric matrices and x ∈ R. Given a strictly positive vector ω = (ω1, ω2, . . . , ωq), the weighted analytic center xac(ω) is the minimizer of the strictly convex function φω(x) := q ∑ j=1 ωj log det[A(j)(x)]−1 overR. The region of weighted analytic centers, W , is a subset ofR. We give several examples for which W 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 ofW , which contains the boundary points ofW which are not boundary points of R. The frame has the same dimension as the boundary of W and is therefore easier to compute than W itself. Furthermore, we develop a Newton-based algorithm that uses a Monte Carlo technique to compute the frame points of W as well as the boundary points of W that are also boundary points of R.