A Weighted Analytic Center for Linear Matrix Inequalities

Let R be the convex subset of IR defined by q simultaneous linear matrix inequalities (LMI) A 0 + ∑n i=1 xiA (j) i 0, j = 1, 2, . . . , q. Given a strictly positive vector ω = (ω1, ω2, · · · , ωq), the weighted analytic center xac(ω) is the minimizer argmin (φω(x)) of the strictly convex function φω(x) = ∑q j=1 ωj log det[A (j)(x)]−1 over R. We give a necessary and sufficient condition for a point of R to be a weighted analytic center. We study the argmin function in this instance and show that it is a continuously differentiable open function. In the special case of linear constraints, all interior points are weighted analytic centers. We show that the region W = {xac(ω) | ω > 0} ⊆ R of weighted analytic centers for LMI’s is not convex and does not generally equal R. These results imply that the techniques in linear programming of following paths of analytic centers may require special consideration when extended to semidefinite programming. We show that the regionW and its boundary are described by real algebraic varieties, and provide slices of a non-trivial real algebraic variety to show that W isn’t convex. Stiemke’s Theorem of the alternative provides a practical test of whether a point is in W . Weighted analytic centers are used to improve the location of standing points for the Stand and Hit method of identifying necessary LMI constraints in semidefinite programming.