90C25,49Q99,46C05,14P25

Let be the convex subset of defined by simultaneous linear matrix inequalities (LMI). Given a strictly positive vector, the weighted analytic center is the minimizerargmin of the strictly convex function over . We give a necessary and sufficient condition for a point of 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 of weighted analytic centers for LMI's is not convex and does not generally equal . 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 region and its boundary are described by real algebraic varieties, and provide slices of a non-trivial real algebraic variety to show that isn't convex. Stiemke's Theorem of the alternative provides a practical test of whether a point is in . 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.