Determinant maximization with linear matrix inequality constraints

The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many elds, including computational geometry, statistics, system identi cation, experiment design, and information and communication theory. It can also be considered as a generalization of the semide nite programming problem. We give an overview of the applications of the determinant maximization problem, pointing out simple cases where specialized algorithms or analytical solutions are known. We then describe an interior-point method, with a simpli ed analysis of the worst-case complexity and numerical results that indicate that the method is very e cient, both in theory and in practice. Compared to existing specialized algorithms (where they are available), the interior-point method will generally be slower; the advantage is that it handles a much wider variety of problems. Submitted to the SIAM Journal on Matrix Analysis and Applications. Research supported in part by AFOSR (under F49620-95-1-0318), NSF (under ECS-9222391 and EEC-9420565), and MURI (under F49620-95-1-0525). Associated software will be available at URL http://www-isl.stanford.edu/people/boyd and from anonymous FTP to isl.stanford.edu in pub/boyd/maxdet. This will include an implementation in Matlab and C, with a user interface (matlab, sdpsol). Future versions of the paper will also be made available at the same URL and FTP-site (pub/boyd/reports/maxdet.ps.Z).