Determinant Maximization with Linear Matrix Inequality Constraints

Abstract

Abstract. 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 a n o v erview 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.

Learning Sparse Causal Gaussian Networks With Experimental Intervention: Regularization and Coordinate Descent

“…Much of the research has focused on estimating undirected graphs with the L 1 penalty. Yuan and Lin (2007) proposed to maximize an L 1 -penalized log-likelihood based on the "max-det" problem considered by Vandenberghe, Boyd, and Wu (1998), while Banerjee, El Ghaoui, and d'Aspremont (2008) employed a blockwise coordinate descent (CD) algorithm to solve the optimization problem. Friedman, Hastie, and Tibshirani (2008) built on the method of Banerjee, El Ghaoui, and d'Aspremont (2008) a remarkably efficient algorithm called the graphical lasso.…”

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“…For a given ∈ C, can be typically chosen to maximize the minimum eigenvalue of L( , ) or to maximize the (log of the) determinant of L( , ). Both of these problems are convex: maximizing the minimum eigenvalue can be reduced to solve a semidefinite program (SDP) [27] and maximizing the determinant can be reduced to solving a MAXDET problem [28].…”

“…where A, B, C, D are the matrices introduced in (28) and (29). Here (A c , B c , C c , D c ) and P correspond, respectively, to and introduced in Section 2.2.…”

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“…Notice that this optimization problem is a semidefinite program (SDP), if f (P ) = trace (P ), and a MAXDET problem, if f (P ) = log det(P ). In both cases the problem can be efficiently solved in polynomial-time by interior point methods for convex programming, [39,40]. We also remark that Lemma 2.1 can be used for directly determining optimized bounds on individual elements of the solution vector x.…”