Abstract. The function that maps a positive semidefinite matrix to the trace of one of its nonnegative integer power is semidefinite representable. In this note, we reduce the size of this semidefinite representation from O(kn) linear matrix inequalities of dimension n, where k is the desired power and n× n the size of the matrix to O(log 2 (k)) linear matrix inequalities of dimension 2n. We al...

A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general met...

A is further called positive definite, symbolized A > 0, if the strict inequality in 1.1 holds for all nonzero x ∈ C. An equivalent condition forA ∈ Mn to be positive definite is thatA is Hermitian and all eigenvalues of A are positive real numbers. Given a positive semidefinite matrix A and p > 0, A denotes the unique positive semidefinite pth power of A. Let A and B be two Hermitian matrices ...

We introduce the T -restricted weighted generalized inverse of a singular matrix A with respect to a positive semidefinite matrix T , which defines a seminorm for the space. The new approach proposed is that since T is positive semidefinite, the minimal seminorm solution is considered for all vectors perpendicular to the kernel of T .

Consider a convex set S = {x ∈ D : G(x) o 0} where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D ⊆ R is a domain on which G(x) is defined, and G(x) o 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set which is defined by a linear matrix inequality (LMI). This paper s...

We introduce the T -restricted weighted generalized inverse of a singular matrix A with respect to a positive semidefinite matrix T , which defines a seminorm for the space. The new approach proposed is that since T is positive semidefinite, the minimal seminorm solution is considered for all vectors perpendicular to the kernel of T .

Chordal graphs play a central role in techniques for exploiting sparsity in large semidefinite optimization problems and in related convex optimization problems involving sparse positive semidefinite matrices. Chordal graph properties are also fundamental to several classical results in combinatorial optimization, linear algebra, statistics, signal processing, machine learning, and nonlinear op...

We discuss three types of sparse matrix nearness problems: given a sparse symmetric matrix, find the matrix with the same sparsity pattern that is closest to it in Frobenius norm and (1) is positive semidefinite, (2) has a positive semidefinite completion, or (3) has a Euclidean distance matrix completion. Several proximal splitting and decomposition algorithms for these problems are presented ...