Unless specified otherwise, all vectors in this lecture live in Rn, and all matrices are symmetric and live in Rn×n. For two vectors v,w, let v · w = ∑ i viwi denote their inner product, and v 0 indicate that all vi ≥ 0. For two matrices A and B, denote by A •B their inner product thinking of them as vectors in Rn2 , i.e. A • B = ∑ ij AijBij = Tr(A >B). Here Tr(·) denotes the trace of a matrix....

In Smoothed Aggregation Algebraic Multigrid, the prolongator is defined by smoothing of the output of a simpler tentative prolongator. The weak approximation property for the tentative prolongator is known to give a bound on the convergence factor of the two-level and even multilevel method. It is known how to bound the constants in the weak approximation property when the system matrix is give...

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the ...

During recent decades, there have been a great number of research articles studying interior-point methods for solving problems in mathematical programming and constrained optimization. Stewart and O’Leary obtained an upper bound for scaled pseudoinverses sup W∈P ‖(W 12X)+W 2 ‖2 of a matrix X where P is a set of diagonal positive definite matrices. We improved their results to obtain the suprem...

Let f be a function from R+ into itself. A classic theorem of K. Löwner says that f is operator monotone if and only if all matrices of the form [ f(pi)−f(pj) pi−pj ] are positive semidefinite. We show that f is operator convex if and only if all such matrices are conditionally negative definite and that f(t) = tg(t) for some operator convex function g if and only if these matrices are conditio...

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main...