Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear nonlinear systems theory. However, the scale of problems that can be addressed reliably efficiently is still limited. In this paper, we introduce a new notion block factor-width-two matrices build hierarchy inner outer approximations cone positive semidefinite (PSD) matrices. T...

We study the convex hull of SO(n), the set of n × n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and ...

We study the convex relaxation of a polynomial optimization problem, maximizing product linear forms over complex sphere. show that this program is also permanent Hermitian positive semidefinite (HPSD) matrices. By analyzing constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to HPSD matrices, as well computationally efficient certificates for a...

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many app...

We consider the smallest eigenvalue problem for symmetric or Hermitian matrices by properties of semidefinite matrices. The work is based on a floating-point Cholesky decomposition and takes into account all possible computational and rounding errors. A computational test is given to verify that a given symmetric or Hermitian matrix is not positive semidefinite, so it has at least one negative ...