Related papers: A Semidefinite Hierarchy for Containment of Spectr…
We propose a method for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of…
We give the first approximation algorithm for mixed packing and covering semidefinite programs (SDPs) with polylogarithmic dependence on width. Mixed packing and covering SDPs constitute a fundamental algorithmic primitive with recent…
We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…
We discuss the general method for obtaining full positivity bounds on multi-field effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed)…
An alternative formulation for the controllability problem of single input linear positive systems is presented. Driven by many industrial applications, this formulations focuses on the case where the region of interest is only a subset of…
We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner)…
In this paper we consider the problem of how to computationally test whether a matrix inequality is positive semidefinite on a semialgebraic set. We propose a family of sufficient conditions using the theory of matrix Positivstellensatz…
We study the problem of whether all bipartite quantum states having a prescribed spectrum remain positive under the reduction map applied to one subsystem. We provide necessary and sufficient conditions, in the form of a family of linear…
The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels.…
We continue the study of real polynomials acting entrywise on matrices of fixed dimension to preserve positive semidefiniteness, together with the related analysis of order properties of Schur polynomials. Previous work has shown that,…
We study the closure of the projection of the (nonconvex) cone of rank restricted positive semidefinite matrices onto subsets of the matrix entries. This defines the feasible sets for semidefinite completion problems with restrictions on…
We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy a linear matrix inequality, is itself a spectrahedron. The main application is that derivative relaxations of…
A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This…
The Schur-Horn theorem is a classical result in matrix analysis which characterizes the existence of positive semidefinite matrices with a given diagonal and spectrum. In recent years, this theorem has been used to characterize the…
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory.…
The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate…
The Helton-Nie Conjecture (HNC) is the proposition that every convex semialgebraic set is a spectrahedral shadow. Here we prove that HNC is equivalent to another propo- sition related to quadratically constrained quadratic programming.…
We propose a numerical method, based on the shift-and-invert power iteration, that answers whether a symmetric matrix is positive definite ("yes") or not ("no"). Our method uses randomization. But, it returns the correct answer with high…
This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of…
Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten theorems involving {\epsilon}-conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and…