Related papers: A Matrix Positivstellensatz with lifting polynomia…
This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set $D_L$, called a free Hilbert spectrahedron, of the linear operator inequality…
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…
This article introduces an iterative method for solving nonsingular non-Hermitian positive semidefinite systems of linear equations. To construct the iteration process, the coefficient matrix is split into two non-Hermitian positive…
Recently, non-SOS Positivstellens\"atze for polynomials on compact semialgebraic sets, following the general form of Schm\"{u}dgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of…
Deciding whether the union of two convex polyhedra is itself a convex polyhedron is a basic problem in polyhedral computations; having important applications in the field of constrained control and in the synthesis, analysis, verification…
The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite…
The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…
Let $p$ be a real zero polynomial in $n$ variables. Then $p$ defines a rigidly convex set $C(p)$. We construct a linear matrix inequality of size $n+1$ in the same $n$ variables that depends only on the cubic part of $p$ and defines a…
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…
Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating…
Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $S\subseteq\mathbb R^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is…
Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other…
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and its full…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite…
In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the…
This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the…
Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions…
The purpose of this paper is to give a characterization for polynomials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative positivstellens\"atze for…