Related papers: Disjunctive Sum of Squares
We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation…
Given an $\mathcal{H}$-polytope $P$ and a $\mathcal{V}$-polytope $Q$, the decision problem whether $P$ is contained in $Q$ is co-NP-complete. This hardness remains if $P$ is restricted to be a standard cube and $Q$ is restricted to be the…
We introduce a novel semi-supervised version of the least squares classifier. This implicitly constrained least squares (ICLS) classifier minimizes the squared loss on the labeled data among the set of parameters implied by all possible…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$…
We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar-Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if $f$ is a polynomial of degree at most $2 d_f$ nonnegative on…
A polynomial that is a sum of squares (SOS) of other polynomials is evidently positive. The converse is not true, there are positive polynomials which are not SOS. This note focuses on the problem of certifying, in exact arithmetic, that a…
We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce…
In 2008, M. Marshall settled a long-standing open problem by showing that if f(x,y) is a polynomial that is non-negative on the strip [0,1] x R, then there exist sums of squares s(x,y) and t(x,y) such that f(x,y) = s(x,y) + (x - x^2)…
Motivated by scheme theory, we introduce strong nonnegativity on real varieties, which has the property that a sum of squares is strongly nonnegative. We show that this algebraic property is equivalent to nonnegativity for nonsingular real…
For fixed degree and increasing number of variables the dimension of the vector space of $n$-variate real symmetric homogeneous polynomials (forms) of degree $d$ stabilizes. We study the limits of the cones of symmetric nonnegative…
In this work, the combine the theory of generalized critical values with the theory of iterated rings of bounded elements (real holomorphy rings). We consider the problem of computing the global infimum of a real polynomial in several…
A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by…
We have recently presented a method to solve an overdetermined linear system of equations with multiple right hand side vectors, where the unknown matrix is to be symmetric and positive definite. The coefficient and the right hand side…